论文标题
布尔值的矩阵期间和竞争期
Matrix periods and competition periods of Boolean Toeplitz matrices
论文作者
论文摘要
在本文中，我们研究了矩阵时期和二进制布尔环矩阵的竞争期限$ \ mathbb {b} = \ {0,1 \} $。给定的子集$ s $和$ t $的$ \ {1，\ ldots，n-1 \} $，a $ n \ times n $ toeplitz matrix $ a = t_n \ langle s; t \ rangle $定义为$ 1 $作为$（i，j）$ - 当时且仅在s $ in s $ in s $ in t $中的$ j-i \ in t $。我们表明，如果$ \ max s+\ min t \ le n $和$ \ min s+\ max t \ le n $，则$ a $具有矩阵周期$ d/d $，并且竞争期$ 1 $ $ d = \ gcd（s+t \ gcd（s+t \ t \ t \ in mid s s，in s in s，in t in t）$ and $ d'= \ gccd（$ d'= \ gcd（d）此外，结果表明，矩阵序列$ \ {a^m（a^t）^m \} _ {m = 1}^\ infty $的极限是除零角性零以外的所有矩阵的有向矩阵总和。在许多文献中，我们看到图理论方法可用于证明矩阵的强结构特性。同样，我们从图理论的角度发展了我们的工作。
In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring $\mathbb{B} = \{0,1\}$. Given subsets $S$ and $T$ of $\{1,\ldots,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n\langle S ; T \rangle$ is defined to have $1$ as the $(i,j)$-entry if and only if $j-i \in S$ or $i-j \in T$. We show that if $\max S+\min T \le n$ and $\min S+\max T \le n$, then $A$ has the matrix period $d/d'$ and the competition period $1$ where $d = \gcd (s+t \mid s \in S, t \in T)$ and $d' = \gcd(d, \min S)$. Moreover, it is shown that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view.
