Table of contents

In this article the author studies the boundary behavior of a one-dimensional complex analytic set A in a neighborhood of a totally real manifold M in Cⁿ with smoothness greater than 1. He proves that the limit points of A on M form a set of locally finite length and that near almost every limit point the closure of A is either a manifold with boundary (with smoothness corresponding to M) or a union of two manifolds with boundary. He investigates the structure of the tangent cone to A at the limit points and proves a theorem concerning the boundary regularity of holomorphic discs "glued" to M. Bibliography: 22 titles.

Let the integer and the modulus of continuity be fixed, and let be the class of all functions continuous on the closed unit disk , analytic on its interior , and having an -continuous th derivative on . Consider for each and each fixed the polynomial in

(the st partial sum of the Taylor series of in a neighborhood of ). Then for any two points

(1.1)

Let be a closed subset of . This article contains a solution of the problem of free interpolation in , formulated as follows: find necessary and sufficient conditions on such that for each collection of th-degree polynomials satisfying conditions of the type (1.1) for all there is a function with . Bibliography: 13 titles.

This paper considers a boundary value problem for a quasilinear parabolic equation. In terms of Sobolev and Besov spaces the author determines a solution space  and a space  of initial conditions and right hand members such that the operator corresponding to the boundary value problem is a diffeomorphism, analytic in the Frechet sense, of the whole space  and a domain  in the space . The behavior of the inverse operator of the problem around the boundary of  is studied, and it is shown that for different problems the domain  can coincide with the whole function space or be a strict subset of it. Bibliography: 13 titles.

Using a rather general theorem on -functions proved in this paper, the author establishes the existence of an effective upper bound for the solutions of certain Diophantine equations, such as those of the form

where and are natural numbers and is a polynomial of small degree. The upper bound has the form

where depends on and and can be written out explicitly, and is an effective positive constant. Bibliography: 17 titles.

A set E with 0 < meas E < +∞ is constructed for which the Fourier transform of its characteristic function vanishes on an interval. The set is the union of a sequence of intervals whose lengths can be estimated asymptotically above and below. The construction is based on an infinite-dimensional version of the implicit function theorem. Bibiography: 6 titles.

For a function defined by a Dirichlet series that converges in a right half-plane, we introduce the -order in the half-plane and the -order in a half-strip . Under certain restrictions on the width of the half-strip, we obtain the inequalities , where is defined by a sequence of powers. The two extreme inequalities are sharp. Bibliography: 3 titles.