Let S be a family of subsets of a set X of cardinality m and VC-dim(S) be the Vapnik-Chervonenkis dimension of S. Haussler, Littlestone, and Warmuth (Inf. Comput., 1994) proved that if G1(S) = (V,E) is the subgraph of the hypercube Qm induced by S (called the 1-inclusion graph of S), then |E| |V | ≤ VC-dim(S). Haussler (J. Combin. Th. A, 1995) presented an elegant proof of this inequality using...

A matrix M is nilpotent of index 2 if M = 0. Let V be a space of nilpotent n × n matrices of index 2 over a field k where card k > n and suppose that r is the maximum rank of any matrix in V . The object of this paper is to give an elementary proof of the fact that dim V ≤ r(n−r). We show that the inequality is sharp and construct all such subspaces of maximum dimension. We use the result to fi...

A systematic analysis is made of the character of the free and free abelian topological groups on uniform spaces and on topological spaces. In the case of the free abelian topological group on a uniform space, expressions are given for the character in terms of simple cardinal invariants of the family of uniformly continuous pseudometrics of the given uniform space and of the uniformity itself....

This paper examines the social dimension of collective storytelling around a particular ludic object. We study the action role-playing game Skyrim (2011) through a mixed methodology that integrates the humanities and the social sciences. Based on extensive analysis of player generated videos, we demonstrate that the social dimension plays an all-important role in the shaping of storytelling by ...

background: although hemodialysis has a therapeutic effect on esrd, but these patients encounter many physical, psychological, and social stress full factors that lead to a decrease in their quality of life (qol). one of the factors that are effective on increasing of qol is self care ability. review of literature demonstrates that there have been done a few studies on different aspects of qol ...