Matrix periods and competition periods of Boolean Toeplitz matrices II

IF 1 3区 数学 Q1 MATHEMATICS
Gi-Sang Cheon , Bumtle Kang , Suh-Ryung Kim , Homoon Ryu
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引用次数: 0

Abstract

This paper is a follow-up to the paper of Cheon et al. (2023) [2]. Given subsets S and T of {1,,n1}, an n×n Toeplitz matrix A=TnS;T is defined to have 1 as the (i,j)-entry if and only if jiS or ijT. In the previous paper, we have shown that the matrix period and the competition period of Toeplitz matrices A=TnS;T satisfying the condition (⋆) maxS+minTn and minS+maxTn are d+/d and 1, respectively, where d+=gcd(s+t|sS,tT) and d=gcd(d,minS). In this paper, we claim that even if (⋆) is relaxed to the existence of elements sS and tT satisfying s+tn and gcd(s,t)=1, the same result holds. There are infinitely many Toeplitz matrices that do not satisfy (⋆) but the relaxed condition. For example, for any positive integers k,n with 2k+1n, it is easy to see that Tnk,nk;k+1,nk1 does not satisfy (⋆) but satisfies the relaxed condition. Furthermore, we show that the limit of the matrix sequence {Am(AT)m}m=1 is Tnd+,2d+,,n/d+d+.

布尔托普利兹矩阵的矩阵周期和竞争周期 II
本文是 Cheon 等人 (2023) [2] 论文的后续。给定{1,...,n-1}的子集 S 和 T,n×n 托普利兹矩阵 A=Tn〈S;T〉的定义是,当且仅当 j-i∈S 或 i-j∈T 时,(i,j)项为 1。在前一篇论文中,我们已经证明了满足条件(⋆)maxS+minT≤n 和 minS+maxT≤n 的托普利兹矩阵 A=Tn〈S;T〉的矩阵周期和竞争周期分别为 d+/d 和 1,其中 d+=gcd(s+t|s∈S,t∈T) 和 d=gcd(d,minS)。在本文中,我们声称,即使将 (⋆) 放宽到存在满足 s+t≤n 且 gcd(s,t)=1 的元素 s∈S 和 t∈T ,结果也同样成立。有无限多的托普利兹矩阵不满足 (⋆) 但满足放宽条件。例如,对于任何 2k+1≤n 的正整数 k、n,很容易看出 Tn〈k,n-k;k+1,n-k-1〉不满足 (⋆),但满足松弛条件。此外,我们还证明矩阵序列 {Am(AT)m}m=1∞ 的极限是 Tn〈d+,2d+,...,⌊n/d+⌋d+〉。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.20
自引率
9.10%
发文量
333
审稿时长
13.8 months
期刊介绍: Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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