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Seminar第1909期 凸多面体和非负符号矩阵的最小秩

创建日期 7/10/2019 福平   浏览次数  119 返回    
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报告主题:凸多面体和非负符号矩阵的最小秩
报告人:Zhongshan Li 教授 (Georgia State University)
报告时间:2019年7月26日(周五)16:00
报告地点:校本部G507
邀请人:谭福平

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

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