{"id":247,"date":"2014-11-06T20:07:32","date_gmt":"2014-11-06T20:07:32","guid":{"rendered":"http:\/\/pages.charlotte.edu\/colloquium\/?p=247"},"modified":"2014-11-06T20:07:32","modified_gmt":"2014-11-06T20:07:32","slug":"friday-november-21-2014-at-300pm-in-the-fret-106","status":"publish","type":"post","link":"https:\/\/pages.charlotte.edu\/colloquium\/blog\/2014\/11\/06\/friday-november-21-2014-at-300pm-in-the-fret-106\/","title":{"rendered":"Friday, November 21, 2014 at 3:00pm, in the FRET 106"},"content":{"rendered":"<table>\n<tbody>\n<tr>\n<td>\n<div>\n<div><a href=\"http:\/\/www2.gsu.edu\/~matzli\/\"><span lang=\"en-US\"><span style=\"font-family: Times New Roman,serif;font-size: medium\">Zhongshan\u00a0 Li<\/span><\/span><\/a>, Georgia State University, Atlanta, GA<\/div>\n<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td>Title:\u00a0 <span dir=\"ltr\"><span lang=\"en-US\"><span style=\"font-family: Times New Roman,serif;font-size: medium\">Sign vectors of subspaces of R^n and minimum ranks of sign patterns<\/span><\/span><br \/>\n<\/span><\/td>\n<\/tr>\n<tr>\n<td>\n<div>Abstract: 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 smallest possible rank of a real matrix whose entries have signs indicated by A. A direct connection between an m by n sign pattern with minimum rank r&gt;=2 and an m point&#8211;n hyperplane configuration in R^{r-1} is established. A possibly smallest example of a sign pattern (with minimum rank 3) whose minimum rank cannot be realized rationally is given. It is shown that for every sign pattern with at most 2 zero entries in each column, the minimum rank can be realized rationally. Using a new approach involving sign vectors of subspaces and oriented matroid duality, it is shown that for every m by n sign pattern with minimum rank &gt;= n-2, rational realization of the minimum rank is possible. It is also shown that for every integer n&gt;=9, there is a positive integer m, such that there exists an m by n sign pattern with minimum rank n-3 for which rational realization is not possible. A characterization of m by n sign patterns A with minimum rank n-1 is given, 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 R^n are discussed; this maximum number is equal to the total number of cells of a generic central hyperplane arrangement in R^k. In particular, it is shown that the maximum number of sign vectors of a 2-dimensional subspace of R^n is 4n+1 and the maximum number of sign vectors of a 3-dimensional subspace of R^n is 4n(n &#8211; 1) + 3. Related results and open problems are stated along the way.<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Zhongshan\u00a0 Li, Georgia State University, Atlanta, GA Title:\u00a0 Sign vectors of subspaces of R^n and minimum ranks of sign patterns Abstract: 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 smallest possible rank of a real matrix [&hellip;]<\/p>\n","protected":false},"author":16,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[1],"tags":[],"class_list":["post-247","post","type-post","status-publish","format-standard","hentry","category-spring-2022"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p3kCtT-3Z","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/pages.charlotte.edu\/colloquium\/wp-json\/wp\/v2\/posts\/247","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pages.charlotte.edu\/colloquium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pages.charlotte.edu\/colloquium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pages.charlotte.edu\/colloquium\/wp-json\/wp\/v2\/users\/16"}],"replies":[{"embeddable":true,"href":"https:\/\/pages.charlotte.edu\/colloquium\/wp-json\/wp\/v2\/comments?post=247"}],"version-history":[{"count":1,"href":"https:\/\/pages.charlotte.edu\/colloquium\/wp-json\/wp\/v2\/posts\/247\/revisions"}],"predecessor-version":[{"id":248,"href":"https:\/\/pages.charlotte.edu\/colloquium\/wp-json\/wp\/v2\/posts\/247\/revisions\/248"}],"wp:attachment":[{"href":"https:\/\/pages.charlotte.edu\/colloquium\/wp-json\/wp\/v2\/media?parent=247"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pages.charlotte.edu\/colloquium\/wp-json\/wp\/v2\/categories?post=247"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pages.charlotte.edu\/colloquium\/wp-json\/wp\/v2\/tags?post=247"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}