Volume 17, Issue 3

For a sequence X = { X 1 , ... , X n , ... } of independent identically distributed random variables, we construct the s -tuples Y i ( s ) = ( X i , ... , X i + s -1 ), i = 1, 2, ... , n , and consider the random variables F i = f ( Y i ( s )), i = 1, 2, ... , where f is a function defined on the set R s and taking non-negative integer values. We consider the sequence F = { F 1 , F 2 , ... } and study two random variables, the variable equal to the number of matches of symbols on a segment of length n of the sequence F (here I {·} stands for the indicator of a random event), and the variable equal to the number of matches of values of the function f of non-overlapping s -tuples of a segment of the sequence X of length n + s – 1. With the use of the Stein method, we find sufficient conditions for the distribution of the random variables Z n ( F ) and Z ′ n ( F ) to converge to the compound Poisson law for the function f of a general form. As corollaries to these results we obtain both known and new limit theorems for the number of matches of values of a function of segments of sequences in a polynomial scheme for a series of particular types of the function f .

We consider a vector (multicriteria) problem of Boolean programming in the case where the partial criteria are the absolute values of linear functions. We study the limit level of disturbances of the coefficients of criterion functions in the space with metrics l ∞ which preserves the Pareto optimality of the solution. We obtain a necessary and sufficient condition for the stability radius of such a solution to be infinite.

This study is devoted to the problem of lifting of a solution of an exponential congruence in rings of integer algebraic numbers. To lift a solution in rings of integer rational numbers, Riesel suggested to use the Fermat quotients apparatus. With their use, the problem reduces to solution of a linear congruence modulo a prime number, and this congruence appears to be irreducible. In this paper we construct analogues of Fermat quotients in rings of integer algebraic numbers which also yield irreducible linear congruences for the problem of lifting of a solution in this case.

We construct a sequence of graphs of large degree with growing number of vertices for which the number of independent sets is substantially greater than the number of all subsets of the independent set of maximal cardinality.

In this paper, we study morphisms (automorphisms, endomorphisms) of linear and alinear quasigroups, give results on endomorphisms of an arbitrary quasigroup which generalise some results of a Shchukin’s paper. We give necessary and sufficient conditions of homomorphisms of linear and alinear quasigroups in terms of endomorphisms of the corresponding group and present the form of an arbitrary endomorphism of linear, alinear, and T -quasigroups.

A subgroup H of a group G is called complemented in G if a subgroup K exists in G such that G = HK and H ∩ K = 1. A group is called completely factorisable if each subgroup of the group is complemented. Let D ( G ) be the subgroup of a group G generated by all subgroups of G which have no complements in G, Z ( G ) be the centre of the group G , and Φ( G ) be the Frattini subgroup of the group G . If all subgroups of G are complemented in G , then we set D (G) = 1. Each cyclic subgroup of the Frattini subgroup Φ( G ) of the group G has no complement in G , therefore Φ( G ) ⊆ D ( G ). In the paper, we obtain a complete description of the structure of a finite group G such that D (G) ⊆ Z ( G )Φ( G ).

We consider a random ( n + s ) × n matrix A with independent rows over a field of q elements. In terms of the Fourier coefficients of distributions of the rows of this matrix we obtain expressions of upper and (in the case where the Fourier coefficients are non-negative quantities) lower bounds for probabilities of values of its rank. We find an upper bound for the distance in variation between the distributions of ranks of the matrix A and a random equiprobable matrix. We present a condition for this distance to tend to zero as and s is fixed and demonstrate that this condition, in some natural sense, cannot be weakened.

The set of variables of a k -valued logic function f ( x 1 , ... , x n ) is partitioned into t parts, t > 1, and a polynomial representation of the function f is considered where the terms are products of all possible subfunctions corresponding to the partitioning. We analyse conditions under which an arbitrary function admits a representation in such a polynomial form.

Let F k ( n , m ) be a random k -conjunctive normal form obtained by selecting uniformly and independently m out of all possible k -clauses on n variables. We prove that if F 4 ( n, rn ) is unsatisfiable with probability tending to one as n → ∞, then r ≥ 8.09. This sharpens the known lower bound r ≥ 7.91.

We consider a homogeneous Markov chain of order s with r partial connections for which the transition probabilities from any state to the next depend not on all s preceding states but on r selected states only. We construct statistical estimators of parameters and statistical tests for values of these parameters of the model and investigate their asymptotic properties. Algorithms to calculate these estimators are suggested. The results of computer simulations are also given.