查看更多>>摘要：In this note we discuss a unified approach to the unconditional versus absolute summability of sequences by means of combinatorial properties of Hadamard matrices. In particular we derive a constructive proof of the classical Macphail's Theorem on the existence of series in l(1) that converge unconditionally but not absolutely. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：For two integers k >= 0 and q >= 1, consider symmetric matrices Mwith knegative eigenvalues counted with multiplicities and qpairwise distinct values of entries such that the rows of Mare mutually distinct and the largest diagonal entry of Mis less than or equal to the smallest off-diagonal entry of M. It is shown that the number of such matrices is finite when kand qare fixed. This generalizes some known results on the adjacency matrices of graphs. It is conjectured that any twinfree graph on nvertices with no isolated vertices has at least -1 + log(2)( n + 2) negative adjacency eigenvalues. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：Let K[HK Theta] denote the Hecke-Kiselman algebra of a finite oriented graph Theta over an algebraically closed field K. All irreducible representations, and the corresponding maximal ideals of K[HK Theta], are characterized in case this algebra satisfies a polynomial identity. The latter condition corresponds to a simple condition that can be expressed in terms of the graph Theta. The result shows a surprising similarity to the classical results on representations of finite semigroups; namely every representation either comes form an idempotent in the Hecke-Kiselman monoid HK Theta(and hence it is 1-dimensional), or it comes from certain semigroup of matrix type. The case when Theta is an oriented cycle plays a crucial role; the prime spectrum of K[HK Theta] is completely characterized in this case. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：The edge-connectivity matrix of a weighted graph is the matrix whose off-diagonal v-w entry is the weight of a minimum edge cut separating verticesvandw. Its computation is a classical topic of combinatorial optimization since at least the seminal work of Gomory and Hu. In this article, we investigate spectral properties of these matrices. In particular, we provide tight bounds on the smallest eigenvalue and the energy. Moreover, we study the eigenvector structure and show in which cases eigenvectors can be easily obtained from matrix entries. These results in turn rely on a new characterization of those nonnegative matrices that can actually occur as edgeconnectivity matrices. (C) 2022 The Authors. Published by Elsevier Inc.
查看更多>>摘要：In this paper we study finite support convolutional codes over Z(pr) by means of an input-state-output representation. We show that the set of finite weight input-state-output trajectories associated to this type of representations has the structure of a Z(pr)-submodule of Z(pr)(n) and therefore is a (finite support) convolutional code. Fundamental system-theoretical properties such as observability, reachability or minimality, are investigated in this context. (C) 2021 Published by Elsevier Inc.
查看更多>>摘要：Let R be a ring and let (a(1), . . . , a(n)). Rnbe a unimodular vector, where n >= 2 and each a(i) is in the center of R. Consider the linear equation a(1)X(1)+ . . . + a(n)X(n) = 0, with solution set S. Then S = S-1+ . . . + S-n, where each S-i is naturally derived from (a(1), . . . , a(n)), and we give a presentation of Sin terms of generators taken from the S-i and appropriate relations. Moreover, under suitable assumptions, we elucidate the structure of each quotient module S/S-i. Furthermore, assuming that R is a principal ideal domain, we provide a simple way to construct a basis of Sand, as an application, we determine the structure of the quotient module S/U-i, where each U-i is a specific module containing S-i. (C) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：Free spectrahedra are natural objects in the theories of operator systems and spaces and completely positive maps. They also appear in various engineering applications. In this paper, free spectrahedra satisfying a Reinhardt symmetry condition are characterized graph theoretically. It is also shown that, for a simple class of such spectrahedra, automorphisms are trivial. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：In this paper, we consider structure-preserving eigenvalue embedding problem (SEEP) for quadratic regular matrix polynomials with symmetry structures. First, we determine perturbations of a quadratic matrix polynomial, unstructured or structured, such that the perturbed polynomials reproduce a desired invariant pair while maintaining the invariance of another invariant pair of the unperturbed polynomial. If the latter is unknown, it is referred to as no spillover perturbation. Then we use these results for solving the SEEP for structured quadratic matrix polynomials that include: symmetric, Hermitian, *-even and *-odd quadratic matrix polynomials. Finally, we show that the obtained analytical expressions of perturbations can realize existing results for structured polynomials that arise in real-world applications, as special cases. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：A set grading on the split simple Lie algebra of type D-13, that cannot be realized as a group-grading, is constructed by splitting the set of positive roots into a disjoint union of pairs of orthogonal roots, following a pattern provided by the lines of the projective plane over GF(3). This answers in the negative [3, Question 1.11]. Similar non-group gradings are obtained for types D-n with n = 1(mod12), by substituting the lines in the projective plane by blocks of suitable Steiner systems. (C) 2022 The Author(s). Published by Elsevier Inc.
查看更多>>摘要：Let S-n,S-2 be the graph obtained by joining each vertex of K-2 to n - 2 isolated vertices, and let S-n,2(-) be the graph obtained from S-n,S-2 by deleting an edge incident to a vertex of degree two. Recently, Zhai, Lin and Shu [20] showed that rho(G) <= 1+root 4m-3/2 for any C-5-free graph of size m >= 8 or C-6-free graph of size m >= 22, with equality if and only if G congruent to S-m+3/2,S-2 (possibly, with some isolated vertices). However, this bound is sharp only for odd m. Motivated by this, we want to obtain a sharp upper bound of rho(G) for C-5-free or C-6-free graphs with medges. In this paper, we prove that if Gis a C-5-free graph of even size m >= 14 or C-6-free graph of even size m >= 74, and G contains no isolated vertices, then rho(G) <= (rho) over tilde (m), with equality if and only if G congruent to S-m+4/2,2(-), where (rho) over tilde (m) is the largest root of x(4) - mx(2) - (m - 2)x + ( m/2 - 1) = 0. (c) 2022 Elsevier Inc. All rights reserved.
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