Volume 17, Issue 1

We consider a subcritical branching process with immigration, infinite number of types T 1 , T 2 , ... of particles, and discrete time. The state of the process at the moment of time t is the set of vectors where ξi ( t ) is the number of particles of type T i at the moment of time t , i = 1, 2, ... It is assumed that at each moment of time only particles of type T 1 immigrate and each particle of type T i turns into a set of particles of types T i and T i + 1 . It is proved that the probability distributions of the vectors ( r, t ) converge as t → ∞ to discrete limit distributions.

We consider extremal values of the mathematical expectation of the number of cells containing exactly r particles in the set of polynomial schemes of allocating n particles to N cells. We describe the properties of the distributions which supply the extremal values and obtain two-sided bounds for the extremal values.

We investigate the asymptotic behaviour of the distribution of the number ξ( B ) of the solutions of a system of homogeneous random linear equations Ax = 0 (the T × n matrix A is composed of independent random variables a i , j uniformly distributed on a set of elements of a finite field K ) which belong to some given set B of non-zero n -dimensional vectors over the field K . We consider the case where, under a concordant growth of the parameters n , T → ∞ and variations of the sets B 1 , . . . , B s such that the mean values converge to finite limits, the limit distribution of the vector (ξ( B 1 ), ... , ξ( B s )) is an s -dimensional compound Poisson distribution. We give sufficient conditions for this convergence and find parameters of the limit distribution. We consider in detail the special case where B k is the set of vectors which do not contain a certain element k ∈ K .

In this paper we give necessary and sufficient conditions for validity of the central limit theorem for sums of independent random variables in terms of characteristic functions in a nonclassical setup where an essential contribution of individual variables to the sum is allowed. These conditions are formulated with the use of the Stein–Tikhomirov operator.

We consider sequences of random variables x ( N ) = ζ 1 · ζ 2 · . . . · ζ N , ω ( N ) = ξ 1 · ζ 1 · ξ 2 · ζ 2 · . . . · ξ N · ζ N , where (ξ N , ζ N ), N ≥ 1, is a sequence of independent identically distributed random variables with values in the Cartesian product G × G of a finite group ( G ; ·). We investigate the degree of dependence of the random variables x ( N ) and ω ( N ) . Such problems arise in the study of a class of information security algorithms. In connection to this problem, we study the random variable with values in G whose distribution coincides with the conditional distribution of the random variable ω ( N ) under condition that x ( N ) = a , where a is such that P ( x ( N ) = a ) > 0. We give conditions of convergence and limit distributions of as N → ∞, where s N is a sequence of integers tending to infinity in such a way that P ( x ( sN ) = a ) > 0.

We consider the problem to manage two stacks in two-level memory. It is assumed that the tops of the stacks grow towards one another in the fast memory to which several processors are allowed to have simultaneous access, and the size of the stacks exceeds that of the fast memory. The fast memory stores the tops of the stacks only, while the remaining parts are stored in the second level memory. If the top of one of the stacks becomes empty or the stacks fill all the fast memory, that is, the stack overflow occurs, then a swapping to the second level memory is performed in such a way that each time a certain state of the memory is set and the next step starts. We study how to choose such a state of the memory in order to maximise the average time before the next swapping to the second level memory.

We obtain a bound for the number of product-free sets in finite groups containing a normal subgroup of prime index.

An r -digraph is an orientation of a multigraph which has no loops and contains at most r edges between any pair of distinct vertices. We give a simple proof of necessary and sufficient conditions for a sequence of non-negative integers arranged in nondecreasing order to be a sequence of numbers, called marks or r -scores, attached to the vertices of an r -digraph.

We consider an application of regular partitions of a space to the solution of Euclidean combinatorial optimisation problems, in particular, conditional optimisation problems on arrangements. We introduce the notion of points of the space equivalent with respect to arrangements and show that the introduced relation between points is an equivalence relation. We give algorithms to search for an element of the set of arrangements which is a representative of the combinatorial class closest to a given class with respect to the lexicographic order (increasing or decreasing). We consider also a new class of optimisation problems, namely, problems of linear conditional lexicographic maximisation on arrangements. We suggest and analyse algorithms for solving the problems of this class. The algorithms are based on the ordered checking of admissible points in the lexicographic (increasing or decreasing) order and use the algorithms to search for the closest element of the set of arrangements mentioned above.

We consider linear recurring sequences of maximal period over a Galois field and study their representations over a Galois ring. Minimal and characteristic polynomials of these representations are described.