Band 21, Heft 4

We consider a two-parameter model of random n -permutations, which is a generalisation of the classical model of A -permutations, and investigate the joint distribution of the number of A -cycles and Ā -cycles under various realisations of the subset A ⊂ X n = {1, 2, . . . , n }.

We suggest a new example of a free subgroup of a group of automaton substitutions. We present two automata with three states which generate this subgroup; the inner semigroups of these automata are not groups.

We consider the classes in the zero layer of the set of infinite hereditary classes of graphs defined by two forbidden subgraphs. One of these subgraphs is K 1, s + O p and the other is K q . We give an upper bound for the number of graphs in these classes.

We give an asymptotic upper bound for the chromatic index of a random hypergraph in the case where the edge length of the hypergraph is an increasing function of the number of vertices of the hypergraph.

We introduce the notion of steepness of a coordinate-convex function of discrete argument on an ordinal-convex set. In terms of guaranteed estimates it is shown that in problems of optimisation of coordinate-convex functions on an ordinal-convex set the gradient coordinatewise lifting algorithm is stable under small perturbations of the utility function. As corollaries we obtain improved guaranteed estimates for accuracy of the gradient algorithm, and also new sufficient conditions for the values of the utility function of the problem under consideration to coincide in the global and gradient extrema.

We describe the rings over which all modules are completely integrally closed.

We consider circuits of functional elements of a finite depth whose elements are arbitrary Boolean functions of any number of arguments. We suggest a method of finding nonlinear lower bounds for complexity applicable, in particular, to the operator of cyclic convolution. The obtained lower bounds for the circuits of depth d ≥ 2 are of the form Ω( nλ d –1 ( n )). In particular, for d = 2, 3, 4 they are of the form Ω( n 3/2 ), Ω( n log n ), and Ω( n log log n ) respectively; for d ≥ 5 the function λ d –1 ( n ) is a slowly increasing function. These lower bounds are the greatest known ones for all even d and for d = 3. For d = 2, 3, these estimates have been obtained in earlier studies of the author.