Volume 19, Issue 3

We consider a finite cooperative game of several players with parametric principle of optimality such that the relations between players in a coalition are based on the Pareto maximum. The introduction of this principle allows us to find a link between such classical concepts as the Pareto optimality and the Nash equilibrium. We carry out a quantitative analysis of the stability of the game situation which is optimal for the given partition method with respect to perturbations of parameters of the payoff functions in the space with the Hölder l p -metric, 1 ≤ p ≤ ∞. We obtain a formula for the radius of stability for such situation, so we are able to point out the limiting level for perturbations of the game parameters such that the optimality of the situation is preserved.

We consider new approaches to the investigation and introduce parameters for some new classes of perfectly balanced Boolean functions.

We consider the problem on completeness of sets of S -functions, the determinate functions such that the automaton calculating them realises in each state functions which emanate no value. We assume that each set of S -functions whose completeness is checked in this paper contains all S -functions depending on at most one variable. We describe all A -precomplete classes of such sets. It is shown that there exists an algorithm recognising A -completeness of S -sets of one-place determinate functions containing all one-place determinate S -functions.

We investigate representations of Boolean functions by formulas. In terms of the residual functions, we describe Boolean functions realised by repetition-free formulas in pre-elementary bases {∨, ·, –, 0, 1, x 1 ( x 2 ∨ . . . ∨ x n ) ∨ x 2 · . . . · x n } where n ≥ 4.

We investigate some properties of multiaffine, bijunctive, weakly positive and weakly negative Boolean functions. The following results are proved: for any integer k ≥ 1 the maximal group of transformations of the domain of definition of a function of k variables with respect to which the set of multiaffine Boolean functions is invariant is the complete affine group AGL ( k , 2); for the bijunctive functions of k ≥ 3 variables it is the group of transformations each of which is a combination of a permutation and an inversion of the variables of the function; and for a weakly positive (weakly negative) function of k ≥ 2 variables it is the group of transformations each of which is a permutation of the variables of the function.

We prove the asymptotic normality of the number of absent noncontinuous chains (chains with gaps) of outcomes of independent trials. The asymptotic normality of the number of absent chains of identical outcomes, which has been proved for equiprobable outcomes, is proved in the nonequiprobable case.

We consider M ≥ 1 independent samples each of which is a realisation of some polynomial scheme. The number of outcomes N and the number of samples M are fixed, the sample sizes grow without bound. We study the asymptotic properties of statistics of the form g (), where g () is a differentiable function of MN real-valued variables, is the vector of relative frequencies of outcomes in samples. Statistics of such a kind are playing an important part in the applied statistical analysis. In this research we expand the capabilities of the well-known δ -method (the linearisation method) in the case of polynomial samples. We prove the asymptotic normality and convergence in distribution of the statistics g () to quadratic forms of normal random variables (both for the fixed probabilities of outcomes in the case of the null hypothesis and for the case of contigual alternatives to it). We give conditions for both types of convergence.

Let m, n, h and k be integers such that m ≥ h > 1 and n ≥ k > 1. An [ h-k ]-bipartite hypertournament on m + n vertices is a triple ( U, V, E ), with two vertex sets U and V , | U | = m , | V | = n , together with an arc set E , a set of ( h + k )-tuples of vertices, with exactly h vertices from U and exactly k vertices from V , called arcs, such that for any h -subset U 1 of U and k -subset V 1 of V , E contains exactly one of the ( h + k )! ( h + k )-tuples whose h entries belong to U 1 and k entries belong to V 1 . We obtain necessary and sufficient conditions for a pair of nondecreasing sequences of nonnegative integers to be the losing score lists or score lists of some [ h-k ]-bipartite hypertournament.

We consider a statistical model of the so-called steganography, which is a new rapidly developing field of information protection. We suggest one possible variant of a hidden embedding of a secret information into nonsecret data transmitted and analyse probabilistic properties of the model.