NCTS(South)/ NCKU Math Colloquium | |
| DATE | 2010-10-15¡@16:10-17:00 |
| PLACE | R204, 2F, NCTS, NCKU |
| SPEAKER | Bit-Shun Tam (ÃÓ¥²«H)¡]Tamkang university¡^ |
| TITLE | Maximal exponents of polyhedral cones |
| ABSTRACT |
Let K be a proper (i.e., closed, pointed, full convex) cone in Rn. An n £ n matrix A is said to be K-primitive if AK µ K and there exists a positive integer k such that Ak(Knf0g) µ intK; the least such k is referred to as the exponent of A and is denoted by ¢X(A). For a polyhedral cone K, the maximum value of ¢X(A), taken over all K-primitive matrices A, is denoted by ¢X(K). It is proved that for any positive integers m; n; 3 ¡P n ¡P m, the maximum value of ¢X(K), as K runs through all n-dimensional polyhedral cones with m extreme rays, equals (n ¡ 1)(m ¡ 1) + 1 2 ¡ 1 + (¡1)(n¡1)m ¢ . For the cases when m = n or n = 3, the cones K and the corresponding K-primitive matrices A such that ¢X(K) and ¢X(A) attain the maximum value are identified up to respectively linear isomorphism and cone-equivalence modulo positive scalar multiplication. |