Volume 20, Issue 3

In this article, a random permutation τ n is considered which is uniformly distributed on the set of all permutations of degree n whose cycle lengths lie in a fixed set A (the so-called A -permutations). It is assumed that the set A has an asymptotic density σ > 0, and | k : k ≤ n , k ∈ A , m – k ∈ A |/ n → σ 2 as n → ∞ uniformly in m ∈ [ n , Cn ] for an arbitrary constant C > 1. The minimum degree of a permutation such that it becomes equal to the identity permutation is called the order of permutation. Let Z n be the order of a random permutation τ n . In this article, it is shown that the random variable ln Z n is asymptotically normal with mean l ( n ) = ∑ k ∈ A ( n ) ln( k )/ k and variance σ ln 3 ( n )/3, where A ( n ) = { k : k ∈ A , k ≤ n }. This result generalises the well-known theorem of P. Erdős and P. Turán where the uniform distribution on the whole symmetric group of permutations S n is considered, i.e., where A is equal to the set of positive integers N .

Dropping the assumption of the stochastic independence of players' randomised choices in non-cooperative games, we introduce the notion of a type of dependence. It is proved that the stochastic independence is the unique type of dependence for which any finite non-cooperative game has a Nash equilibrium in mixed strategies.

We consider matrices of arbitrary sizes (including infinite matrices) over a distributive lattice L and prove that if L = 2 X is a lattice of all subsets of a set X , then the potential divisibility of matrices (from the left or from the right) of one of the matrices by the other matrix is equivalent to the usual divisibility. In particular, in the semigroup of square matrices over the lattice 2 X the Green relation ℒ coincides with the generalised Green relation ℒ*

The problem of learning a function in the context of the exact model of learning using membership queries consists in reconstruction of this function table of values using membership queries. Here we obtain the order of complexity of learning monotone Boolean functions with irrelevant variables.

We continue the study of properties of perfectly balanced Boolean functions with the use of the concept of barriers of Boolean functions.

We consider systems of the form M = F ∪ ν, where F is some Post class and ν is a finite system of definite automata. We divide the Post classes into those for which the problem of A -completeness of such systems of definite automata is algorithmically decidable and those for which the problem of A -completeness is algorithmically undecidable.