Table of contents

A parabolic equation is exhibited for which Conjecture A of the authors' recent paper (MR 42 #8087) is false. Bibliography: 4 items.

Pseudodifferential operators with symbols satisfying

(1)

for all multi-indices , , where and , are considered. For operators of this class a priori estimates (in part as well as all of the variables) are established. Necessary and sufficient conditions are found for some classes of pseudodifferential operators with symbols satisfying (1) to have the Fredholm property. Bibliography: 11 items.

In this paper we consider groups of the form , where is a verbal subgroup of a normal divisor of a group , and is either free or the free product of certain groups. In the latter case we assume that is contained in the Cartesian subgroup. We prove that the factors of the lower central series of are torsion-free or even free abelian if the corresponding property is possessed by the factors of the lower central series of and . Bibliography: 7 items.

In this paper elliptic problems in exterior domains polynomially depending on a spectral parameter are considered. These problems are obtained from a mixed problem for hyperbolic equations by substituting for . For such elliptic problems analytic properties of the resolvent are studied in the neighborhood of the point , which permits, for the corresponding nonstationary problem, a complete asymptotic expansion of solutions as . Bibliography: 11 items.

In this paper some questions concerning the removal of singular sets for quasiconformal mappings are considered. Unlike previously existing results, in which it was required that the mapping be a homeomorphism or that the capacity of the singular points be zero, in this paper the restriction is weaker: the Hausdorff measure of the singular points is less than

n – 1

. A series of examples is given which show how to construct a set of singular points. In addition, theorems on removable singular sets are proved in which the quasiconformal mapping always has a continuous extension. In particular, the principle of symmetry for quasiconformal mappings is proved. Bibliography: 16 items.

In this paper the question is considered of uniform approximation by polynomials and smooth functions of uniformly continuous functions given on the unit ball of countable-dimensional Hilbert space. Bibliography: 8 items.

A class of so-called special polyhedra is defined for each . The following theorems are proved: 1. Every piecewise linear manifold with boundary can be collapsed to some -dimensional special polyhedron. 2. The manifold is uniquely determined by this special polyhedron. 3. If , then any special polyhedron can be thickened to an -dimensional manifold. The author also gives applications of the results obtained to a series of questions connected with the Zeeman conjecture about the collapsibility of , where is a contractible polyhedron. Figures: 4. Bibliography: 6 items.

An analog of the usual integral on a group is considered for the case when the domain of values is an abelian topological group. Necessary and sufficient conditions are obtained for the existence of the integral on a compact group with values in a locally compact abelian group. Bibliography: 4 items.

In this work the author investigates the relationship between the homology of an inverse limit of spaces and the limit of the homology of the spaces (for an exact homology theory), and also the relationship of the cohomology of limits of certain pairs of spaces to the limit of the cohomology of the pairs. Applications of the results of these investigations are presented. Bibliography: 11 items.

This paper considers the Banach algebra with the usual norm and convolution as multiplication. A characterization is given for closed ideals of which are rotation invariant and have as spectrum, in terms of annihilators of certain collections of pseudomeasures. The main result of the paper is connected with a construction which yields an uncountable chain of closed ideals intermediate between neighboring invariant closed ideals with spectrum . This construction associates an ideal with a closed subset . It is shown that if then . Another result is the lack of a continuous projection from the largest to the smallest ideal when , and when , from an invariant ideal onto the neighboring smaller invariant ideal. A certain algebra of functions on the sphere which arises naturally in the construction of the intermediate ideals is also studied. Bibliography: 18 items.