应我校数学科学学院邀请，美国乔治亚州立大学的李忠善教授于2015年6月19日来我校进行短期访问讲学。欢迎数学科学学院及全校相关教师、博士生、硕士生参加!
报告题目：Sign vectors of subspaces and minimum ranks of sign patterns
报告人：李忠善教授（Prof. Li Zhongshan）
报告人单位：美国乔治亚州立大学
时 间：2014年6月19日（周五）下午15：30
地 点：8号楼七楼会议室
李忠善教授简介：1983年毕业于兰州大学数学系，1990年获得美国北卡罗来纳州立大学博士学位。自1991年起在美国乔治亚州立大学数学与统计系任教，1998年评为美国乔治亚州立大学副教授，2007年评为美国乔治亚州立大学终身教授。主要从事组合矩阵论方面的研究。在《Linear Algebra and Its Applications》、《SIAM Journal on Discrete Mathematics》等国际高水平学术期刊上发表论文四十余篇。担任国际期刊《JP Journal of Algebra, Number Theory and Applications》编委，曾受邀参与撰写《Handbook of Linear Algebra》，入选山西省百人计划。
李忠善个人主页：www2.gsu.edu/~matzli
报告摘要：A sign pattern matrix is a matrix whose entries are from the set $\{+,-, 0\}$. The minimum rank of a sign pattern matrix $A$ is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of $A$. It is shown in this talk that for any $m \times n$ sign pattern $A$ with minimum rank $n-2$, rational realization of the minimum rank is possible. This is done using a new approach involving sign vectors and duality. It is also shown that for each integer $n\geq 9$, there exists a nonnegative integer $m$ such that there exists an $n\times m$ sign pattern matrix with minimum rank $n-3$ for which rational realization is not possible. A characterization of $m\times n$ sign patterns $A$ with minimum rank $n-1$ is given (which solves an open problem posed by Brualdi et al.), along with a more general description of sign patterns with minimum rank $r$, in terms of sign vectors of certain subspaces. A number of results on the maximum and minimum numbers of sign vectors of $k$-dimensional subspaces of $\mathbb R^n$ are obtained. In particular, it is shown that the maximum number of sign vectors of a $2$-dimensional subspace of $\mathbb R^n$ is $4n+1$ and the maximum number of sign vectors of a $3$-dimensional subspace of $\mathbb R^n$ is $4n(n - 1) + 3$.
Several related open problems are stated along the way
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