Volume 26, Issue 1

We describe a method of partitioning the given set of graphs into non-intersecting subsets each consisting of graphs with common frame. Sets of graphs having such representations are presented. The possibility to apply these representations to the derivation of asymptotic expansions in some problems of statistical mechanics is discussed.

The paper is concerned with problems in steganography on the detection of embeddings and statistical estimation of positions at which message bits are embedded. Binary stationary Markov chains with known or unknown matrices of transition probabilities are used as mathematical models of cover sequences (container hles). Based on the runs statistics and the likelihood ratio statistic, statistical tests are constructed for detecting the presence of embeddings. For a family of contiguous alternatives, the asymptotic power of statistical tests based on the runs statistics is found. An algorithm of polynomial complexity is developed for the statistical estimation of positions with embedded bits. Results of computer experiments are presented.

We study the circuit complexity of Boolean functions in an infinite basis consisting of all characteristic functions of antichains over the Boolean cube. For an arbitrary symmetric function we obtain the exact value of its circuit complexity in this basis. In particular, we prove that the circuit complexities of the parity function and the majority function of n variables for all integers n ≥ 1 in this basis are ⌊n+12⌋$\left\lfloor {\tfrac{{n + 1}}{2}} \right\rfloor $ and ⌊n2⌋+1$\left\lfloor {\tfrac{n}{2}} \right\rfloor + 1$ respectively. For the maximum circuit complexity of n -variable Boolean functions in this basis, we show that its order of growth is equal to n . The research is supported by the Russian Foundation for Basic Research, project 14–01–00598.

The paper is concerned with n -place Boolean functions not admitting implicents of k variables, 1 ≤ k < n . Estimates for the minimal possible weight w ( n , k ) of such functions are obtained. It is shown that w ( n , 1) = 2, n = 2, 3, ..., and w ( n , 2) ~ log 2 n as n → ∞, and moreover, for k > 2 there exists n 0 such that w ( n , k ) > 2 k −2 · log 2 n for all n > n 0 .

We consider critical Bellman – Harris processes with two types of particles. The tail of the lifetime distribution of the first type particles decreases as o ( t −2 ), the tail of the lifetime distribution of the second type particles is regularly varying with the index in (−1, 0). Such processes are connected with the matrix renewal functions of special two-dimensional renewal processes. V. A. Vatutin and the author have used the asymptotics of these matrix renewal functions and their first and second order increments in the proofs of several limit theorems for branching processes. Here we investigate the properties of such matrix renewal functions under significantly weaker conditions on the lifetime distributions and apply the results to the description of the asymptotics of several moments of branching processes and of their increments.