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[第201516号]_李忠善教授_美国乔治亚州立大学
2015-07-19 08:09   审核人:

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2015619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                      

人:李忠善教授Prof. Li Zhongshan                                  

报告人单位:美国乔治亚州立大学                      

   间:2014619日(周五)下午1530                      

   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                      

                     

欢迎各位老师同学届时参加!

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