Vol 39, No 5 (2018)

The Sobolev spaces with continuous and discrete coherent pairs of weights are considered. The positivity of the inner product is equivalent to the Markov–Bernstein inequality for the weighted integral norm. Asymptotics of the sharp constants for these inequalities, when the degree of polynomials goes to infinity, are obtained.

Questions of definability of computably enumerable degrees in the difference hierarchy (degrees of sets from finite levels of the Ershov difference hierarchy) are studied. Several approaches to the solution of this problem are outlined.

An experience of designing integrated hardware and software solutions for highperformance computing in solving modern geophysical problems on the basis of full-wave inversion is described. Problems of designingmass high-performance software systems intended for extensive use in industry are discussed.

For (n + 1)-ly connected planar domain D with analytic boundary we construct the function F(w,w₀) = (w − w₀)f(w,w₀) which maps D conformally onto the unit disk with circular and radial slits. We show that if n ≥ 2, then Mityuk’s function, M(w) = −(2π)⁻¹ ln |f(w,w)|, representing the generalized reduced module of the domain D has at least one stationary point in D.

For finite-dimensional saddle point problem with a nonlinear monotone operator and constraints on direct variables, iterative methods are developed, which in the potential case can be viewed as preconditioned Uzawa methods or as Uzawa-block relaxation methods. Convergence conditions of the iterative methods are formulated in the form of operator inequalities connecting the operator of the problem and the preconditioning matrix. When applied to mesh problems, this allows us to construct suitable preconditioners that ensure the convergence and effective implementation of iterative methods and to obtain the admissible intervals of iterative parameters which don’t depend on mesh parameters. The presented results are based on the general theory developed by the author with co-authors in recent years.

We study a particular class of switching systems called systems with partially freezed components. Fix an autonomous system of differential equations and consider a family of systems obtained from it by replacing some of components of the right-hand side by zeros (thus, “freezing” some components of a solution). We obtain the flow of a system with partially freezed components by fixing a sequence of switching times and of the corresponding sets of “active” components (i.e., the components that are not “freezed”). Such a sequence is called a freezing sequence. Systems with partially freezed components appear, for example, in the study of generalized Turing machines. In this paper, we are mostly interested in stability properties of such systems; our main goal is to relate such properties to properties of the generating system and freezing sequences.

Recently the author obtained a series of sharp estimates of convolution classes by spaces of shifts of a single function. Those estimates generalize the well-known classical inequalities of Favard, Akhiezer and Krein. In the present paper we compute the average dimension of shift spaces. It appears that this dimension coincides with the average dimension of the spaces of entire functions of exponential type and of equidistant splines.