Table of contents

The paper is concerned with the new spaces of functions of smoothness zero defined on the -dimensional Euclidean space  or on a subdomain of . These spaces are compared with the spaces and . The embedding theorems for Sobolev spaces are refined in terms of the space with the limiting exponent.

Bibliography: 8 titles.

The difference scheme which approximates the equations of large-scale ocean dynamics in a unit cube to the second degree in the space variables is investigated. It is shown that the solutions converge to the solution of the differential problem. Namely, under the assumption that the solution is sufficiently smooth it is proved that

where is the grid -norm with respect to the space variables, is the solution of the grid problem, and is the solution of the differential problem.

Bibliography: 7 titles.

We prove a generalization of Bertrand's theorem to the case of abstract surfaces of revolution that have no 'equators'. We prove a criterion for exactly two central potentials to exist on this type of surface (up to an additive and a multiplicative constant) for which all bounded orbits are closed and there is a bounded nonsingular noncircular orbit. We prove a criterion for the existence of exactly one such potential. We study the geometry and classification of the corresponding surfaces with the aforementioned pair of potentials (gravitational and oscillatory) or unique potential (oscillatory). We show that potentials of the required form do not exist on surfaces that do not belong to any of the classes described.

Bibliography: 33 titles.

A test for the convergence of the generalized spherical and Bochner-Riesz means in the Hardy spaces , , is obtained, where is the unit polydisc. Precise orders of the approximation of functions by the generalized Bochner-Riesz means in terms of the -functional and special moduli of smoothness are found.

Bibliography: 31 titles.

Boundary value and spectral problems for an elliptic differential equation with rapidly oscillating coefficients in a thin perforated region with rapidly changing thickness are investigated. Descriptions of asymptotic algorithms for solutions of such problems in thin regions with different limiting dimensions are combined. For a mixed inhomogeneous boundary value problem a corrector is constructed and an asymptotic estimate in the corresponding Sobolev space is established. Asymptotic bounds for eigenvalues and eigenfunctions of the Neumann spectral problems are also found. Full asymptotic expansions for the eigenvalues and eigenfunctions are constructed under certain symmetry assumptions about the structure of the thin perforated region and the coefficients of the equations.

Bibliography: 21 titles.

Graph-links arise as the intersection graphs of turning chord diagrams of links. Speaking informally, graph-links provide a combinatorial description of links up to mutations. Many link invariants can be reformulated in the language of graph-links. Khovanov homology, a well-known and useful knot invariant, is defined for graph-links in this paper (in the case of the ground field of characteristic two).

Bibliography: 14 titles.

The Lindenbaum algebra of Peano PA can be enriched by the -consistency operators which assign, to a given formula, the statement that the formula is compatible with the theory PA extended by the set of all true -sentences. In the Lindenbaum algebra of PA, a lower semilattice is generated from  by the -consistency operators. We prove the undecidability of the elementary theory of this semilattice and the decidability of the elementary theory of the subsemilattice (of this semilattice) generated by the 0-consistency and -consistency operators only.

Bibliography: 16 titles.