Volume 18, Issue 1

We consider monic (with higher coefficient 1) polynomials of fixed degree n over an arbitrary finite field GF ( q ), where q ≥ 2 is a prime number or a power of a prime number. It is assumed that on the set F n ={ ƒ n } of all q n such polynomials the uniform measure is defined which assigns the probability q -n to each polynomial. For an arbitrary polynomial ƒ n ∈ F n , its local structure K n = K ( ƒ n ) is defined as the set of multiplicities of all irreducible factors in the canonical decomposition of ƒ n , and various structural characteristics of a polynomial (its exact and asymptotic as n → ∞ distributions) which are functionals of K n are studied. Directions of possible further research are suggested.

We consider the class of all permutations of degree n whose cycle lengths are elements of a fixed finite set A ⊂ N such that card A ≥ 2 and gcd. Under the assumption that the permutation X is equiprobably chosen from this class, we obtain a multidimensional local normal theorem for the joint distribution of the numbers of cycles of given sizes in this permutation. The obtained results are utilised and sharpened in the case where X is an equiprobably chosen solution of the equation X r = e , where e is an identity permutation of degree n , r ≥ 2 is a fixed positive integer.

In this paper we study the weak convergence of distributions for products of random variables with values in a compact group provided that the distributions of the factors are defined by a finite simple homogeneous irreducible Markov chain. We show that after an appropriate shift the sequence of distributions of these products converges weakly to the normalised Haar measure on some closed subgroup of the initial group, in other words, the convergence principle due to B. M. Kloss holds true, which has been established earlier for products of independent factors. We describe conditions on the Markov chain and on the initial distributions which guarantee that the limit behaviour of the distribution of the products is similar to the limit behaviour of the distributions of some products of independent random variables.

We study the numbers S mn of labelled connected graphs with n vertices among which m are cutpoints. We obtain an explicit expression for the exponential generating function S m ( Z ) of these numbers and find the asymptotics of the numbers S mn as n → ∞ and m = o ( n ).

Let A be a ring and let ϕ be an automorphism of A . Then the skew Laurent series ring A (( x, ϕ )) is a right serial ring with the maximum condition on right annihilators if and only if A is a right Artinian right serial ring.

We suggest a new block algorithm for solving sparse systems of linear equations over GF (2) of the form where A is a symmetric matrix, F = GF (2) is a field with two elements. The algorithm is constructed with the use of matrix Padé approximations. The running time of the algorithm with the use of parallel calculations is max{ O ( dN 2 / n ), O ( N 2 )}, where d is the maximal number of nonzero elements over all rows of the matrix A . If d < Cn for some absolute constant C , then this estimate is better than the estimate of the running time of the well-known Montgomery algorithm.

In this paper we suggest a new algorithm of anti-unification of logic terms represented by acyclic directed graphs and estimate its complexity. The anti-unification problem consists of the following: for two given terms find the most specific term that has the given terms as instances. We suggest an anti-unification algorithm whose complexity linearly depends on the size of the most specific term it computes. It is thus established that the anti-unification problem is of almost the same complexity as the unification problem. It is also shown that there exist terms whose most specific term is of size O ( n 2 ), where n is the size of the graphs representing these terms.

The description of the (v, 3)-groups was given previously, it appears that all such groups are Abelian. It is known that there are non-Abelian (v, 5)-groups. In this paper, a description of the Abelian (v, 5)-groups is given.