Table of contents

The following assertion is proved. Let be an automorphism of a Lebesgue space , preserving the (finite or infinite) measure , and let , , be actions of a countable amenable group by automorphisms on , such that , where is the normalizer of the full group . For the existence of an automorphism such that (the outer conjugacy of the actions and ), where , , it is necessary and sufficient that

The proof uses properties of cocycles of approximable groups of automorphisms.

Bibliography: 25 titles.

On every compact symmetric space, Riemannian metrics are constructed for which the geodesic flow is completely integrable in the sense of Liouville. Some of the metrics constructed in the case of a sphere are conformally equivalent to metrics on n-axial ellipsoids.

Bibliography: 22 titles.

Duality methods in the theory of averaging of nonlinear variational problems are developed. The questions of a general nature that are discussed include a detailed analysis of the concept of regularity, an example of a nonregular Lagrangian, and the derivation of duality formulas that take account of the regularity problem. The main content is concerned with the averaging of variational problems with stochastic Lagrangians. Three groups of questions are investigated: 1) averaging of Lagrangians of a general form; 2) averaging of the Lagrangians of plasticity (the theory of the limit load); and 3) averaging of degenerate Lagrangians (problems with random soft or rigid inclusions).

Bibliography: 13 titles.

A new metric with absolutely convex balls is introduced on a metrizable locally convex space. Necessary and sufficient conditions are given for all closed hypersubspaces and all nonnormable closed subspaces of a Fréchet space to be proximal, i.e., to have the property that there exist elements of best approximation with respect to this metric. In particular, these conditions are expressed in terms of the topologies of the original space and the strong dual space. It is proved that the Fréchet spaces have the proximality property, where B is a reflexive Banach space and is the nuclear Fréchet space of all numerical sequences. Questions of Albinus and Wriedt are answered.

Bibliography: 23 titles.

The author introduces a notion of the minimal object of a field K finitely generated over C which generalizes the notion of a minimal model of K. In the case of an algebraic surface not isomorphic to a rational or ruled surface, it coincides with the minimal model. The minimal objects of three-dimensional algebraic spaces are investigated.

Bibliography: 6 titles

The Cauchy problem with rapidly oscillating initial conditions is investigated for the class of nonstrictly hyperbolic equations with nonsmooth characteristics. The domain of applicability of the Maslov canonical operator method is determined.

Bibliography: 12 titles.

It is proved that for a complete discrete valuation ring of zero characteristic with residue field of positive characteristic and maximal ideal , the natural homomorphism of -groups with coefficients

is an isomorphism for all positive and .

For the ring of integers in a local field , the groups are finite.

Bibliography: 13 titles.

Estimates are obtained for the resolvents of Toeplitz operators under certain restrictions on their symbols. Conditions are found for the existence of nontrivial invariant subspaces and for a Toeplitz operator to be similar to a unitary operator. A theorem on inclusion of spectra is obtained for Toeplitz operators with unimodular symbols.

Bibliography: 26 titles.

It is shown, using the technique of switching toral subalgebras, that in finitedimensional Lie -algebras every Cartan subalgebra with maximal toral part has dimension equal to the rank of the algebra. As is known, every Cartan subalgebra of a Lie -algebra  is of the form , where is the nilspace of the endomorphism , . It is proved that there exists a Zariski-open subset  such that for every  the subspace is a Cartan subalgebra with maximal toral part. A further result is the proof that the class of Cartan subalgebras with maximal toral part is the same as the class of Cartan subalgebras with minimal nilpotent part. The results are used to settle a question concerning anisotropic forms of Lie algebras over finite fields.

Bibliography: 12 titles.

Dynamical systems with new spectral properties are constructed using approximation theory. It is proved that these properties are generic (in a metric and topological sense) and realized within the class of smooth systems preserving a smooth measure.

Bibliography: 21 titles.

Both local and global contour-solid theorems for holomorphic functions of a complex variable are presented, with proofs, under essentially weakened restrictions on the functions, for arbitrary bilogarithmically concave (in a generalized sense) majorants, in arbitrary open sets. Some properties of subharmonic functions that are used to prove the contour-solid results are also established.

Bibliography: 16 titles.

Let G be a compact Lie group, and A a C*-algebra with identity. A K-theory of G-equivariant A-vector bundles is developed along with a corresponding theory of Fredholm operators, and the analytic and topological indices of an elliptic equivariant pseudodifferential operator over a C*-algebra A are defined. An index theorem generalizing the Mishchenko–Fomenko theorem is proved.

Bibliography: 19 titles.

It is said that the Hartogs phenomenon occurs for a complex manifold if every holomorphic mapping of a domain over into extends to a holomorphic mapping of the envelope of holomorphy into . In this paper it is proved that a holomorphically convex Kähler manifold exhibits the Hartogs phenomenon if and only if contains no rational curves.

Bibliography: 10 titles.