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The Cauchy relations distinguish between rari- and multi-constant linear elasticity theories. These relations are treated in this paper in a form that is invariant under two groups of transformations: indices permutation and general linear transformations of the basis. The irreducible decomposition induced by the permutation group is outlined. The Cauchy relations are then formulated as a requirement of nullification of an invariant subspace. A successive decomposition under rotation group allows to define the partial Cauchy relations and two types of elastic materials. We explore several applications of the full and partial Cauchy relations in physics of materials. The structure's deviation from the basic physical assumptions of Cauchy's model is defined in an invariant form. The Cauchy and non-Cauchy contributions to Hooke's law and elasticity energy are explained. We identify wave velocities and polarization vectors that are independent of the non-Cauchy part for acoustic wave propagation. Several bounds are derived for the elasticity invariant parameters. [ABSTRACT FROM AUTHOR]

This paper is concerned with the iterative solutions to a class of generalized Sylvester quaternion tensor equations. From the Kronecker product and the properties of quaternion tensors, we propose the biconjugate A-orthogonal residual stabilized and the generalized product-type biconjugate A-orthogonal residual methods in their tensor forms for solving the original tensor equation. As an application, the proposed methods are tested with some colour video restoration problems. Numerical results show the effectiveness of our methods compared with some existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we study that the algebraic multiplicity of the zero Laplacian eigenvalue of a connected uniform hypergraph. We give the algebraic multiplicity of the zero Laplacian eigenvalue of a hyperstar. For a loose hyperpath, we characterize the algebraic multiplicity of the zero Laplacian eigenvalue by the multiplicities of points in the affine variety defined by the Laplacian eigenvalue equations. We compute the algebraic multiplicities of the zero Laplacian eigenvalue of a loose hyperpath. We also show that the algebraic multiplicity of the zero Laplacian eigenvalue is not smaller than the number of irreducible components of the eigenvariety associated with the zero Laplacian eigenvalue for a loose hyperpath. [ABSTRACT FROM AUTHOR]

In magnetic resonance imaging, high angular resolution diffusion imaging (abbr. HARDI) is used to characterize non-Gaussian diffusion processes. One approach to analyzing HARDI data is to model the apparent diffusion coefficient with higher order diffusion tensors from a diffusivity function. An intrinsic property of the diffusivity function is positive semi-definite, which reflects the phenomenon of water molecular diffusion in complicated biological tissue environments. In this paper, we provide a workable criterion for judging the positive semi-definiteness of a diffusivity function and shows that it is effective via two numerical examples. [ABSTRACT FROM AUTHOR]

Recently, solving tensor equations (or multilinear systems) has attracted a lot of attention. This paper investigates the tensor form of the Bi-CGSTAB and Bi-CRSTAB methods, by employing Kronecker product, vectorization, and bilinear operator, to solve the generalized coupled Sylvester tensor equations ∑ i = 1 n (X × 1 A i 1 × 2 A i 2 + Y × 1 B i 1 × 2 B i 2) = E 1 , ∑ i = 1 n (X × 1 C i 1 × 2 C i 2 + Y × 1 D i 1 × 2 D i 2) = E 2 , with no matricization. Also some properties of the new methods are presented. By applying multilinear operator, the proposed methods are extended to the general form. Some numerical examples are provided to compare the efficiency of the investigated methods with some existing popular algorithms. Finally, some concluding remarks are given. [ABSTRACT FROM AUTHOR]

In this paper, we propose several improved uniqueness conditions of solution for multilinear PageRank with order-3 irreducible stochastic tensors. By considering upper and lower bounds of solutions, the refined uniqueness conditions are more effective than the existing ones. Furthermore, as an application, sharp perturbation bounds for multilinear PageRank are also developed basing on the new uniqueness conditions. Numerical examples are given to verify the efficiency of the proposed uniqueness conditions. [ABSTRACT FROM AUTHOR]

Efficient and fast computation of a tensor singular value decomposition (t-SVD) with a few passes over the underlying data tensor is crucial because of its many potential applications. The current/existing subspace randomized algorithms need (2 q + 2) passes over the data tensor to compute a t-SVD, where q is a non-negative integer number (power iteration parameter). In this paper, we propose an efficient and flexible randomized algorithm that can handle any number of passes q, which not necessary need be even. The flexibility of the proposed algorithm in using fewer passes naturally leads to lower computational and communication costs. This advantage makes it particularly appropriate when our task calls for several tensor decompositions or when the data tensors are huge. The proposed algorithm is a generalization of the methods developed for matrices to tensors. The expected/average error bound of the proposed algorithm is derived. Extensive numerical experiments on random and real-world data sets are conducted, and the proposed algorithm is compared with some baseline algorithms. The extensive computer simulation experiments demonstrate that the proposed algorithm is practical, efficient, and in general outperforms the state of the arts algorithms. We also demonstrate how to use the proposed method to develop a fast algorithm for the tensor completion problem. [ABSTRACT FROM AUTHOR]

The strong ellipticity condition (abbr. SE-condition) of the displacement equations of equilibrium for general nonlinearly elastic materials plays an important role in nonlinear elasticity and materials. Existing literature shows that it can be equivalently transformed into the SE-condition of an elasticity tensor, and that the SE-condition of an elasticity tensor holds if and only if all of its M-eigenvalues are positive. In order to judge the strong ellipticity of an elasticity tensor, we in this paper propose a direct method for calculating all M-eigenvalues of an elasticity tensor and show that the direct method is workable by a numerical example. [ABSTRACT FROM AUTHOR]

The Jacobian scheme of a reduced, singular projective plane curve is the zero-dimensional scheme, whose homogeneous ideal is generated by the partials of its defining polynomial. The degree of such a scheme is called the global Tjurina number and, if the curve is not a set of concurrent lines, some upper and lower bounds depending on the degree of the curve and the minimal degree of a Jacobian syzygy, have been given by A.A. du Plessis and C.T.C. Wall. In this paper, we give a complete geometric characterization of conic-line arrangements, with global Tjurina number attaining the upper bound. Furthermore, we characterize conic-line arrangements attaining the lower bound for the global Tjurina number, among all curves with a linear Jacobian syzygy. As an application, we characterize conic-line arrangements with Jacobian scheme equal to an eigenscheme of some ternary tensor, and we study the geometry of their polar maps. [ABSTRACT FROM AUTHOR]

The smallest M-eigenvalue τ M (A) of a fourth-order partial symmetric tensor A plays an important role in judging the strong ellipticity condition (abbr. SE-condition) in elastic mechanics. Specifically, if τ M (A) > 0 , then the SE-condition of A holds. In this paper, we establish lower and upper bounds of τ M (A) via extreme eigenvalues of symmetric matrices and tensors constructed by the entries of A . In addition, when A is an elasticity Z-tensor, we establish lower bounds for τ M (A) via the extreme C-eigenvalues of piezoelectric-type tensors. Finally, numerical examples show the efficiency of our proposed bounds in judging the SE-condition of A . [ABSTRACT FROM AUTHOR]

Sparse representation has garnered significant attention across multiple fields, including signal processing, statistics, and machine learning. The fundamental concept of this technique is that we can express the signal as a linear combination of only a few elements from a known basis. Compressed sensing (CS) is an interesting application of this technique. It is valued for its potential to improve data collection and ensure efficient acquisition and recovery from just a few measurements. In this paper, we propose a novel approach for the high-order CS problem based on the Einstein product, utilizing a tensor dictionary instead of the commonly used matrix-based dictionaries in the Tucker model. Our approach provides a more general framework for compressed sensing. We present two novel models to address the CS problem in the multidimensional case. The first model represents a natural generalization of CS to higher-dimensional signals; we extend the traditional CS framework to effectively capture the sparsity of multidimensional signals and enable efficient recovery. In the second model, we introduce a complexity reduction technique by utilizing a low-rank representation of the signal. We extend the OMP and the homotopy algorithms to solve the high-order CS problem. Through various simulations, we validate the effectiveness of our proposed method, including its application to solving the completion tensor problem in 2D and 3D colored and hyperspectral images. [ABSTRACT FROM AUTHOR]

In this paper, an alternative version of the quantum phase estimation is proposed, in which the Hadamard gates at the beginning are substituted by a quantum Fourier transform. This new circuit coincides with the original one, when the ancilla is initialized with 0 . With the help of a projection-based tensor decomposition and closed-form expressions of its exponential, this new method can be interpreted as a multiplier coupled to the Hamiltonian of the corresponding target unitary operator. Based on this observation, a recursive decomposition is derived. [ABSTRACT FROM AUTHOR]

Many numerical problems with input x and output y can be formulated as a system of equations F (x , y) = 0 where the goal is to solve for y . The condition number measures the change of y for small perturbations to x . From this numerical problem, one can derive a (typically underdetermined) relaxation by omitting any number of equations from F. We propose a condition number for underdetermined systems that relates the condition number of a numerical problem to those of its relaxations, thereby detecting the ill-conditioned constraints. We illustrate the use of our technique by computing the condition of two problems that do not have a finite condition number in the classic sense: two-factor matrix decompositions and Tucker decompositions. [ABSTRACT FROM AUTHOR]

The main aim of this article is to introduce the weighted numerical range and the weighted numerical radius for an even-order square tensor via the Einstein product and establish their various properties. Also, the proof of convexity of the numerical range of a tensor is revisited. The notions of weighted unitary tensor, weighted positive definite tensor, and weighted positive semi-definite tensor are then discussed. The spectral decomposition for normal tensors is also provided. This is then used to present the equality between the weighted numerical radius and the spectral radius of a weighted normal tensor. As applications of the above fact, a few equalities of weighted numerical radius and weighted tensor norm are obtained. [ABSTRACT FROM AUTHOR]

We answer to a question posed recently in reference [Lovitz B, Petrov F. A generalization of Kruskal's theorem on tensor decomposition. Available at arXiv 2103.15633; 2021], proving the conjectured sufficient minimality and uniqueness condition of the 3-tensor decomposition. [ABSTRACT FROM AUTHOR]

The rank of a tensor is analysed in the context of quantum entanglement. A pure quantum state $ \bf v $ v of a composite system consisting of d subsystems with n levels each is viewed as a vector in the d-fold tensor product of n-dimensional Hilbert space and can be identified with a tensor with d indices, each running from 1 to n. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for the characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state $ \mathbf{v} $ v is called entangled, if it cannot be written in the product form, $ \mathbf{v} \ne \mathbf{v}_1 \otimes \mathbf{v}_2 \otimes \cdots \otimes \mathbf{v}_d $ v ≠ v 1 ⊗ v 2 ⊗ ⋯ ⊗ v d , what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed. [ABSTRACT FROM AUTHOR]

Let G be a k-uniform hypergraph with vertex set [n] and edge set E(G), where k ≥ 2 . For i ∈ [ n ] , d i denotes the degree of vertex i in G. The ABC spectral radius of G is max k ∑ e ∈ E (G) ∑ i ∈ e d i - k ∏ i ∈ e d i k ∏ i ∈ e x i : x ∈ R + n , ∑ i = 1 n x i k = 1. We give tight lower and upper bounds for the ABC spectral radius, and determine the maximum ABC spectral radii of uniform hypertrees, uniform non-hyperstar hypertrees and uniform non-power hypertrees of given size, as well as the maximum ABC spectral radii of uniform unicyclic hypergraphs and linear uniform unicyclic hypergraphs of given size, respectively. We also characterize those uniform hypergraphs for which the maxima for the ABC spectral radii are actually attained in all cases. [ABSTRACT FROM AUTHOR]

Starting from the problem of d-tensor isomorphism (d- TI ), we study the relation between various code equivalence problems in different metrics. In particular, we show a reduction from the sum-rank metric ( CE sr ) to the rank metric ( CE rk ). To obtain this result, we investigate reductions between tensor problems. We define the monomial isomorphism problem for d-tensors (d- TI ∗ ), where, given two d-tensors, we ask if there are d - 1 invertible matrices and a monomial matrix sending one tensor into the other. We link this problem to the well-studied d- TI and the TI -completeness of d- TI ∗ is shown. Due to this result, we obtain a reduction from CE sr to CE rk . In the literature, a similar result was known, but it needs an additional assumption on the automorphisms of matrix codes. Since many constructions based on the hardness of Code Equivalence problems are emerging in cryptography, we analyze how such reductions can be taken into account in the design of cryptosystems based on CE sr . [ABSTRACT FROM AUTHOR]

Dynamical low-rank approximation (DLRA) for the numerical simulation of Vlasov–Poisson equations is based on separation of space and velocity variables, as proposed in several recent works. The standard approach for the time integration in the DLRA model uses a splitting of the tangent space projector for the low-rank manifold according to the separated variables. It can also be modified to allow for rank-adaptivity. A less studied aspect is the incorporation of boundary conditions in the DLRA model. In this work, a variational formulation of the projector splitting is proposed which allows to handle inflow boundary conditions on spatial domains with piecewise linear boundary. Numerical experiments demonstrate the principle feasibility of this approach. [ABSTRACT FROM AUTHOR]

Motivated by the PageRank model for graph partitioning, we develop an extension of PageRank for partitioning uniform hypergraphs. Starting from adjacency tensors of uniform hypergraphs, we establish the multi-linear pseudo-PageRank (MLPPR) model, which is formulated as a multi-linear system with nonnegative constraints. The coefficient tensor of MLPPR is a kind of Laplacian tensors of uniform hypergraphs, which are almost as sparse as adjacency tensors since no dangling corrections are incorporated. Furthermore, all frontal slices of the coefficient tensor of MLPPR are M-matrices. Theoretically, MLPPR has a solution, which is unique under mild conditions. An error bound of the MLPPR solution is analyzed when the Laplacian tensor is slightly perturbed. Computationally, by exploiting the structural Laplacian tensor, we propose a tensor splitting algorithm, which converges linearly to a solution of MLPPR. Finally, numerical experiments illustrate that MLPPR is effective and efficient for hypergraph partitioning problems. [ABSTRACT FROM AUTHOR]

We develop and study incremental algorithms for truncated higher-order singular value decompositions. By combining the SVD updating and different truncated higher-order singular value decompositions, two incremental algorithms are proposed. Not only the factor matrices but also the core tensor are updated in an incremental style. The costs of these algorithms are compared and the approximation errors are analyzed. Numerical results demonstrate that the proposed incremental algorithms have advantages in online computation. [ABSTRACT FROM AUTHOR]