Table of contents

In this paper we consider questions of whether a compact space can be embedded in a Euclidean space. The problem of embedding an '-like' compact space in is solved affirmatively under certain restrictions on the dimension . We clarify the relations between the realization problem and areas of homotopy theory and differential topology.

Approximation by the rectangular Fourier sums is studied for classes of functions of several variables defined in terms of the orders of decrease of the mixed moduli of smoothness. The problem of the existence of a single (independent of ) function on which the order of the approximation in the corresponding class is realized is solved.

The structure of ideals of a cone P in a group G is studied. The case when there is a single completely prime ideal in P is studied in full. Corresponding examples are constructed using the universal covering group of the special linear group of 2×2-matrices over .

It is that a three-dimensional variety that is a conic bundle in the Mori sense has a base with at most double rational singularities of type . A rationality criterion is proved subject to this assumption in the case when the discriminant curve is large enough, for example, for the case when .

A proof is given of a theorem stating that there is a correspondence between the irreducible complex representations of a finite p-group and the irreducible representations of its associated nilpotent Lie algebra over a field of characteristic p. As a corollary it is found that the sets of degrees of the irreducible representations are the same.

An incidence system consisting of points and lines is called an -partial geometry of order if each line contains points, each point lies on lines (the lines intersect in at most one point), and for any point a not lying on a line there are exactly lines passing through and intersecting (this geometry is denoted by ). The point graph of the partial geometry is strongly regular with parameters: , , and . A graph with the indicated parameters is called a pseudogeometric graph of the corresponding geometry. It is proved that a pseudogeometric graph of a partial geometry in which the -subgraphs are regular graphs without triangles is the triangular graph , the quotient of the Johnson graph , or the McLaughlin graph.

Questions on the differentiability of integrals of functions in the Hölder classes with respect to fairly general bases (including the basis of convex sets) are considered.

A problem of Ehrenpreis on factorization in the convolution algebra of smooth functions with compact support is considered. It was proved at the beginning of the 1980s that not every smooth function with compact support in () can be represented as a convolution of two smooth functions with compact support. Dickson proved that a smooth function of one variable with compact support can be represented as a convolution of two smooth functions with compact support if all the zeros of the Fourier-Laplace transform of this function are located in some horizontal strip and

It is proved in the present paper that the factorization is possible if all the zeros of the Fourier-Laplace transform are located in a domain of the following form: