Volume 21, Issue 1

For a random unlabelled unrooted forest consisting of N trees and n vertices we obtain limit distributions of the maximum tree size in all domains where N and n tend to infinity. We formulate conditions for emergence of a giant tree in the random forest.

We consider a stationary sequence x 1 , x 2 , . . . , x t , . . . of random variables each of which takes values in the set . Let μ r = μ r ( n, N ) be the number of values in the set which occur r times in the first n elements of the sequence. In this paper, we obtain asymptotic formulas for the mean value E μ r ( n, N ) and the variance D μ r ( n, N ) and give sufficient conditions for asymptotic normality of the random variables μ r ( n, N ) as n, N → ∞ under the condition that the sequence consists of m -dependent random variables and the parameters of the distribution of the sequence vary in the so-called central domain.

We consider the mapping C N,n of a set with n numbered elements into itself, which has N ≤ n connected components and is uniformly distributed on the set of all such mappings. We denote the number of such mappings by a ( n, N ). In addition to the known estimates we derive some new estimates of the number a ( n, N ) under the condition that n → ∞ and N = N ( n ). Let η 1 , . . . , η N be the sizes of connected components of the random mapping C N,n , numbered in one of the N ! possible ways. We obtain limit theorems estimating the distribution of the random vector ( η 1 , . . . , η N ) as n, N →∞ including the domain of large deviations. A new asymptotic estimate of the local probabilities for a sum of independent identically distributed random variables which determine the corresponding generalised allocation scheme is obtained.

We describe the marginal and joint distributions of the numbers of values and the numbers of runs in ternary Markov sequences. A relative simplicity of such sequences permits to derive exact formulas for means, variances, and covariances with the use of direct combinatorial calculations.

Let be a class of finite groups, τ be a subgroup functor; an ω -foliated τ -closed formation of finite groups 𝔉 with direction δ is called the minimal ω -foliated τ -closed non--formation with direction δ , or, in other words, ωτδ -critical formation if 𝔉 ⊈ , but all proper ω -foliated τ -closed subformations with direction δ in 𝔉 are contained in the class . In this paper we investigate the structure of the minimal ω -foliated τ -closed non--formations with bp -direction δ satisfying the condition δ ≤ δ 3 in the case where τ is a regular δ -radical subgroup functor.

In this paper it is proved that the semigroup of endomorphisms of a commutative Moufang loop (CML) which admits a decomposition into the direct product of its subloops is isomorphic to the semigroup of M , M -matrices. We introduce the notion of the holomorph of a CML which admits a direct decomposition and give the matrix representation of the holomorph.

The paper contains a detailed description of the Voronoi polyhedra P V ( E 6 ) of the rooted lattice E 6 and of the lattice dual to E 6 . For these polyhedra, tables of types of all faces and the number of faces of each type are given. It is known that the polyhedron P V ( E 6 ) is the union of the Schläfli polyhedron P Schl and its antipodal polyhedron – P Schl . In this paper, it is proved that is the intersection of these polyhedra.

We consider efficient algorithms of calculation of the characteristic polynomials of matrices over commutative rings. We give estimates of complexity treated as the number of ring operations, and for the ring of integers the estimates are presented in terms of the number of multiplication operations over the machine words. We suggest a new algorithm to calculate the characteristic polynomial which has the best estimate of complexity in the ring operations. We give recommendations concerning applications of the algorithm of calculation of the characteristic polynomials depending on the size of the matrix, in particular, the algorithm suggested in this paper is recommended to be applied to integer-element matrices of size greater than 60.

We consider algebraic models of sequential programs where the semantics of operators is defined on the base of a semigroup. In the paper, for the first time an example is constructed of a solvable semigroup with indecomposable neutral element for which the stopping problem for a Turing machine reduces to the problem of equivalence of programs over a given semigroup. All known examples of models of insoluble problem of equivalence of programs were based on groups. Thus, in the paper we succeeded in refining the boundary between soluble and insoluble cases of the problem of equivalence of programs in algebraic models. The obtained result supports also the importance of some sufficient conditions of solubility of the problem of equivalence of programs.