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1. Akbari, Saieed

et al.

Friedland, Shmuel

Markström, Klas

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Zare, Sanaz

On 1-sum flows in undirected graphs2016In: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 31, p. 646-665Article in journal (Refereed)

Abstract [en]

Let G = (V, E) be a simple undirected graph. For a given set L subset of R, a function omega: E -> L is called an L-flow. Given a vector gamma is an element of R-V , omega is a gamma-L-flow if for each v is an element of V, the sum of the values on the edges incident to v is gamma(v). If gamma(v) = c, for all v is an element of V, then the gamma-L-flow is called a c-sum L-flow. In this paper, the existence of gamma-L-flows for various choices of sets L of real numbers is studied, with an emphasis on 1-sum flows. Let L be a subset of real numbers containing 0 and denote L* := L \ {0}. Answering a question from [S. Akbari, M. Kano, and S. Zare. A generalization of 0-sum flows in graphs. Linear Algebra Appl., 438:3629-3634, 2013.], the bipartite graphs which admit a 1-sum R* -flow or a 1-sum Z* -flow are characterized. It is also shown that every k-regular graph, with k either odd or congruent to 2 modulo 4, admits a 1-sum {-1, 0, 1}-flow.

Let F be a field of characteristic different from 2. It is shown that the problems of classifying

(i) local commutative associative algebras over F with zero cube radical,

(ii) Lie algebras over F with central commutator subalgebra of dimension 3, and

(iii) finite p-groups of exponent p with central commutator subgroup of order are hopeless since each of them contains

• the problem of classifying symmetric bilinear mappings UxU → V , or

• the problem of classifying skew-symmetric bilinear mappings UxU → V ,

in which U and V are vector spaces over F (consisting of p elements for p-groups (iii)) and V is 3-dimensional. The latter two problems are hopeless since they are wild; i.e., each of them contains the problem of classifying pairs of matrices over F up to similarity.

3.

Dmytryshyn, Andrii

et al.

Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).

Kågström, Bo

Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).

The set of all solutions to the homogeneous system of matrix equations (X-T A + AX, X-T B + BX) = (0, 0), where (A, B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A, B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. Kagstrom, and V. V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375-3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.

4.

Kågström, Bo

et al.

Umeå University, Faculty of Science and Technology, Department of Computing Science.

Karlsson, Lars

Umeå University, Faculty of Science and Technology, Department of Computing Science.

A generalized matrix product can be formally written as Lambda(sp)(p) Lambda(sp-1)(p-1) ... Lambda(s2)(2) Lambda(s1)(1) where s(i) is an element of {- 1,+ 1} and ( A(1), ..., A(p)) is a tuple of ( possibly rectangular) matrices of suitable dimensions. The periodic eigenvalue problem related to such a product represents a nontrivial extension of generalized eigenvalue and singular value problems. While the classification of generalized matrix products under eigenvalue-preserving similarity transformations and the corresponding canonical forms have been known since the 1970's, finding generic canonical forms has remained an open problem. In this paper, we aim at such generic forms by computing the codimension of the orbit generated by all similarity transformations of a given generalized matrix product. This can be reduced to computing the so called cointeractions between two different blocks in the canonical form. A number of techniques are applied to keep the number of possibilities for different types of cointeractions limited. Nevertheless, the matter remains highly technical; we therefore also provide a computer program for finding the codimension of a canonical form, based on the formulas developed in this paper. A few examples illustrate how our results can be used to determine the generic canonical form of least codimension. Moreover, we describe an algorithm and provide software for extracting the generically regular part of a generalized matrix product.