Abstract An improvement of the algebraic method of computing the Hadamard product of rational functions is given. Some applications of the Hadamard product to combinatorial and probabilistic problems are considered.

Abstract We consider average-case complexity of computing monotone Boolean functions by straight-line programs with a conditional stop over the basis of all Boolean functions of at most two variables. For the set of all n -ary monotone Boolean functions new Shannon-type upper and lower bounds for the average-case complexity as n → ∞ are established.

Abstract The paper gives a complete justification of the modular algorithm for reducing matrices to the Hermitian normal form, which enables one to construct a new modular algorithm for reducing to the Smith normal form that may simultaneously calculate the left matrix of the transformations. The main term in the estimate of the number of operations is 2(n 3 log D), where n is the size and D is the determinant (or a multiple of it) of the matrix under consideration.

Abstract A class of graphs is called monotone if it is closed under deletion of vertices and edges. Any such class may be defined in terms of forbidden subgraphs. The chromatic index of a graph is the smallest number of colors required for its edge-coloring such that any two adjacent edges have different colors. We obtain a complete classification of the complexity of the chromatic index problem for all monotone classes defined in terms of forbidden subgraphs having at most 6 edges or at most 7 vertices.

Abstract We consider a sequence of almost critical branching processes with immigration supposing that the immigration process is weakly stationary. The rate of growth and asymptotic properties of fluctuations of such branching processes are investigated.

Abstract We propose and investigate new algorithms permitting to find an identical object in the database using the number of operations not depending on the volume of the database. One algorithm requires memory size that depends linearly on the database volume in the average.

Abstract In a sequence of independent positive random variables with the same continuous distribution function a monotonic subsequence of record values is chosen. A corresponding sequence of record times divides the initial sequence into interrecord intervals. Let α i j ( i ⩾ 1 , j = 1 , … , i ) $\alpha_i^j \ (i\geqslant 1, \, j = 1, \ldots, i)$ be the number of random variables in the interval between i-th and (i + 1)-th record moments with values between (j − 1)-th and j-th records. Explicit formulas for the joint distributions of the random variables α i j , 1 ⩽ j ⩽ i ⩽ n $\alpha_i^j,\,1\leqslant j\leqslant i\leqslant n$ , are derived, limit theorems for the distributions of α i j $\alpha_i^j$ for i − j → ∞ are proved.

Abstract We study finite automata representations of numerical rings. Such representations correspond to the class of linear p-adic automata that compute homogeneous linear functions with rational coefficients in the ring of p-adic integers. Finite automata act both as ring elements and as operations. We also study properties of transition diagrams of automata that compute a function f(x)= cx of one variable. In particular we obtain precise values for the number of states of such automata and show that for c > 0 transition diagrams are self-dual (this property generalises self-duality of Boolean functions). We also obtain the criterion for an automaton computing a function f(x)= cx to be a permutation automaton, and fully describe groups that are transition semigroups of such automata.

Abstract We give a survey of the enumeration of independent sets in graphs and of some related problems.

Abstract The paper is concerned with checking problems for failure and state diagnostics of N gates implementing in a working condition a given Boolean function f(x