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International audience; We uncover new behaviors of the transmission of information by three quantum channels in superposition of causal orders subject to some level of noise. We find that the transmission can exhibit three different behaviors as the level of noise is varied. This multifold behavior can be explained by the different equivalence classes of quantum switch matrices related to specific combinations of causal orders. We classify these matrices using their characteristic polynomials and matrix invariants, and we calculate analytical expressions for the Holevo information in three representative cases. Our results are a step forward to understand and harness quantum control of causal orders with different levels of noise. We also study the Holevo information as function of a continuous order parameter and analyse transitions at integer values.

An S-type singular value inclusion set for rectangular tensors is given. Based on the set, new upper and lower bounds for the largest singular value of nonnegative rectangular tensors are obtained and proved to be sharper than some existing results. Numerical examples are given to verify the theoretical results.

A new Z-eigenvalue localization set for tensors is given and proved to be tighter than those in the work of Wang et al. (Discrete Contin. Dyn. Syst., Ser. B 22(1):187-198, 2017). Based on this set, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

In this work we derive important properties regarding matrix invariants which occur in the theory of differential equations with reflection., Preprint

We consider two group actions on $m$-tuples of $n \times n$ matrices. The first is simultaneous conjugation by $\operatorname{GL}_n$ and the second is the left-right action of $\operatorname{SL}_n \times \operatorname{SL}_n$. We give efficient algorithms to decide if the orbit closures of two points intersect. We also improve the known bounds for the degree of separating invariants in these cases., Better bounds for separating invariants and improved exposition in some sections

In this paper we introduce three new classes of nonnegative forms (or equivalently, symmetric tensors) and their extensions. The newly identified nonnegative symmetric tensors constitute distinctive convex cones in the space of general symmetric tensors (order six or above). For the special case of quartic forms, they collapse into the set of convex quartic homogeneous polynomial functions. We discuss the properties and applications of the new classes of nonnegative symmetric tensors in the context of polynomial and tensor optimization. Numerical experiments for solving certain polynomial optimization models based on the new classes of nonnegative symmetric tensors are presented.

In this paper, we give a new eigenvalue localization set for tensors and show that the new set is tighter than those presented by Qi (2005) and Li et al. (2014). As applications, we give a new sufficient condition of the positive (semi-)definiteness for an even-order real symmetric tensor and new lower and upper bounds of the minimum eigenvalue for $ { \mathcal M } $-tensors.

Hierarchical tensors can be regarded as a generalisation, preserving many crucial features, of the singular value decomposition to higher-order tensors. For a given tensor product space, a recursive decomposition of the set of coordinates into a dimension tree gives a hierarchy of nested subspaces and corresponding nested bases. The dimensions of these subspaces yield a notion of multilinear rank. This rank tuple, as well as quasi-optimal low-rank approximations by rank truncation, can be obtained by a hierarchical singular value decomposition. For fixed multilinear ranks, the storage and operation complexity of these hierarchical representations scale only linearly in the order of the tensor. As in the matrix case, the set of hierarchical tensors of a given multilinear rank is not a convex set, but forms an open smooth manifold. A number of techniques for the computation of low-rank approximations have been developed, including local optimisation techniques on Riemannian manifolds as well as truncated iteration methods, which can be applied for solving high-dimensional partial differential equations. In a number of important cases, quasi-optimality of approximation ranks and computational complexity have been analysed. This article gives a survey of these developments. We also discuss applications to problems in uncertainty quantification, to the solution of the electronic Schr��dinger equation in the strongly correlated regime, and to the computation of metastable states in molecular dynamics.

Similar to static systems, periodically driven systems can host a variety of topologically non-trivial phases. Unlike the case of static Hamiltonians, the topological indices of bulk Floquet bands may fail to describe the presence and robustness of edge states, prompting the search for new invariants. We develop a unified description of topological phases and their invariants in driven systems, by using scattering theory. We show that scattering matrix invariants correctly describe the topological phase, even when all bulk Floquet bands are trivial. Additionally, we use scattering theory to introduce and analyze new periodically driven phases, such as weak topological Floquet insulators, for which invariants were previously unknown. We highlight some of their similarities with static systems, including robustness to disorder, as well as some of the features unique to driven systems, showing that the weak phase may be destroyed by breaking translational symmetry not in space, but in time., 11 pages, 8 figures. New title and references, expanded discussion. Accepted for publication in PRB

In this article, we present various new results on Cauchy tensors and Hankel tensors. { We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries.} As a consequence, we show that Cauchy tensors are positive semi-definite if and only if they are SOS (Sum-of-squares) tensors.} Furthermore, we prove that all positive semi-definite Cauchy tensors are completely positive tensors, which means every positive semi-definite Cauchy tensor can be decomposed { as} the sum of nonnegative rank-1 tensors. We also establish that all the H-eigenvalues of nonnegative Cauchy tensors are nonnegative. Secondly, we present new mathematical properties of Hankel tensors. { We prove that an even order Hankel tensor is Vandermonde positive semi-definite if and only if its associated plane tensor is positive semi-definite. We also show that, if the Vandermonde rank of a Hankel tensor $\mathcal{A}$ is less than the dimension of the underlying space, then positive semi-definiteness of $\mathcal{A}$ is equivalent to the fact that $\mathcal{A}$ is a complete Hankel tensor, and so, is further equivalent to the SOS property of $\mathcal{A}$. Lastly, we introduce a new structured tensor called Cauchy-Hankel tensors, which is a special case of Cauchy tensors and Hankel tensors simultaneously.} Sufficient and necessary conditions are established for an even order Cauchy-Hankel tensor to be positive definite. Final remarks are listed at the end of the paper.

The purpose of this paper is to establish the admitted region for five simultaneous, functionally independent invariants of the strain rate tensorSand rotation rate tensorΩand calculate some simultaneous invariants of these tensors which are encountered in the theory of constitutive relations for turbulent flows. Such a problem, as far as we know, has not yet been considered, though it is obviously an integral part of any problem in which scalar functions of the tensorsSandΩare studied. The theory provided inside this paper is the building block for a derivation of new algebraic constitutive relations for three-dimensional turbulent flows in the form of expansions of the Reynolds-stress tensor in a tensorial basis formed by the tensorsSandΩ, in which the scalar coefficients depend on simultaneous invariants of these tensors.

Multilocal higher-order invariants, which are higher-order invariants defined at distinct points of representation space, for the classical groups are derived in a systematic way. The basic invariants for the classical groups are the well-known polynomial or rational invariants as derived from the Capelli identities. Higher-order invariants are then constructed from the former ones by means of total derivatives. At each order, it appears that the invariants obtained in this way do not generate all invariants. The necessary additional invariants are constructed from the invariant polynomials on the Lie algebra of the Lie transformation groups.

In this paper, we extend some classes of structured matrices to higher order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a structured tensor is still a structured tensor in the same class, with a lower dimension. The potential links of such structured tensors with optimization, nonlinear equations, nonlinear complementarity problems, variational inequalities and the nonnegative tensor theory are also discussed., arXiv admin note: text overlap with arXiv:1405.1288 by other authors

International audience; In this paper, we explore new conditions for an elasticity tensor to belong to a given symmetry class. Our goal is to propose an alternative approach to the identification problem of the symmetry class, based on polynomial invariants and covariants of the elasticity tensor C, rather than on spectral properties of the Kelvin representation. We compute a set of algebraic relations which describe precisely the orthotropic ([D2]), trigonal ([D3]), tetragonal ([D4]), transverse isotropic ([SO(2)]) and cubic ([O]) symmetry classes in H4, the higher irreducible component in the decomposition of Ela. We provide a bifurcation diagram which describes how one "travel" in H4 from a given isotropy class to another. Finally, we study the link between these polynomial invariants and those obtained as the coefficients of the characteristic or the Betten polynomials.We show, in particular, that the Betten invariants do not separate the orbits of the elasticity tensors.

Considering arbitrary conformal field theories in general (non-conformally flat) backgrounds, we adopt a dimensional regularization approach to obtain stress tensors from Weyl anomalies. The results of type A anomaly-induced stress tensors in four and six-dimensions generalize the previous results calculated in a conformally flat background. On the other hand, regulators are needed to have well-defined type B anomaly-induced stress tensors. We also discuss ambiguities related to type D anomalies, Weyl invariants and order of limit issues., 15 pages

The action of the orthogonal group \(O(2)\) on the space of plane elasticity tensors has been the subject of some recent investigations. It is shown here that the approach based on the “harmonic decomposition” technique, which is also used in a three-dimensional setting, gives a unified perspective on this issue. We construct explicit relationships between invariants and quantities derived from such an approach and what was found earlier by Tsai and Pagano and, more recently, through the “polar method” and the use of complex variables.

International audience; In both theoretical and applied mechanics, the modeling of nonlinear constitutive relations of materials is a topic of prime importance. To properly formulate consistent constitutive laws some restrictions need to be impose on tensor functions. To that aim representations theorems for both isotropic and anisotropic functions have been extensively investigated since the middle of the XXth century. Nevertheless, in three-dimensional physical space, most of the results are restricted to sets of tensors up to second-order. The purpose of the present paper is thus to get one step further and to provide an integrity basis for isotropic polynomial functions of a completely symmetric third-order tensor. To explicitly construct this basis, the link that exists between the O(3)-action on harmonic tensors and the SL(2,C)-action on the space of binary forms is exploited. We believe that such an integrity basis may found interesting applications both in continuum mechanics and in other fields of theoretical physics.

Let $\mathbb{P}_n$ be the set of all matrices which have the same zero patterns with some permutation matrix of order $n$. In this paper, we prove the following result: Let $\mathbb{I}$ be the unit tensor of order $m\ge3$ and dimension $n\ge2$. Suppose that $P$ and $Q$ are two matrices with $P\mathbb{I}Q=\mathbb{I}$, then $P,Q\in \mathbb{P}_n$. This gives a characterization for the similarities of tensors with order $m\ge3$., 7 pages

International audience; In this paper we provide a complete and irreducible representation for transversely isotropic sixth order tensors having minor symmetries. Such tensors appear in some practical problems of elasticity for which their inversion is required. For this kind of tensors, we provide an irreducible basis which possesses some remarkable properties, allowing us to provide a representation in a compact form which uses two scalars and three matrices of dimension 2, 3 and 4. It is shown that the calculation of sum, product and inverse of transversely isotropic sixth order tensors is greatly simplified by using this new formalism and appears to be appropriate for deriving new various solutions to some practical problems in mechanics which use such kinds of higher order tensors. For instance, we derive the fields within a cylindrical inhomogeneity submitted to remote gradient of strain. The method of resolution uses the Eshelby equivalent inclusion method extended to the case of a polynomial type eigenstrain. It is shown that the approach leads to a linear system involving a sixth order tensor whose closed form solution is derived by means of the tensorial formalism introduced in the first part of the paper.

The problem discussed in this paper is the symmetric best low multilinear rank approximation of third-order symmetric tensors. We propose an algorithm based on Jacobi rotations, for which symmetry is preserved at each iteration. Two numerical examples are provided indicating the need for such algorithms. An important part of the paper consists of proving that our algorithm converges to stationary points of the objective function. This can be considered an advantage of the proposed algorithm over existing symmetry-preserving algorithms in the literature.

We have constructed graphical (qualitative and visual) representations of DNA sequences as 2D maps and their numerical (quantitative and computational) analysis. The maps are obtained by transforming the four-letter sequences (where letters represent the four nucleic bases) via a spiral representation over triangular and square cells grids into a four-color map. The so constructed maps are then represented by distance matrices. We consider the use of several matrix invariants as DNA descriptors for determining the degree of similarity of a selection of DNA sequences. (doi: 10.5562/cca2338)

We give another characterization of the annihilator of the space of (dual) harmonic tensors in the group algebra of symmetric group.

International audience; In the present paper we propose new results concerning linear tensorial algebra for third-order and non symmetric isotropic sixth-order tensors in the most general case (i.e. having not the major and minor symmetries). Such tensors are used, for instance, in the theory of microstructured elastic media. A formalism based on an irreducible basis for isotropic sixth-order tensors is introduced, which is useful for performing the classical tensorial operations. Specially, a condensed expression for the product between two isotropic sixth-order tensors is provided, which allows the obtaining of a condition on these tensors for being invertible and a closed form expression of the inverse of such a tensor. Finally, the condition of positiveness of third-order tensor-valued quadratic functions is derived. For instance, such conditions are required for computing the elastic energy of microstructured media.

Cartesian tensors of various orders have been employed for either modeling the diffusivity or the orientation distribution function in Diffusion-Weighted MRI datasets. In both cases, the estimated tensors have to be positive-definite since they model positive-valued functions. In this paper we present a novel unified framework for estimating positive-definite tensors of any order, in contrast to the existing methods in literature, which are either order-specific or fail to handle the positive-definite property. The proposed framework employs a homogeneous polynomial parametrization that covers the full space of any order positive-definite tensors and explicitly imposes the positive-definite constraint on the estimated tensors. We show that this parametrization leads to a linear system that is solved using the non-negative least squares technique. The framework is demonstrated using synthetic and real data from an excised rat hippocampus.

The concept of tensor rank was introduced in the 1920s. In the 1970s, when methods of Component Analysis on arrays with more than two indices became popular, tensor rank became a much studied topic. The generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor. We explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions, based on Terracini's lemma, and compare it with the algebraic results already known in the real or complex fields. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, complex tensors enjoying Hermitian symmetries, or merely tensors with free entries. (C) 2009 Elsevier Inc. All rights reserved.

International audience; In condensed matter physics tensors are used to model physical properties of materials. Some well known examples are elasticity which is modeled by a 4th-order tensor or piezoelectricity by a 3rdorder tensor. Some tensors of higher order might occur in theory of generalized continuum, for example in Mindlin's strain-gradient elasticity [2] in which the physical state of the material is described in each point using three tensors from 4th-order to 6th-order one. Damage could also be modeled by an 8th-order tensor. When ones studying such complex models the first step is to focus attention in its bi-dimensional version. Taking a complex problem at a lower level help to better understand the physics of the behavior. But we have to wonder in which measure the phenomenon we study is independent of the dimension of the space we are working in. In this paper we will focus our attention on the issue of symmetry classes of linear constitutive behavior, and show that in a two dimension anomalies could occur.

We generalize the Perron–Frobenius Theorem for nonnegative matrices to the class of nonnegative tensors.

International audience; The concept of tensor rank, introduced in the twenties, has been popularized at the beginning of the seventies. This has allowed to carry out Factor Analysis on arrays with more than two indices. The generic rank may be seen as an upper bound to the number of factors that can be extracted from a given tensor, with certain uniqueness conditions. We explain how to obtain numerically the generic rank of tensors of arbitrary dimensions, and compare it with the rare algebraic results already known at order three. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, or tensors with free entries. Related applications include antenna array processing.

We present a new proof of the representation theorem for fourth-order isotropic tensors that does not assume the tensor to have major or minor symmetries at the outset.

We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly useful in generalizing certain areas where the spectral theory of matrices has traditionally played an important role. For illustration, we will discuss a multilinear generalization of the Perron-Frobenius theorem.

This chapter introduces the notion of tensors in physics (field tensors and material property tensors), starting from the matrix of physical properties. The basic properties of vector spaces and the elementary mathematical properties of tensors are then recalled. The most important part of the chapter, Section 1.1.4, is devoted to the symmetry properties of tensors: Neumann's principle, Curie laws and the reduction of the components of polar and axial tensors of rank up to 4 for the 32 crystallographic groups. The relations between the thermodynamic functions and the physical property tensors are developed in Section 1.1.5.

We define and characterize weakly isotropic tensors in terms of properties of invariance with respect to some sets of rotations, for instance, those in the cubic group. Hence, in particular, we characterize weakly isotropic tensors which are isotropic or skew-isotropic depending on whether their order is even or odd, respectively. Restricting our attention to tensors of order less than or equal to 7, we characterize a minimal set of rotations such that the invariance with respect to it implies the property of weak isotropy. Then we show that any weakly isotropic tensor of order 7 can be represented in terms of 36 independent arbitrary scalars. This number of independent scalars is minimal, in the sense that any representation for such a tensor involves at least 36 scalars. We produce two such minimal representations, firstly by finding a suitable choice of independent components, in terms of which we express the remaining tensor components. Secondly, we find 36 linearly independent weakly isotropic tensors which are products of a Ricci symbol with two Kronecker symbols and are such that any other weakly isotropic tensor can be expressed as a linear combination of them.

Operations with tensors, or multiway arrays, have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-1 outer products using either the CANDECOMP/PARAFAC (CP) or the Tucker models, or some variation thereof. Such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array. The specifics of the approximate representation (such as how many terms to use in the sum, orthogonality constraints, etc.) depend on the application. In this paper, we explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem. Reminiscent of matrix factorizations, we present a new factorization of a tensor as a product of tensors. To derive the new factorization, we define a closed multiplication operation between tensors. A major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications. Specifically, this new multiplication allows us to introduce concepts such as tensor transpose, inverse, and identity, which lead to the notion of an orthogonal tensor. The multiplication also gives rise to a linear operator, and the null space of the resulting operator is identified. We extend the concept of outer products of vectors to outer products of matrices. All derivations are presented for third-order tensors. However, they can be easily extended to the order-p ( p > 3 ) case. We conclude with an application in image deblurring.

It is shown that the full symmetry class of tensors associated with the irreducible character [2,1 n −2 ] of S n does not have an orthogonal basis consisting of decomposable symmetrized tensors.

We prove new versions of A r λ (Ω) -weighted imbedding inequalities for A -harmonic tensors locally and globally. The results are used to estimate the integrals of a homotopy operator T from the Banach space L s (Ω,∧ l ) to the Sobolev space W 1,s (Ω,∧ l−1 ) , l=1,2,…,n , and Sobolev–Poincare imbedding inequality with A r λ (Ω) weight.

An extension of Polya's Theorem inventories the sequence set corresponding to an induced basis in a higher degree symmetry class of tensors.

A recently proposed tensor-tensor multiplication (M.E. Kilmer, C.D. Martin, L. Perrone, A Third-Order Generalization of the Matrix SVD as a Product of Third-Order Tensors , Tech. Rep. TR-2008-4, Tufts University, October 2008) opens up new avenues to understanding the action of n × n × n tensors on a space of n × n matrices. In particular it emphasizes the need to understand the space of objects upon which tensors act. This paper defines a free module and shows that every linear transformation on that module can be represented by tensor multiplication. In addition, it presents a generalization of ideas of eigenvalue and eigenvector to the space of n × n × n tensors.

Here we use invariant theory to describe the Brauer–Severi scheme of the fibers of trace rings of generic matrices with an algebraically closed base field of characteristic zero when the trace ring is viewed as a sheaf of algebras over the variety of matrix invariants. Using this approach, we first prove that the Brauer–Severi scheme of a trace ring is isomorphic to Proj Q , for a graded ring Q whose generators we describe in the first section. This description also has a relevant interpretation over base fields of arbitrary characteristic. In the second section of this paper we show that the Brauer–Severi scheme of the fiber of a trace ring over a point that is not too degenerate will have smooth irreducible components meeting transversally and describe these irreducible components as Brauer–Severi schemes of certain algebras.

The trace algebra C n d is generated by all traces of products of d generic n × n matrices. Minimal generating sets of C n d and their defining relations are known for n 3 and n = 3 , d = 2 . This paper states a minimal generating set and their defining relations for n = d = 3 . Furthermore the computations yield a description of C 33 as a free module over the ring generated by a homogeneous system of parameters.

We suggest a different point of view on some aspects of classical invariant theory. A tensor is regarded as a multiply subscripted array of numbers which represents a multilinear form in a natural way. Two tensors are equivalent if they represent the same symmetric multilinear form. Canonical forms for symmetric 2×2×2 tensors are derived from this point of view and can be regarded as a generalization of Sylvester's law of inertia. In addition, some problems concerning the classification of cubic forms and cubic functionals under the action of various groups are suggested.

In this paper, a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors. What is more, we prove this method has an explicit linear convergence rate for indirectly positive tensors. Interestingly, the algorithm is exactly the NQZ method (proposed by Ng, Qi and Zhou in Finding the largest eigenvalue of a non-negative tensor SIAM J Matrix Anal Appl 31:1090–1099, 2009) by taking a specific parameter. Furthermore, we give a modified NQZ method, which has an explicit linear convergence rate for nonnegative tensors and has an error bound for nonnegative tensors with a positive Perron vector. Besides, we promote an inexact power-type algorithm. Finally, some numerical results are reported.

A necessary and sufficient condition for the existence of the orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with dicyclic group and dihedral group were studied by M. R. Darafsheh and M. R. Pournaki (in press, Linear and Multilinear Algebra) and R. R. Holmes and T. Y. Tam (1992, Linear and Multilinear Algebra32, 21–31). These authors used a certain permutation structure of these groups to prove the necessary condition. In this article we show that the necessary condition found in these previous works is independent of the permutation structures of these groups.

In solid mechanics of isotropic and anisotropic materials representing scalar-valued tensor functions or symmetric second-order tensor-valued tensor functions is of major concern, For instance, the plastic potential is scalar-valued, whereas constitutive equations are tensor-valued. In this paper scalar-valued functions have been represented. Anisotropic effects have been characterized by material tensors of rank two and four. For instance, the effect of damage or the behaviour of oriented solids have been characterized by second-order tensors, and with respect to more general cases inborn anisotropy has been described by using a rank four. In representing scalar-valued tensor functions, a set of irreducible invariant involving the above mentioned tensor variables has been constructed. The central problem is: to find an integrity basis for the argument tensors. Together with the invariants of the single argument tensors the system of simultaneous or joint invariants is considered. Such invariants are given by traces of outer products formed from different argument tensors considered. In finding irreducible invariants of a fourth-order tensor the characteristic polynomial for a fourth-order tensor is derived by the definition of the eigenvektor . Furthermore, Hamilton-Cayley's theorem is applied to a fourth-order tensor. Thus irreducible invariants of a fourth-order tensor can be expressed through sums of principal minors.

We prove that the rings of invariants of 2×2 matrices over an infinite field are Cohen–Macaulay. This result generalizes the similar theorem of Mehta and Ramadas in odd characteristics. Our approach is more elementary and it uses only some standard facts from the theory of modules with good filtrations and the theory of determinantal rings.

Lagrangian field theories of geometric objects are the most natural framework for investigating the notion of general covariance. We discuss here geometric theories of interacting fields, depending on Lagrangians of arbitrary order, and we give general definitions of energy flow, partial energy flows, energy‐momentum tensors, and stress tensors. We also investigate the role which energy‐momentum tensors and stress tensors play in formulating the natural conservation laws associated with the second theorem of Noether. Examples of application may be found elsewhere.

The decomposition of tensors into irreducible representations of the orthogonal groups is calculated for three and four dimensions. The connection is shown with the problem of the allowed values of ordinary and isotopic spin for a given symmetry of the spacial eigenfunction of a nuclear system.