So, 〈(A ∗B)ei, ej〉 = 〈Aei, ej〉 〈Bei, ej〉 . The Schur product theorem states that if A,B ∈ L (C) are positive then their Hadamard product A ∗B is positive. 1John B. Conway, A Course in Functional Analysis, second ed., p. 33, Proposition 2.12. 2John B. Conway, A Course in Functional Analysis, second ed., p. 34, Proposition 2.13. 3Ward Cheney and Will Light, A Course in Approximation Theory, p. 81...

We study certain Schur functors which preserve singularity categories of rings and we apply them to study the singularity category of triangular matrix rings. In particular, combining these results with Buchweitz–Happel’s theorem, we can describe singularity categories of certain non-Gorenstein rings via the stable category of maximal Cohen–Macaulay modules. Three concrete examples of finitedim...

The main result of this paper is the extension of the Schur–Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequences ξ and η that converge to 0, there exists a positive compact operator A with eigenvalue list η and diagonal sequence ξ if and only if ∑n j=1 ξj ∑n j=1 ηj for every n if and only if ξ = Qη for some orthostochastic matrix Q. When ξ and η are summable, requirin...

is irreducible. Irreducibility here and throughout this paper refers to irreducibility over the rationals. Some condition, such as ja0j = janj = 1, on the integers aj is necessary; otherwise, the irreducibility of all polynomials of the form above would imply every polynomial inZ[x] is irreducible (which is clearly not the case). In this paper, we will mainly be interested in relaxing the condi...

We prove a conjecture of Cuttler et al. (2011) on the monotonicity of normalized Schur functions under the usual (dominance) partialorder on partitions. We believe that our proof technique may be helpful in obtaining similar inequalities for other symmetric functions. © 2015 Elsevier Ltd. All rights reserved. We prove a conjecture of Cuttler et al. [1] on themonotonicity of normalized Schur fun...

This paper presents a comprehensive exploration of probabilistic adaptation the Eneström–Kakeya theorem, applied to random polynomials featuring various coefficient distributions. Unlike deterministic rendition our study dispenses with necessity any specific order. Instead, we consider coefficients drawn from spectrum sets diverse probability distributions, encompassing finite, countable, and u...

In this paper we present a comparison of the service disciplines in real-time queueing systems (the customers have a deadline before which they should enter the service booth). We state that the more a service discipline gives priority to customers having an early deadline, the least the average stationary lateness is. We show this result by comparing adequate random vectors with the Schur-Conv...

In this paper, using a relation between Schur multipliers of pairs and triples of groups, the fundamental group and homology groups of a homotopy pushout of Eilenberg-MacLane spaces, we present among other things some behaviors of Schur multipliers of pairs and triples with respect to free, amalgamated free, and direct products and also direct limits of groups with topological approach.

We prove that every infinite sequence of skew-symmetric or symmetric matrices M1, M2, . . . over a fixed finite field must have a pair Mi, Mj (i < j) such that Mi is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in Mj , if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitt...