Finiteness of a set of non-collinear vectors generated by a family of linear operators

✍ Scribed by R. Mubarakzjanov

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 195 KB
• Volume: 294
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(99)00031-2

✦ Synopsis

Let R n be a real n-dimensional space, let fex j x P g be a family of m j j linear operators in R n , and let u r be a sharp polyhedral cone formed by a set of rvectors, u r & R n X Let u r be invariant under fex j x P g, i.e. u r ex u r , for x P . We study a maximum set of non-collinear vectors derived from a vector h P u r by the family fex j x P g in this paper. It is shown that there is a function f nY mY r such that this set of non-collinear vectors is ®nite i the cardinality of this set is not greater than f nY mY r. This result can be used for solving the following problem: when does a channel simulated by a probabilistic automaton have a ®nite set of states? Ó 1999 Elsevier Science Inc. All rights reserved.