查看更多>>摘要：In this work, linearized multivariate skew polynomials over division rings are introduced. Such polynomials are right linear over the corresponding centralizer and generalize linearized polynomial rings over finite fields, group rings or differential polynomial rings. Their natural evaluation is connected to the remainder-based evaluation of free multivariate skew polynomials. It is shown that P-independent sets are those given by right linearly independent sets when partitioned into conjugacy classes. Hence finitely generated P-closed sets correspond to lists of finite-dimensional right vector spaces, extending Lam and Leroy's results on univariate skew polynomials. It is also shown that products of free multivariate skew polynomials translate into coordinate-wise compositions of linearized multivariate skew polynomials, which in turn translate into matrix products over the corresponding centralizers. Later, linearized multivariate Vandermonde matrices are introduced, which generalize multivariate Vandermonde, Moore and Wronskian matrices. The previous results explicitly give their ranks in general. PGalois extensions of division rings are then introduced, which generalize classical (finite) Galois extensions. Three Galoistheoretic results are generalized to such extensions: Artin's theorem on extension degrees, the Galois correspondence and Hilbert's Theorem 90.(c) 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
查看更多>>摘要：For a graph G on the vertex set {0, 1, ..., n}, the G -parking function ideal M-G is a monomial ideal in the polynomial ring R = K[x(1), ..., x(n)] such that the vector space dimension of R/M-G is given by the determinant of its reduced Laplacian. For any integer k, the k -skeleton ideal M-G((k)) is the subideal of M-G, where the monomial generators correspond to nonempty subsets of [n] of size at most k + 1. For a simple graph G, Dochtermann conjectured that the vector space dimension of R/M-G((1)) is bounded below by the determinant of the reduced signless Laplacian. We show that the Dochtermann conjecture holds for any (multi) graph G. More generally, we prove that this bound holds for ideals ,J(H) defined by a larger class of symmetric positive semidefinite n x n matrices H. (c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：For a fixed integer n >= 2, denote by M-n the algebra of all n x n complex matrices. Consider also an integer k between 1 and n -1. For k >= 2, we characterize linear maps on M-n compressing the polynomial numerical hull of order k of each matrix T is an element of M-n. We obtain the same type of characterization when k = 1 with an extra assumption on the preserving map. (c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：We study the faces of the convex polytope of all n x n doubly substochastic matrices, denoted by omega(n). We give the necessary and sufficient conditions of a face being nonempty. We also describe all 1-dimensional faces, 2-dimensional faces, and facets of omega(n). Moreover, we explore the relation between the faces of omega(n) and the faces of Omega(n), the convex polytope of all n x n doubly stochastic matrices. (c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：Let K be a field and P a partially ordered set (poset). Let F I(P, K) and I(P, K) be the finitary incidence algebra and the incidence space of P over K, respectively, and let D(P, K) = F I(P, K)circle plus I(P, K) be the idealization of the F I(P, K)-bimodule I(P, K). In the first part of this paper, we show that D(P, K) has an anti-automorphism (involution) if and only if P has an anti-automorphism (involution). We also present a characterization of the anti-automorphisms and involutions on D(P, K). In the second part, we obtain the classification of involutions on D(P, K) to the case when char K (sic) 2 and P is a connected poset such that every multiplicative automorphism of F I(P, K) is inner and every derivation from F I(P, K) to I(P, K) is inner (in particular, when P has an element that is comparable with all its elements). (c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：In further pursuit of a solution to the celebrated nonnegative inverse eigenvalue problem, Loewy and London ((1978/1979) [8]) posed the problem of characterizing all polynomials that preserve all nonnegative matrices of a fixed order. If p(n) denotes the set of all polynomials that preserve all n by -n nonnegative matrices, then it is clear that polynomials with nonnegative coefficients belong to p(n). However, it is known that gin contains polynomials with negative entries. In this work, novel results for p(n) with respect to the coefficients of the polynomials belonging to p(n). Along the way, a generalization for the even-part and odd-part are given and shown to be equivalent to another construction that appeared in the literature. Implications for further research are discussed. (c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：In this paper we give a complete classification of two-step nilpotent Leibniz algebras in terms of Kronecker modules associated with pairs of bilinear forms. Among these, we describe the complex and the real case of the indecomposable Heisenberg Leibniz algebras as a generalization of the classical (2n + 1)-dimensional Heisenberg Lie algebra h(2n+1). Then we use the Leibniz algebras-Lie local racks correspondence proposed by S. Covez to show that nilpotent real Leibniz algebras have always a global integration. As an application, we integrate the indecomposable nilpotent real Leibniz algebras with one-dimensional commutator ideal. Finally we show that every Lie quandle integrating a Leibniz algebra is induced by the conjugation of a Lie group and the Leibniz algebra is the Lie algebra of that Lie group. (c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：For a simple graph G the signless Laplacian matrix of G is defined as D(G) + A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. By the smallest signless Laplacian eigenvalue of G, denoted by q'(G), we mean the smallest eigenvalue of the signless Laplacian matrix of G. In this paper we study the smallest signless Laplacian eigenvalue of graphs and find some relations between this and the chromatic number of graphs. We prove that if G is a graph of order n and with chromatic number chi(G), then q'(G) <= q'(T(n, chi(G))), where T(n, t) is the Turan graph on n vertices and t parts. Using this inequality we obtain some bounds for q'(G) that improve the known previous bounds. (c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：Let the regions circle minus(n) and omega(n) be the subsets of the complex planes that consist of all eigenvalues of all n x n stochastic and doubly stochastic matrices, respectively. Also, let Pi(n) denote the convex hull of the nth roots of unity. Levick, Pereira and Kribs (2015) [10] made the following conjectures on the relations between circle minus(n), omega(n) and Pi(n): omega(n) = circle minus(n-1) boolean OR Pi(n) and circle minus(n-1) subset of omega(n). These two conjectures are known to be true for n = 2, 3, 4. In this paper, we will show that these two conjectures are not true for n >= 5. (c) 2021 Elsevier Inc. All rights reserved.
查看更多>>摘要：The Symmetric Tensor Approximation problem (STA) consists of approximating a symmetric tensor or a homogeneous polynomial by a linear combination of symmetric rank-1 tensors or powers of linear forms of low symmetric rank. We present two new Riemannian Newton-type methods for low rank approximation of symmetric tensor with complex coefficients.The first method uses the parametrization of the set of tensors of rank at most r by weights and unit vectors. Exploiting the properties of the apolar product on homogeneous polynomials combined with efficient tools from complex optimization, we provide an explicit and tractable formulation of the Riemannian gradient and Hessian, leading to Newton iterations with local quadratic convergence. We prove that under some regularity conditions on non-defective tensors in the neighborhood of the initial point, the Newton iteration (completed with a trust-region scheme) is converging to a local minimum. The second method is a Riemannian Gauss-Newton method on the Cartesian product of Veronese manifolds. An explicit orthonormal basis of the tangent space of this Riemannian manifold is described. We deduce the Riemannian gradient and the Gauss-Newton approximation of the Riemannian Hessian. We present a new retraction operator on the Veronese manifold.We analyze the numerical behavior of these methods, with an initial point provided by Simultaneous Matrix Diagonalisation (SMD). Numerical experiments show the good numerical behavior of the two methods in different cases and in comparison with existing state-of-the-art methods.(c) 2021 Elsevier Inc. All rights reserved.
NSTL
Elsevier