Table of contents

Let be a continuous real functional on the space . Continuity of the operator from into itself is considered, where for each . In particular, in the case of a normed space the following is proved. Write

and let be the totality of all closed convex sets in . A set is called approximately compact if every minimizing sequence in contains a subsequence converging to an element of . Suppose is reflexive, is convex and the set is bounded for and contains interior points. Then the following assertions are equivalent: a) , , . b) every set is approximately compact. Bibliography: 15 titles.

Under general assumptions on the functions and it is proved that the inequality

where is the distance from to the nearest integer and , , has only a finite number of solutions in integers for almost all . This establishes the extremality of the surface . Bibliography: 11 titles.

In this paper a theory of the angular potential in Hölder classes is constructed, and several of its applications to problems in the theory of harmonic functions are considered. Bibliography: 9 titles.

The Riemann-Hilbert problem on a complex analytic manifold is as follows. Consider an analytic submanifold of codimension 1 in and a representation : . Does there exist a Pfaffian system of Fuchs type on whose solution space realizes the representation ? This paper is devoted to the study of conditions for the solvability of the Riemann-Hilbert problem on with a given reducible algebraic variety of codimension 1 on it, whose irreducible components are nonsingular and cross each other normally. Bibliography: 15 titles.

In this paper analytic continuation and functional equations are proved for Eisenstein series on the symplectic group associated to forms that are not cusp forms. Bibliography: 9 titles.

It is proved that in the upper semilattice of recursively enumerable btt-degrees, every upper bound of the set of minimal elements coincides with the unit of the semilattice. In any recursively enumerable nonrecursive w-degree there exist sets having minimal m- and btt-degrees. Bibliography: 8 titles.

Estimates are obtained of the rate of approximation almost everywhere as a function of the modulus of continuity of the approximated functions in , and of the set from which the approximating functions are chosen. From this point of view the author studies the approximation of functions by Steklov means, partial sums of Fourier-Haar series, arbitrary sequences of polynomials in the Haar and Faber-Schauder systems, and piecewise monotone functions with variable intervals of monotonicity. The estimates of the rate of approximation almost everywhere that are obtained are distinguished from approximation estimates in an integral metric (i.e. from estimates of the type of Jackson's theorem in ) by unbounded factors depending on the modulus of continuity and the approximating functions. Estimates of the growth of these factors are obtained, and it is established that in a number of cases these estimates are best possible, or almost so. Bibliography: 17 titles.

In this paper we study some classes of Markov random fields. In particular, it is proved that the solution of a linear stochastic partial differential equation is a Markov field. Bibliography: 7 titles.

The first boundary value problem is considered for an elliptic selfadjoint operator of order in a domain of complicated structure of the form , where is a comparatively simple domain in and is a closed, connected, highly fragmented set in . The asymptotic behavior of the resolvent of this problem is studied for when the set becomes ever more fragmented and is disposed volumewise in so that the distance from to any point tends to zero. It is shown that converges in norm to the resolvent of an operator , which is considered in the simple domain under null conditions in . A massivity characteristic of the sets (of capacity type) is introduced, which is used to formulate necessary and sufficient conditions for convergence, and the function is described. Bibliography: 7 titles.