查看更多>>摘要：In this paper, the well-known Faulkner construction is revisited and adapted to include the super case, which gives a bijective correspondence between generalized Jordan (super)pairs and faithful Lie (super)algebra (super)modules, under certain constraints (bilinear forms with properties analogous to the ones of a Killing form are required, and only finite-dimensional objects are considered). We always assume that the base field has characteristic different from 2. It is also proven that associated objects in this Faulkner correspondence have isomorphic automorphism group schemes. Finally, this correspondence will be used to transfer the construction of the tensor product to the class of generalized Jordan (super)pairs with "good" bilinear forms.
查看更多>>摘要：We study the inverse problem of recovering a tree graph together with the weights on its edges (equivalently a metric tree) from the knowledge of the Dirichlet-to-Neumann matrix associated with the Laplacian. We prove an explicit formula which relates this matrix to the pairwise weighted distances of the leaves of the tree and, thus, allows to recover the weighted tree. This result can be viewed as a counterpart of the Calderon problem in the analysis of PDEs. In contrast to earlier results on inverse problems for metric graphs, we only assume knowledge of the Dirichlet-to-Neumann matrix for a fixed energy, not of a whole matrix-valued function. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：Instability in Jacobians is determined by the presence of an eigenvalue lying in the right half plane. The coefficients of the characteristic polynomial contain information related to the specific matrix elements that play a greater destabilising role. Yet the destabilising circuits, or cycles, constructed by multiplying these elements together, form only a subset of all the cycles comprising a given system. This paper looks at the destabilising cycles in three sign-restricted forms in terms of sets of the matrix elements to explore how sign structure affects how the elements contribute to instability. This leads to quite rich combinatorial structure among the destabilising cycle sets as set size grows within the coefficients of the characteristic polynomial. (c) 2022 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：We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determined by at most three parameters. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license
查看更多>>摘要：We prove that for all integers k >= 1, there exists a constant Ck depending only on k, such that for all q > C-k, and for n = 1, 2 every matrix in Mn(F-q) is a sum of two kth powers and for all n >= 3 every matrix in Mn(F-q) is a sum of at most three kth powers. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：For two n x n complex matrices A and B, we define the q-deformed commutator as [A, B](q) := AB - qBA for a real parameter q. In this paper, we investigate a generalization of the Bottcher-Wenzel inequality which gives the sharp upper bound of the (Frobenius) norm of the commutator. In our generalisation, we investigate sharp upper bounds on the q-deformed commutator. This generalization can be studied in two different scenarios: firstly bounds for general matrices, and secondly for traceless matrices. For both scenarios, partial answers and conjectures are given for positive and negative q. In particular, denoting the Frobenius norm by parallel to.parallel to(F), when A or B is normal, we prove the following inequality to be true and sharp: parallel to[A, B](q) parallel to(2)(F) <= (1 + q(2))parallel to A parallel to(2)(F) parallel to B parallel to(2)(F) for positive q. Also, we conjecture that the same bound is true for positive q when A or B is traceless. For negative q, we conjecture other sharp upper bounds to be true for the generic scenarios and the scenario when A or B is traceless. All conjectures are supported with numerics and proved for n = 2. (c) 2022 The Authors. 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：We prove that the inner products of spherical s-distance t designs with t >= 2s - 2 (Delsarte codes) and s >= 3 are rational with the only exception being the icosahedron. In other formulations, we prove that all sharp configurations have rational inner products and all spherical codes which attain the Levenshtein bound, have rational inner products, except for the icosahedron. (C) 2022 Elsevier Inc. All rights reserved.
Mai Hoang BienTruong Huu DungNguyen Thi Thai HaTran Nam Son...
13页
查看更多>>摘要：Let D be a division algebra of degree m. The first aim of this paper is to show that if D is tame and totally ramifield and if the center of D is Henselian, then there exists a positive d depending on m such that every element in the commutator subgroup D? of the unit group D* = D \ {0} is a product at most d commutators, which answers a problem of P. Draxl ([5], Problem 1, Page 102) for tame and totally ramifield division algebras whose centers are Henselian. The second goal is to prove that if D is infinite and every element in D? is a product at most alpha commutators in D*, then every matrix in the special linear group SLn(D) of degree n > 1 is a product of at most 2 + 6 alpha commutators of involutions. (C) 2022 Elsevier Inc. All rights reserved.
查看更多>>摘要：As is well known, any complex cyclic matrix A is similar to the unique companion matrix associated with the minimal polynomial of A. On the other hand, a cyclic matrix over a division ring F is similar to a companion matrix of a polynomial which is defined up to polynomial similarity. In this paper we study more rigid canonical forms by embedding a given cyclic matrix over a division ring F into a controllable or an observable pair. Using the characterization of ideals in F [z] in terms of controllable and observable pairs we consider ideal interpolation schemes in F [z] which merge into polynomial interpolation problems containing both left and right interpolation conditions.(c) 2022 Published by Elsevier Inc.
查看更多>>摘要：This paper considers feedback methods for ensemble reachability of parameter-dependent linear systems (A(theta), B(theta)), where the parameter theta is varying over a compact Jordan arc in the complex plane. Recently, pointwise testable sufficient conditions for uniform ensemble reachability have been developed. Beside the necessity of pointwise reachability these conditions put restrictions on the spectra of the matrices A(theta) and the Hermite indices of the pair (A(theta), B(theta)). In this paper we show that these conditions can be ensured by applying a suitable feedback transformation if the pair (A(theta), B(theta)) is pointwise reachable and its Kronecker indices are independent of the parameter. (c) 2022 Elsevier Inc. All rights reserved.
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