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Summary Consideration of fabric anisotropy is crucial to gaining an improved understanding of the behavior of granular materials. This paper presents a constitutive model to describe the sand behavior associated with fabric anisotropy within a framework of a strain space multiple mechanism model. In the proposed model, a second‐order fabric tensor is extended by incorporating a new function that represents the effect of inherent (or initial fabric) anisotropy, along with three additional parameters: two of them, a1 and a2, control the degree of anisotropy, and the second mode of inherent anisotropy can be expressed by introducing the parameter a2 as well as the first mode by the parameter a1. The third parameter, θ0, expresses the principal direction of inherent anisotropy (eg, the normal vector direction of bedding planes relative to horizontal axis). The formulation of the dilative component of dilatancy (ie, positive dilatancy) is also extended to consider the effect of inherent anisotropy based on the interlocking mechanism. Experimental data on the complex anisotropic responses of Fraser River sand and Toyoura sand under monotonic loading is used to validate this model. The proposed model is shown to successfully capture anisotropic responses, which become contractive or dilative depending on different principal‐stress directions, with a single set of anisotropy parameters; thus, the model is considered to possess the capability to simulate the anisotropic behaviors of granular materials. In addition to different loadings on the same fabric, the effects of different fabric anisotropies upon the sand behavior under the same loadings are also investigated. [ABSTRACT FROM AUTHOR]

Potential functions can be used for generating potentials of relevant geometric structures for a Riemannian manifold such as the Riemannian metric and affine connections. We study whether this procedure can also be applied to tensors of rank four and find a negative answer. We study this from the perspective of solving the inverse problem and also from an intrinsic point of view. [ABSTRACT FROM AUTHOR]

We show that every limit point of a Zariski dense discrete subgroup Γ of the isometry group of a symmetric space of noncompact type is conical if and only if Γ is convex cocompact. [ABSTRACT FROM AUTHOR]

Continuing the work of [8] and [9] , we derive an analogue of the classical “ k / 12 -formula” for Drinfeld modular forms of rank r ≥ 2 . Here the vanishing order ν ω ( f ) of one modular form at some point ω of the complex upper half-plane is replaced by the intersection multiplicity ν ω ( f 1 , … , f r − 1 ) of r − 1 independent Drinfeld modular forms at some point ω of the Drinfeld symmetric space Ω r . We apply the formula to determine the common zeroes of r − 1 consecutive Eisenstein series E q i − 1 , where n − r < i < n for some n ≥ r . [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel signal characteristic, which is ubiquitous in the vast majority of communication signals. That is, a modulated signal is of an inherent low-rank structure following a reshaping operation. We first use a toy model to develop a framework for modelling this signal characteristic, and theoretically prove the impact on the signals' rank structure by additive white Gaussian noise and inter-symbol interference, which is of great concern in the wireless communication field. Subsequently, the model is generalized to multi-input-multi-output signals, and tensor rank is taken into account. Using multi-linear algebra, we prove that the low-rankness of a reshaped signal only depends on the structure of its embedding subspace, and that its rank measure is upper bounded by the multi-rank of the basis tensor. As an application, we propose a novel adaptive sampling and reconstruction scheme for generic software-defined radio based on the low-rank structure. Numerical simulations demonstrate that the proposed method outperforms compressed sensing-based method, particularly when the modulated signal does not satisfy the sparsity assumption in the time and frequency domains. The results of practical experiments further demonstrate that many types of modulated signals can be effectively reconstructed from very limited observations using our proposed method. [ABSTRACT FROM AUTHOR]

Low-rankness has been widely exploited for the tensor completion problem. Recent advances have suggested that the tensor nuclear norm often leads to a promising approximation for the tensor rank. It treats the singular values equally to pursue the convexity of the objective function, while the singular values for the practical images have clear physical meanings with different importance and should be treated differently. In this paper, we propose a non-convex logDet function as a smooth approximation for tensor rank instead of the convex tensor nuclear norm and introduce it into the low-rank tensor completion problem. An alternating direction method of multiplier (ADMM)-based method is developed to solve the problem. Experimental results have shown that the proposed method can significantly outperform existing state-of-the-art nuclear norm-based methods for tensor completion. [ABSTRACT FROM AUTHOR]

This paper introduces matrix product state (MPS) decomposition as a new and systematic method to compress multidimensional data represented by higher order tensors. It solves two major bottlenecks in tensor compression: computation and compression quality. Regardless of tensor order, MPS compresses tensors to matrices of moderate dimension, which can be used for classification. Mainly based on a successive sequence of singular value decompositions, MPS is quite simple to implement and arrives at the global optimal matrix, bypassing local alternating optimization, which is not only computationally expensive but cannot yield the global solution. Benchmark results show that MPS can achieve better classification performance with favorable computation cost compared to other tensor compression methods. [ABSTRACT FROM PUBLISHER]

For several objects of interest in geometric complexity theory, namely for the determinant, the permanent, the product of variables, the power sum, the unit tensor, and the matrix multiplication tensor, we introduce and study a fundamental SL-invariant function that relates the coordinate ring of the orbit with the coordinate ring of its closure. For the power sums we can write down this fundamental invariant explicitly in most cases. Our constructions generalize the two Aronhold invariants on ternary cubics. For the other objects we identify the invariant function conditional on intriguing combinatorial problems much like the well-known Alon–Tarsi conjecture on Latin squares. We provide computer calculations in small dimensions for these cases. As a main tool for our analysis, we determine the stabilizers, and we establish the polystability of all the mentioned forms and tensors (including the generic ones). [ABSTRACT FROM AUTHOR]

Over a field of characteristic zero, we prove that for each r, there exists a constant C( r) so that the prime ideal of the rth secant variety of any Veronese embedding of any projective space is generated by polynomials of degree at most C( r). The main idea is to consider the coordinate ring of all of the ambient spaces of the Veronese embeddings at once by endowing it with the structure of a Hopf ring, and to show that its ideals are finitely generated. We also prove a similar statement for partial flag varieties and, in fact, arbitrary projective schemes, and we also get multi-graded versions of these results. [ABSTRACT FROM AUTHOR]

The elastoresistivity tensor mi j,kl relates changes in resistivity to strains experienced by a material. As a fourth-rank tensor, it contains considerably more information about the material than the simpler (second-rank) resistivity tensor; in particular, for a tetragonal material, the B1g and B2g components of the elastoresistivity tensor (mx x, x x − mx x, y y and 2mx y, x y, respectively) can be related to its nematic susceptibility. Previous experimental probes of this quantity have focused exclusively on differential longitudinal elastoresistance measurements, which determine the induced resistivity anisotropy arising from anisotropic in-plane strain based on the difference of two longitudinal resistivity measurements. Here we describe a complementary technique based on transverse elastoresistance measurements. This new approach is advantageous because it directly determines the strain-induced resistivity anisotropy from a single transverse measurement. To demonstrate the efficacy of this new experimental protocol, we present transverse elastoresistance measurements of the 2mx y, x y elastoresistivity coefficient of BaFe2As2, a representative iron-pnictide that has previously been characterized via differential longitudinal elastoresistance measurements. [ABSTRACT FROM AUTHOR]

With the increasing availability of various sensor technologies, we now have access to large amounts of multiblock (also called multiset, multirelational, or multiview) data that need to be jointly analyzed to explore their latent connections. Various component analysis methods have played an increasingly important role for the analysis of such coupled data. In this article, we first provide a brief review of existing matrix-based (two-way) component analysis methods for the joint analysis of such data with a focus on biomedical applications. Then, we discuss their important extensions and generalization to multiblock multiway (tensor) data. We show how constrained multiblock tensor decomposition methods are able to extract similar or statistically dependent common features that are shared by all blocks, by incorporating the multiway nature of data. Special emphasis is given to the flexible common and individual feature analysis of multiblock data with the aim to simultaneously extract common and individual latent components with desired properties and types of diversity. Illustrative examples are given to demonstrate their effectiveness for biomedical data analysis. [ABSTRACT FROM AUTHOR]

Mathematica program for various general observer quantities in the general theory of relativity has been developed which uses coordinate system, metric tensor and unit timelike velocity vector as input. [ABSTRACT FROM AUTHOR]

Tensors representations are widely used in multimedia applications. As a key step of tensor processing, the rank-1 tensor decomposition (i.e., the CANDECOMP/PARAFAC (CP) decomposition) always requires the estimation of the tensor rank. The ℓ 2 , 1 -norm has been shown to be effective for tensor rank selection. The existing tensor rank selection algorithm force the same columns of the tensor matrices to simultaneously become zero. However, the real sparse columns for different factor matrices may be different. Such strategy does not really uncover the sparse information of each factor matrix. In this paper, we add a separable ℓ 2 , 1 -norm on multiple factor matrices to obtain real sparse results along to different modes. And then different sparse results are assembled into a joint sparse pattern for tensor rank selection. This added separable regularization term has twofold role in enhancing the effect of regularization for each factor matrix and fully utilizing the knowledge of multiple factor matrices to facilitate decision making. In order to effectively exploit the structure information of multimedia data, we propose a model of tensor bag of words (tBOW) as the direct input of our algorithms. In the experiments, we apply the proposed algorithms to three representative tasks of multimedia analysis, i.e., image classification, video action recognition, and head pose estimation. Experimental results on three open benchmark datasets show that our algorithms are effective to multimedia analysis. [ABSTRACT FROM AUTHOR]

Ricci-flat spacetimes of signature ( 2 , q ) with q = 2 , 3 , 4 are constructed which admit irreducible Killing tensors of rank-3 or rank-4. The construction relies upon the Eisenhart lift applied to Drach's two-dimensional integrable systems which is followed by the oxidation with respect to free parameters. In four dimensions, some of our solutions are anti-self-dual. [ABSTRACT FROM AUTHOR]

A new procedure to detect urea-formaldehyde adhesive in industrial particle board is presented. The method uses thin sections stained with a visible dye (gentian violet) and a fluorescent dye (brilliant sulphaflavine), respectively, in a two-step procedure. Microscope images of a selected area of interest acquired in visible and fluorescence modes are combined to obtain sufficient contrast, enabling semi-automated detection of adhesive by means of image analysis. No addition of dye prior to particle board production is necessary. [ABSTRACT FROM AUTHOR]

The problem of finding the best rank-one approximation to higher-order tensors has extensive engineering and statistical applications. It is well-known that this problem is equivalent to a homogeneous polynomial optimization problem. In this paper, we study theoretical results and numerical methods of this problem, particularly focusing on the 4-th order symmetric tensor case. First, we reformulate the polynomial optimization problem to a matrix programming, and show the equivalence between these two problems. Then, we prove that there is no duality gap between the reformulation and its Lagrangian dual problem. Concerning the approaches to deal with the problem, we propose two relaxed models. The first one is a convex quadratic matrix optimization problem regularized by the nuclear norm, while the second one is a quadratic matrix programming regularized by a truncated nuclear norm, which is a D.C. function and therefore is nonconvex. To overcome the difficulty of solving this nonconvex problem, we approximate the nonconvex penalty by a convex term. We propose to use the proximal augmented Lagrangian method to solve these two relaxed models. In order to obtain a global solution, we propose an alternating least eigenvalue method after solving the relaxed models and prove its convergence. Numerical results presented in the last demonstrate, especially for nonpositive tensors, the effectiveness and efficiency of our proposed methods. [ABSTRACT FROM AUTHOR]

Many real world problems are high-dimensional in that their solution is a function which depends on many variables or parameters. This presents a computational challenge since traditional numerical techniques are built on model classes for functions based solely on smoothness. It is known that the approximation of smoothness classes of functions suffers from the so-called 'curse of dimensionality'. Avoiding this curse requires new model classes for real world functions that match applications. This has led to the introduction of notions such as sparsity, variable reduction, and reduced modeling. One theme that is particularly common is to assume a tensor structure for the target function. This paper investigates how well a rank one function f( x,..., x)= f( x)⋯ f( x), defined on Ω=[0,1] can be captured through point queries. It is shown that such a rank one function with component functions f in $W^{r}_{\infty}([0,1])$ can be captured (in L) to accuracy O( C( d, r) N) from N well-chosen point evaluations. The constant C( d, r) scales like d. The queries in our algorithms have two ingredients, a set of points built on the results from discrepancy theory and a second adaptive set of queries dependent on the information drawn from the first set. Under the assumption that a point z∈ Ω with nonvanishing f( z) is known, the accuracy improves to O( dN). [ABSTRACT FROM AUTHOR]

Abstract: We present a uniqueness theorem for k-graph -algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k-graph -algebra, it is sufficient that the representation be injective on a distinguished abelian -subalgebra. A crucial part of the proof is the application of an abstract uniqueness theorem, which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient -algebra, each of which is the unique extension of a state on the distinguished abelian -subalgebra. [Copyright &y& Elsevier]

In this paper, we show that any Coxeter graph which defines a higher rank Coxeter group must have disjoint induced subgraphs each of which defines a hyperbolic or higher rank Coxeter group where the groups in question do not necessarily act cocompactly or cofinitely. We then use this result to demonstrate several classes of Coxeter graphs which define hyperbolic Coxeter groups and to produce structure theorems for hyperbolic Coxeter groups. [ABSTRACT FROM AUTHOR]

Given a tensor f in a Euclidean tensor space, we are interested in the critical points of the distance function from f to the set of tensors of rank at most k, which we call the critical rank-at-most-k tensors for f. When f is a matrix, the critical rank-one matrices for f correspond to the singular pairs of f. The critical rank-one tensors for f lie in a linear subspace Hf, the critical space of f. Our main result is that, for any k, the critical rank-at-most-k tensors for a sufficiently general f also lie in the critical space Hf. This is the part of Eckart-Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space Hf is spanned by the complex critical rank-one tensors. Since f itself belongs to Hf, we deduce that also f itself is a linear combination of its critical rank-one tensors. [ABSTRACT FROM AUTHOR]

Combining analyses of spatial and temporal characteristics of the ionosphere is of great significance for scientific research and engineering applications. Tensor decomposition is performed to explore the temporal-longitudinal-latitudinal characteristics in the ionosphere. Three-dimensional tensors are established based on the time series of ionospheric vertical total electron content maps obtained from the Centre for Orbit Determination in Europe. To obtain large-scale characteristics of the ionosphere, rank-1 decomposition is used to obtain U(1), U(2), and U(3), which are the resulting vectors for the time, longitude, and latitude modes, respectively. Our initial finding is that the correspondence between the frequency spectrum of U(1) and solar variation indicates that rank-1 decomposition primarily describes large-scale temporal variations in the global ionosphere caused by the Sun. Furthermore, the time lags between the maxima of the ionospheric U(2) and solar irradiation range from 1 to 3.7 h without seasonal dependence. The differences in time lags may indicate different interactions between processes in the magnetosphere-ionosphere-thermosphere system. Based on the dataset displayed in the geomagnetic coordinates, the position of the barycenter of U(3) provides evidence for north-south asymmetry (NSA) in the large-scale ionospheric variations. The daily variation in such asymmetry indicates the influences of solar ionization. The diurnal geomagnetic coordinate variations in U(3) show that the large-scale EIA (equatorial ionization anomaly) variations during the day and night have similar characteristics. Considering the influences of geomagnetic disturbance on ionospheric behavior, we select the geomagnetic quiet GIMs to construct the ionospheric tensor. The results indicate that the geomagnetic disturbances have little effect on large-scale ionospheric characteristics. [ABSTRACT FROM AUTHOR]