Volume 29, Issue 3

Let { S n , n ≥ 0} be integer-valued random walk with zero drift and variance σ 2 . Let ξ ( k , n ) be number of t ∈ {1, …, n } such that S ( t ) = k . For the sequence of random processes ξ (⌊ uσ n⌋ ,n)$\begin{array}{} \xi(\lfloor u\sigma \sqrt{n}\rfloor,n) \end{array}$ considered under conditions S 1 > 0, …, S n > 0 a functional limit theorem on the convergence to the local time of Brownian meander is proved.

We obtain precise values of cardinality of minimal generating sets for all Cartesian products of two-element set with respect to an arbitrary set of Boolean operations from the central part of the Post lattice. For the case of sets containing operations from the remaining classes of the Post lattice we obtain cardinality estimations that are accurate up to one.

The paper is concerned with the problem of existence of periodic structures in words from formal languages. Squares (that is, fragments of the form xx , where x is an arbitrary word) and Δ -squares (that is, fragments of the form xy , where a word x differs from a word y by at most Δ letters) are considered as periodic structures. We show that in a binary alphabet there exist arbitrarily long words free from Δ -squares with length at most 4 Δ +4. In particular, a method of construction of such words for any Δ is given.

A centrally essential ring is a ring which is an essential extension of its center (we consider the ring as a module over its center). We give several examples of noncommutative centrally essential rings and describe some properties of centrally essential rings.

We refine local limit theorems for the distribution of a part of the weight vector of subfunctions and for the distribution of a part of the vector of spectral coefficients of linear combinations of coordinate functions of a random binary mapping. These theorems are used to derive improved asymptotic estimates for the numbers of correlation-immune and k -resilient vectorial Boolean functions.