Table of contents

A simple homotopy equivalence of manifolds splits along a submanifold if it is homotopic to a map that is a simple homotopy equivalence on the transversal preimage of the submanifold and on the complement of this preimage. The problem of splitting along a submanifold with filtration is a natural generalization of this problem. In this paper we define groups of obstructions to splitting along a submanifold with filtration and describe their properties. We apply the results obtained to the problem of the realization of surgery and splitting obstructions by maps of closed manifolds and consider several examples. Bibliography: 36 titles.

Let be a sequence of -subharmonic functions in some domain . Conditions are studied under which the convergence of as a sequence of generalized functions implies its convergence in the Lebesgue spaces . Hörmander studied the case where is a sequence of subharmonic functions and the measure  is the restriction of the Lebesgue measure to a compactum contained in . In this paper a more general case is considered and theorems of two types are obtained. In theorems of the first type it is assumed that . In theorems of the second type it is assumed that the support of the measure is a compactum and . In the second case, is assumed to be the half-plane. Bibliography: 11 titles.

For Borel functions on a perfect normal space and a perfect topological space there are two Baire convergence classifications: one due to Lebesgue and Hausdorff and the other due to Banach. However, neither classification is valid for an arbitrary topological space. In this paper the Baire convergence classification of Borel functions on an arbitrary space is given. This classification of Borel functions uses two classifications of Borel sets: one generalises the Young-Hausdorff classification for a perfect space and the other is new. Bibliography: 17 titles.

The problem of the motion of a dynamically and geometrically symmetric heavy ellipsoid on a smooth horizontal plane is investigated. The problem is integrable and can be considered a generalization of the problem of motion of a heavy rigid body with fixed point in the Lagrangian case. The Smale bifurcation diagrams are constructed. Surgeries of tori are investigated using methods developed by Fomenko and his students. Bibliography: 9 titles.

The Green's function for the de la Vallée-Poussin problem

where , , , and , is investigated. It is defined in the square , and vanishes at the lines , , , ; it is proved that the orders of its zeros have uniform bounds. Bibliography: 27 titles.

Let be a linear uniformly elliptic operator of the second order in , , with bounded measurable real coefficients, that satisfies the weak uniqueness property. The removability of compact subsets of a domain is studied for weak solutions of the equation (in the sense of Krylov and Safonov) in some classes of continuous functions in . In particular, a metric criterion for removability in Hölder classes with small exponent of smoothness is obtained. Bibliography: 20 titles.