Volume 20, Issue 4

We suggest an algorithm calculating the limit probabilities of the distribution of the permanent of a random matrix A mn = ‖ α ij ‖ with m rows and n independent columns over the field GF( p ).

We investigate the distribution of the random variable equal to the number of coincidences of two homogeneous random walks with positive independent increments. This random variable is the length of the subsequence of common elements in two random sequences which are random subsequences of the same random sequence. For the considered random variable we obtain the asymptotic expression for the mathematical expectation and a limit theorem under the assumption that the sequential intervals between coincidences of the two random walks have a finite variance. For the particular case of random walks with increments equal to 1 and 2 we prove a finiteness of this variance and obtain the expression of the variance in terms of the parameters of the random walks.

We consider sums of elements of plane sections of the generalised Pascal pyramide. We evaluate the sums and find recurrence relations which these sums obey, and then suggest enumerative interpretations of the combinatorial objects under investigation.

We consider a generalisation of the classic combinatorial problem of P. Erdős and A. Hajnal in the theory of hypergraphs to the case of prescribed colourings. We investigate the value m pr ( n, r ) equal to the minimum number of edges of a hypergraph in the class of n -uniform hypergraphs with prescribed chromatic number greater than r . We obtain a lower bound for this value which is better than the known results for r ≥ 3. Moreover, we give a sufficient conditions for existence of a prescribed r -colourability of an n -uniform hypergraph in terms of restrictions on the intersections of edges. As a corollary we obtain a new bound for the characteristic equal to the minimum number of edges of a hypergraph in the class of n -uniform simple hypergraphs (in which any two edges have at most one common vertex) with the prescribed chromatic number greater than r .

A new estimate of the number of zeros ϱ n = β n + iγ n of the Riemann zeta-function with ordinates γ n belonging to a given interval and for which the distance to the next zero is sufficiently large in comparison with the mean value 2 π (ln( γ n /(2 π ))) –1 is obtained.

The paper is devoted to the investigation of properties of correlation-immune and stable functions as a whole. The description of these properties is realised with the use of algebraic notions and operations.

Methods of synthesis of easily testable circuits of functional elements over the Zhegalkin basis for arbitrary Boolean functions are suggested. It is assumed that the faults are constant faults of type 0 at outputs of elements. It is proved that any Boolean function can be realised by a circuit allowing a complete test of length 1.

In this paper we complete the solution of the minimisation problem for a set of binary two-tape automata. The beginning of the solution was published in Discrete Mathematics and Applications 18 (2008), pp. 271–292. The minimisation problem consists of finding all minimal automata in each equivalence class of a considered set. The search is performed by means of equivalent transformations of automata. The solution of the problem is based on an appropriate partition of the equivalence class.