Table of contents

The concept of weak generalized localization almost everywhere is introduced. For the multiple Fourier series of a function , weak generalized localization almost everywhere holds on the set ( is an arbitrary set of positive measure ) if the condition , , on implies that the indicated series converges almost everywhere on some subset of positive measure. For a large class of sets , , a number of propositions are proved showing that weak localization of rectangular sums holds on the set in the classes , , if and only if the set has certain specific properties. In the course of the proof the precise geometry and structure of the subset of on which the multiple Fourier series converges almost everywhere to zero are determined.

Bibliography: 13 titles.

Let be a finite extension of the field of -adic numbers and the field of Laurent series for which the are bounded in the norm of and as . In the -dimensional local field a pairing is given in explicit form between the completed Milnor -functor and the multiplicative group with values in the group of th roots of unity.

Bibliography: 14 titles.

This paper establishes isoperimetric inequalities, in the language of the theory of parametrized multivarifolds, for the class of parametrized multidimensional films in a Euclidean space.

Bibliography: 5 titles.

The following theorem is proved. If H≥T^a, where T>T₀>0 and a>27/82, then for 1/2<σ≤1 the estimate

holds uniformly in σ, where N(σ₁,t) denotes the number of zeros s=σ+it, with σ>σ₁ and 0<t<T, of the Riemann zeta-function ζ(s).

For hypersurface singularities f=0, certain rationality conditions are formulated in terms of the Newton diagram of f and the initial terms of a series expansion of f. A classification of compound Du Val singular points of three-dimensional hypersurfaces (cDV-singularities of Reid) is given. A method is indicated for calculating normal forms of equations of those singular points. The method is based on the spectral sequence of the two-term upper Koszul complex of f with the Newton filtration, which generalizes Arnol'd's spectral sequence for the reduction of functions to normal form. Examples of applications of the method are given.

Bibliography: 6 titles.

This paper establishes best possible conditions, on the degree of approximation of functions in () by rational functions, that guarantee that the function has a th mean differential of order everywhere except on a set of zero Hausdorff () measure ().

Bibliography: 11 titles.

Suppose is a maximal order of a totally real algebraic number field ; is a totally positive quadratic form over ; and are ideals of the ring ; ; and . The author proves an asymptotic formula for the number of solutions of the system in numbers . The proof is based on a discrete ergodic method.

Bibliography: 19 titles.

Complex Green functions for the Laplace operator on the background of general Yang-Mills fields are interpreted in terms of cohomology in the space of complex light lines by means of the Penrose-Ward transformation.

Bibliography: 15 titles.

On the basis of homological properties of the homogeneous coordinate ring R(V) of an n-dimensional projective Fano variety V over the field C estimates are obtained for the degrees of generators of the algebra R(V) and the ideal I(V). All possible values are found for the dimension and the degree of a variety V of codimension e in P^N which is not a complete intersection. We give a description of multidimensional Fano varieties of codimension 4 in P^N whose linear sections are canonical curves of genus 6.

Bibliography: 12 titles.