学术报告通知

报告题目：Convex polytopes and minimum ranks of   nonnegative signpattern

                    matrices

报告人：李忠善教授  （美国乔治亚州立大学数学与统计系）

报告时间：6月19 日下午3点-4点

报告地点：数统院307      

 数学与统计学院

2019.06.12

报告摘要：A sign pattern matrix (resp., nonnegative sign patternmatrix) is a matrix whose entries are from the set $\{+, -, 0\}$ (resp., $ \{+, 0 \}$). The minimum rank (resp., rational minimum rank) of a  sign patternmatrix $\cal A$ is the minimum of the ranks of the  matrices (resp.,rational matrices) whose entries have signs equal to the corresponding entriesof $\cal A$. Using a correspondence between sign patterns with minimum rank$r\geq 2$ and  point-hyperplane configurations in $\mathbb R^{r-1}$ andSteinitz's theorem on the rational realizability of 3-polytopes, it is shownthat for every nonnegative sign pattern of minimum rank at most 4, the minimumrank and the rational minimum rank are equal. But there are nonnegative signpatterns with minimum rank 5 whose rational minimum rank is greater than5.  It is established that every $d$-polytope determines a nonnegativesign pattern with minimum rank $d+1$ that has a $(d+1)\times (d+1)$ triangular submatrix with all diagonal entries positive.  It is also shownthat  there are at most  $\min \{ 3m, 3n \}$ zero entries inany  condensed nonnegative $m \times n$ sign pattern of minimum rank 3.Some bounds on the entries of some integer matrices achieving the minimum ranksof nonnegative sign patterns with minimum rank 3 or 4 are established

报告人简介：李忠善，美国佐治亚州立大学数学系终身正教授。兰州大学数学学学士、北京师范大学硕士美国北卡罗来纳州立大学博士。研究兴趣包括组合矩阵理论、代数图论、矩阵理论应用等。