Table of contents

An example is given of a diophantine relation which has exponential growth. This, together with the well-known results of Martin Davis, Hilary Putnam, and Julia Robinson, yields a proof that every enumerable predicate is diophantine. This theorem implies that Hilbert's tenth problem is algorithmically unsolvable.

We examine n-dimensional complex varieties of fundamental type. We prove the existence of a universal set of algebraic deformations of a canonically polarized variety over a field of characteristic zero.

We investigate the question: What is the smallest ring over which the elements of a dense subgroup (in the Zariski topology) of a semisimple algebraic group can be written down simultaneously for various rational linear representations?

We consider algebraic groups defined over a field k and containing a maximal torus T which is defined and anisotropic over k and split over a given quadratic extension K of k. We study certain structural features of such groups, and the results obtained are used to investigate the behavior of these groups over special fields.

Conditions are found for which PL-homeomorphic homology spheres are diffeomorphic. It is shown that compact contractible manifolds with diffeomorphic boundaries are diffeomorphic. An analog of the group Γ_n of pseudoisotopies is investigated for compact contractible manifolds.

The basic results of the paper [3] (MR 33 #2798) on endomorphisms of compact groups are extended to endomorphisms of homogeneous spaces of such groups.

In this work we give an expression for the principal term of deviation of periodic functions belonging to the space L_p,

1 ≤ p ≤ ∞,

from their Fejér sums.

In this paper, we compute upper bounds for the best approximation by trigonometric polynomials in the metrics and on the classes of 2-periodic functions such that , where is a given convex modulus of continuity. In doing this, we obtain a series of results which explain certain new properties of differentiable functions expressed in terms of rearrangements. Also, we obtain precise estimates for functionals of the form , where , and belongs to a certain class of differentiable functions defined by bounds on the norm of and its derivatives in or .

A Dirichlet series is associated, through formulas giving its coefficients, with a function analytic in a closed bounded convex domain. It is shown how to recover the function from the coefficients of the associated Dirichlet series.

For the spectral function of the generalized second order boundary problem

and for the function , which may belong to an extremely large class of positive functions that are nonincreasing on , the problem of characterizing the growth of the function as and of the convergence of the integral is connected with the behavior as of the function . The results that are proved in this article were announced by the author in [5] (MR 37 #5456).

We study classes of correct solvability of boundary value problems for systems of linear equations with constant coefficients of the form in the layer with boundary conditions consisting in prescribing certain components of the vectors and for .

Exact bounds are obtained for the derivatives of the solutions of general boundary-value problems for elliptic equations of order 2m in the neighborhood of a power singularity. The order of the singularity is arbitrary, and it can be at an interior point or on the boundary.

The aim of this article is to obtain two-sided estimates for the norm of an element x in a uniformly convex and uniformly flattened Banach space E in terms of l_p-norms of the sequence of coefficients which occur in the expansion of x in a basis .

In this article we prove some theorems about the sequences of coefficients which occur for expansions relative to a basis in a Banach space, and for a certain type of basis in investigated by K. I. Babenko, namely , . As an application of our results, we prove that there exists no universal basis in a separable Hilbert space.

Problems in the control of continuous Markov processes on a semicompactum by two players with conflicting interests are studied. The basic content of the paper is a derivation of Bellman's equations in the case where control is exercised for an infinite time (Theorem 3), and in the case of a problem of optimal stopping (Theorem 6). The results are illustrated by two examples (Theorems 1 and 2).