Related papers: Tensor norms on ordered normed spaces, polarizatio…
In this paper an idea of soft linear spaces and soft norm on soft linear spaces are given and some of their properties are studied. Soft vectors in soft linear spaces are introduced and their properties are studied. Completeness of soft…
We study symmetric and antisymmetric tensor products of Hilbert-space operators, focusing on norms and spectra for some well-known classes favored by function-theoretic operator theorists. We pose many open questions that should interest…
A tensor network is a type of decomposition used to express and approximate large arrays of data. A given data-set, quantum state or higher dimensional multi-linear map is factored and approximated by a composition of smaller multi-linear…
We establish several mathematical and computational properties of the nuclear norm for higher-order tensors. We show that like tensor rank, tensor nuclear norm is dependent on the choice of base field --- the value of the nuclear norm of a…
We consider mixed finite element methods for linear elasticity where the symmetry of the stress tensor is weakly enforced. Both an a priori and a posteriori error analysis are given for several known families of methods that are uniformly…
We study the symmetric tensor rank of multiplication over finite field extensions using linearized polynomials. Via field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and symmetric matrices, allowing…
We present the basic concepts of tensor products of vector spaces, emphasizing linear algebraic and combinatorial techniques as needed for applied areas of research. The topics include (1) Introduction; (2) Basic multilinear algebra; (3)…
A new method to find the propagation equation system governing the scattering of an electromagnetic wave by a nonlinear medium is proposed. The aim is to let the effects appear spontaneously, deleting as far as possible the phenomenological…
We show that the spectral norm of a $d$-mode real or complex symmetric tensor in $n$ variables can be computed by finding the fixed points of the corresponding polynomial map. For a generic complex symmetric tensor the number of fixed…
The norm of the $m$th derivative of the map that takes an operator to its $k$th antisymmetric tensor power is evaluated. The case $m=1$ has been studied earlier by Bhatia and Friedland [R. Bhatia and S. Friedland, Variation of Grassman…
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…
Extensions of nuclear supersymmetry are discussed, together with a proposal for new, more stringent and precise tests that probe the susy classification and specific two-particle correlations among supersymmetric partners. The combination…
The polarizability measures how the system responds to an applied electrical field. Computationally, there are many different ways to evaluate this tensorial quantity, some of which rely on the explicit use of the external perturbation and…
Specific definitions of the core and core-EP inverses of complex tensors are introduced. Some characterizations, representations and properties of the core and core-EP inverses are investigated. The results are verified using specific…
The tensor-tensor product (t-product) [M. E. Kilmer and C. D. Martin, 2011] is a natural generalization of matrix multiplication. Based on t-product, many operations on matrix can be extended to tensor cases, including tensor SVD, tensor…
The representation theory of tensor functions is a powerful mathematical tool for constitutive modeling of anisotropic materials. A major limitation of the traditional theory is that many point groups require fourth- or sixth-order…
In this work, we investigate the tensor inequalities in the tensor t-product formalism. The inequalities involving tensor power are proved to hold similarly as standard matrix scenarios. We then focus on the tensor norm inequalities. The…
S.Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529--563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs. We prove that each poset limit can be represented as a…
High-order phenomena play crucial roles in many systems of interest, but their analysis is often highly nontrivial. There is a rich literature providing a number of alternative information-theoretic quantities capturing high-order…
We study the optimal transport problem for $d>2$ discrete measures. This is a linear programming problem on $d$-tensors. It gives a way to compute a "distance" between two sets of discrete measures. We introduce an entropic regularization…