We consider m × n random binary matrices, m = m ( n ) . For arbitrary odd integer k we investigate the asymptotic distribution of the random number of sub-matrices of size k × n for which the number of ones in every column satisfies the majority rule. We discuss possible impacts of our result and give examples of applications.

The interchangeability property of Nash equilibria in two-player zero-sum games is well known. This article studies possible generalizations of this property to multi-player zero-sum games.Aform of interchangeability property for strict Nash equilibria in such games is established. It is also shown, by proving a completeness theorem, that strict Nash equilibria do not satisfy any other non-trivial properties.

In this paper we settle an open combinatorial conjecture in artificial neural networks: we show that the bound on the number of dichotomies given by Mitchinson and Durbin [Biological Cybernetics, 60 (1989), pp. 345--365] is tight, and that their structural asymptotics remain unchanged with varying required success probability. In our proof we use Rényi's graph-sieves inequalities, and we derive a contracted version and a sharp bound for a triple-indexed sum of binomial coefficients with dependent indices.