Submodules of Kronecker modules via extension monoid products

Using some results on Reineke's extension monoid product over the Kronecker algebra kQ, we provide numerical criteria in terms of Kronecker invariants for a module in $\mathrm{mod}\text{-}kQ$ to be isomorphic with the submodule of an another module in $\mathrm{mod}\text{-}kQ$. The results can be transferred to matrix pencils in solving an important challenge in matrix pencil theory: characterize in terms of Kronecker invariants when a given pencil is a subpencil of an another one.     

Kronecker products and BIBDs

Recursive constructions are given which permit, under conditions described in the paper, a (v, b, r, k, λ)-configuration to be used to obtain a (v′, b′, r′, k, λ)-configuration.     

Using Kronecker products to construct mimetic gradients

This paper describes how discrete versions of the derivative on the real line induce discrete version of the gradient and divergence in higher dimensions. This is geometrically motivated by results in algebraic graph theory since the grid in n dimensions is the graph Kronecker product of the path on n vertices. The resulting technique relies heavily on the matrix Kronecker product, and is an analogue of a derivation in multilinear algebra.     

模糊有限自动机的kronecker积

在矩阵理论框架下,引入了模糊有限自动机转移矩阵,变换矩阵半群以及覆盖概念.定义了模糊有限自动机Kronecker积,讨论了其转移矩阵性质及变换矩阵半群间的覆盖关系.     

Equitable colorings of Kronecker products of graphs

For a positive integer k, a graph G is equitably k-colorable if there is a mapping f:V(G)→{1,2,…,k} such that f(x)≠f(y) whenever xy∈E(G) and ||f⁻¹(i)|−|f⁻¹(j)||≤1 for 1≤i<j≤k. The equitable chromatic number of a graph G, denoted by χ₌(G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by , is the minimum t such that G is equitably k-colorable for k≥t. The current paper studies equitable chromatic numbers of Kronecker products of graphs. In particular, we give exact values or upper bounds on χ₌(G×H) and when G and H are complete graphs, bipartite graphs, paths or cycles.     

Spectral properties of sums of certain Kronecker products

We introduce two kinds of sums of Kronecker products, and their induced operators. We study the algebraic properties of these two kinds of matrices and their associated operators; the properties include their eigenvalues, their eigenvectors, and the relationships between their spectral radii or spectral abscissae. Furthermore, two projected matrices of these Kronecker products and their induced operators are also studied.     

Exponential functions of Kronecker products and trace calculation

An identity for the trace of an exponential function of Kronecker products of matrices is derived. This identity is important for the calculation of the grand potential for Fermi systems.     

Spline approximation,Kronecker products and multilinear forms

Sums of Kronecker products occur naturally in high‐dimensional spline approximation problems, which arise, for example, in the numerical treatment of chemical reactions. In full matrix form, the resulting non‐sparse linear problems usually exceed the memory capacity of workstations. We present methods for the manipulation and numerical handling of Kronecker products in factorized form. Moreover, we analyze the problem of approximating a given matrix by sums of Kronecker products by making use of the equivalence to the problem of decomposing multilinear forms into sums of one‐forms. Greedy algorithms based on the maximization of multilinear forms over a torus are used to obtain such (finite and infinite) decompositions that can be used to solve the approximation problem. Moreover, we present numerical considerations for these algorithms. Copyright © 2016 John Wiley & Sons, Ltd.     

Characterizations of sums of dyads and of Kronecker products

This research was partially supported by Air Force Office of Scientific Research Grant 75-2858.     

Trace cocharacters and the Kronecker products of Schur functions

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M_r(F) of r×r matrices over a field F of characteristic zero are given by TC_r,n=∑_λΛr(n)χ^λχ^λ where χ^λχ^λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ_r(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ₁λ_k) with kr. We study the behavior of the sequence of trace cocharacters TC_r,n. In particular, we study the behavior of the coefficient of χ^(ν,n−m) in TC_r,n as a function of n where ν=(ν₁ν_k) is some fixed partition of m and n−mν_k. Our main result shows that such coefficients always grow as a polynomial in n of degree r−1.     

Some rank equalities and inequalities for Kronecker products of matrices

A set of rank equalities and inequalities are established for block matrices consisting of Kronecker products. Various consequences are also given.     

Some properties of Hadamard matrices generated recursively by Kronecker products

A natural Hadamard matrix H_n is a 2ⁿ × 2ⁿ matrix defined recursively as

$\text{H}{}_{\mathrm{n+1}}\text{=}\text{1}\text{1}\text{1}\text{-1}\text{? H}{}_{\mathrm{n}}$

, where ? denotes the Kronecker product. Some properties of such matrices which follow from the above definition are shown in this paper. These include a certain relation between defined reduction and expansion properties, parity considerations in the Hadamard domain and some dyadic properties.     

Some rank equalities and inequalities for Kronecker products of matrices

A set of rank equalities and inequalities are established for block matrices consisting of Kronecker products. Various consequences are also given.     

On Butler's Unimodality Result

denote the number of subgroups of order in a finite abelian p-group of type λ. Then is a polynomial in p with nonnegative coefficients, which depends only on λ and i. Butler proved that where has nonnegative coefficients. We prove this fact by using formulas shown by Stehling. Received: August 5, 1997     

Unimodality and Lie Superalgebras

It is well-known how the representation theory of the Lie algebra sl(2, ?) can be used to prove that certain sequences of integers are unimodal and that certain posets have the Sperner property. Here an analogous theory is developed for the Lie superalgebra osp(1,2). We obtain new classes of unimodal sequences (described in terms of cycle index polynomials) and a new class of posets (the “super analogue” of the lattice L(m,n) of Young diagrams contained in an m × n rectangle) which have the Sperner property.