Volume 13, Issue 1

We prove local and integral limit theorems for large deviations of Cramer type for a critical Galton–Watson branching process under the assumption that the radius of convergence of the generating function of the progeny is strictly greater than one. The proof is based on a modified Cramer approach which consists of construction of an auxiliary non-homogeneous in time branching process.

We analyse the modification of an algorithm for finding discrete logarithms over the field GF(p) ( p is a prime number) which has been described by the author previously. It is shown that this modification gives the best estimate at the present time of the complexity of finding discrete logarithms over finite prime fields which coincides with the best known estimate of the complexity of factoring integers obtained by D. Coppersmith.

We suggest a method of synthesis of a scheme of maximal reliability without essential enlargement of complexity in the case of one-type constant faults at the inputs of elements over the basis {&, ∨, ¯}.

We consider automaton bases with complete Boolean part. We construct an algorithm to test completeness of such bases and give upper bounds for its complexity.

An undirected graph on v vertices of valences equal to k , whose each edge belongs to exactly λ triangles is called edge-regular with parameters ( v, k, λ ). Let b 1 = k – λ – 1. We say that a pair of vertices u, w is good if these vertices have exactly k – 2 b 1 + 1 common neighbours. We prove that if k ≥ 3 b 1 – 1, then either for any vertex u at most two vertices in Γ form good pairs with u , or k = 3 b 1 – 1, Γ is a polygon or the icosahedron graph, and any two vertices which are 2 distant from each other form good pairs. We give a new upper bound for the number of vertices in an edge-regular graph of diameter two with k ≥ 3 b 1 – 1. We prove that an edge-regular graph with parameters of the triangular graph T (n), n = 5; 6, the Clebsch graph, or the Schläfli graph coincides with the corresponding graph.

Two necessary conditions for a polynomial to be chromatic, which in a series of cases are stronger than some known conditions, are suggested.