Volume 29, Issue 6

Finite automata transform periodic sequences into periodic ones. The period of the output sequence is bounded from above by a linear function of input period. It is known that pushdown automata also preserve the set of periodic sequences. We prove that the output period for one-counter pushdown automata is bounded from above by a quadratic function of input period. We also give an example of an automaton with a quadratic lower bound on output period.

The paper is concerned with systems of incidence involving a space of points X and lines consisting of q points each. A free space X is defined. For a space X an analogue of the Burnside problem (solved in the negative) and an analogue of the weakened Burnside problem are formulated. In the case q = 3 the positive answer to the analogue of the weakened Burnside problem is equivalent to the existence of a universal finite geometry.

We give an elementary proof that by multiplication on left and right by affine permutations A , B ∈ AGL (3, 2) each permutation π : GF (2) 3 → GF (2) 3 may be reduced to one of the 4 permutations for which the 3 × 3-matrices consisting of the coefficients of quadratic terms of coordinate functions have as an invariant the rank, which is either 3, or 2, or 1, or 0, respectively. For comparison, we evaluate the number of classes of affine equivalence by the Pólya enumerative theory.

For sets of iid random points having a uniform (in a definite sense) distribution on the arbitrary metric space a maximal distance to the nearest neighbour is considered. By means of the Chen–Stein method new limit theorems for this random variable is proved. For random uniform samples from the set of binary cube vertices analogous results are obtained by the methods of moments.

The paper is devoted to the study of equations with the left-hand side having the form of a composition of operations which belong to given sets S 1 , …, S n , … of quasigroup operations. Elementary transformations are described which allow reducing systems of this kind to the form where all equations except one do not depend essentially on the variable x n . A class of systems is said to be Gaussian if every system obtained via such transformations also belongs to this class. It is evident that for Gaussian classes of systems of equations there is an efficient solving algorithm. This motivates the problem of finding conditions under which the class is Gaussian. In this work it is shown that for a class of systems to be Gaussian the operations in the sets S i should satisfy the generalized distributivity law. Sets of operations obeying this condition are to be investigated in the future.

A family ( A 1 , …, A k ) of subsets of a group G is called k -solution-free family if the equation x 1 + … + x k = 0 has no solution in ( A 1 , …, A k ) such that x 1 ∈ A 1 , …, x k ∈ A k . We find the asymptotic behavior for the logarithm of the number of k -solution-free families in Abelian groups.

Let B ( x ) be a multiple power series with nonnegative coefficients which is convergent for all x ∈ (0, 1) n and diverges at the point 1 = (1, …, 1). Random vectors (r.v.) ξ x such that ξ x has distribution of the power series B ( x ) type is studied. The integral limit theorem for r.v. ξ x as x ↑ 1 is proved under the assumption that B ( x ) is regularly varying at this point. Also local version of this theorem is obtained under the condition that the coefficients of the series B ( x ) are one-sided weakly oscillating at infinity.