Related papers: Basis-Free Analysis of Singular Tuples and Eigenpa…

We consider the Dirac equation in flat Minkowski 3-space and rewrite it as the Maxwell equation in Minkowski 4-space with torsion. The torsion tensor is defined as the dual of the electromagnetic vector potential. Our model clearly…

The analytic rank of a tensor, first defined by Gowers and Wolf in the context of higher-order Fourier analysis, is defined to be the logarithm of the bias of the tensor. We prove that it is a subadditive measure of rank: that is, the…

Developed in a series of seminal papers in the early 2010s, the tubal tensor framework provides a clean and effective algebraic setting for tensor computations, supporting matrix-mimetic features such as a tensor Singular Value…

We show how to reduce free independence to tensor independence in the strong sense. We construct a suitable unital *-algebra of closed operators `affiliated' with a given unital *-algebra and call the associated closure `monotone'. Then we…

Tensors, or multidimensional arrays, are data structures that can naturally represent visual data of multiple dimensions. Inherently able to efficiently capture structured, latent semantic spaces and high-order interactions, tensors have a…

One of the main issues in computing a tensor decomposition is how to choose the number of rank-one components, since there is no finite algorithms for determining the rank of a tensor. A commonly used approach for this purpose is to find a…

In this paper, we introduce set-valued tensor complementarity problem where the elements of the involved tensors are defined based on a set-valued mapping. We study several properties of the solution set under the framework of set-valued…

This note deals with a simultaneous approximation of several matrices by a finite family of diagonalizable matrices satisfying an additional condition for the spectrum of a matrix product. That is the simplicity of all eigenvalues.

Factor model is an appealing and effective analytic tool for high-dimensional time series, with a wide range of applications in economics, finance and statistics. This paper develops two criteria for the determination of the number of…

The theory of eigenvalues and eigenvectors is one of the fundamental and essential components in tensor analysis. Computing the dominant eigenpair of an essentially nonnegative tensor is an important topic in tensor computation because of…

We study in this article multiplicities of eigenvalues of tensors. There are two natural multiplicities associated to an eigenvalue $\lambda$ of a tensor: algebraic multiplicity $\operatorname{am}(\lambda)$ and geometric multiplicity…

The rank of a tensor is analyzed in context of quantum entanglement. A pure quantum state $\bf v$ of a composite system consisting of $d$ subsystems with $n$ levels each is viewed as a vector in the $d$-fold tensor product of…

This paper deals with the problem of describing the vector spaces of divergence-free, natural tensors on a pseudo-Riemannian manifold that are second-order; i.e., that are defined using only second derivatives of the metric. The main result…

The tensor track is a background-independent discretization of quantum gravity which includes a sum over all topologies. We discuss how to define a functional renormalization group flow and the Wetterich equation in the corresponding theory…

The mnesor theory is the adaptation of vectors to artificial intelligence. The scalar field is replaced by a lattice. Addition becomes idempotent and multiplication is interpreted as a selection operation. We also show that mnesors can be…

In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have been utilized recently in Statistics and Signal Processing. It is shown why the decompositions well known in linear algebra can hardly be…

I wrote this book in a "do-it-yourself" style so that I give only a draft of tensor theory, which includes formulating definitions and theorems and giving basic ideas and formulas. All other work such as proving consistence of definitions,…

Invariant tensors play an important role in gauge theories, for example, in dualities of N=1 gauge theories. However, for theories with fields in representations larger than the fundamental, the full set of invariant tensors is often…

An $n \times n \times p$ tensor is called a T-square tensor. It arises from many applications, such as the image feature extraction problem and the multi-view clustering problem. We may symmetrize a T-square tensor to a T-symmetric tensor.…

The tensor power method generalizes the matrix power method to higher order arrays, or tensors. Like in the matrix case, the fixed points of the tensor power method are the eigenvectors of the tensor. While every real symmetric matrix has…