Table of contents

The asymptotic behaviour of the spectrum of a self-adjoint second-order differential operator on the axis is investigated. The coefficients of this operator depend on rapid and slow variables and are periodic in the rapid variable. The period of oscillations in the rapid variable is a small parameter. The dependence of the coefficients on the rapid variable is localized, and they stop depending on it at infinity. Asymptotic expansions for the eigenvalues and the eigenfunctions of the operator in question are constructed. It is shown that, apart from eigenvalues convergent to eigenvalues of the homogenized operator as the small parameter converges to zero, the perturbed operator can also have an eigenvalue convergent to the boundary of the continuous spectrum. Necessary and sufficient conditions for the existence of such an eigenvalue are obtained.

Bibliography: 22 titles.

Let be the generator of a uniformly bounded -semigroup in a Banach space  such that has a trivial kernel and a dense range. The question whether is a generator of a -semigroup is considered. It is shown that the answer is negative in general for , . In the case when is a Hilbert space it is proved that there exist -semigroups , , of arbitrarily slow growth at infinity such that the densely defined operator  is not the generator of a -semigroup.

Bibliography: 19 titles.

The problem under consideration is the estimate of the length of the lemniscate

where

It is shown that . A sharp estimate for the variation of a rational function along a curve of bounded rotation of the secant is also obtained.

Bibliography: 15 titles.

All the Fomenko-Zieschang invariants are calculated in the Kovalevskaya-Yehia integrable case of the problem of the motion of a heavy gyrostat in a gravitational field; the topological classification of the non-degenerate equilibria of this system is obtained and the loop molecules of all the singularities are calculated.

Bibliography: 10 titles.

A new construction of symmetric non-commutative signature of non-simply-connected topological manifolds is proposed based on the natural definition of homology and cohomology of a topological manifold using the singular chain and cochain complexes.

Bibliography: 5 titles.

It is proved that a nodal quartic threefold X containing no planes is -factorial if it has at most 12 singular points. An exception here is a quartic with precisely 12 singularities containing a quadric surface. Some geometric constructions relating to such a quartic are presented.

Bibliography: 14 titles.

Let be a (complex) Radon measure in  with compact support and finite variation and let

be the maximal Cauchy integral. Estimates for the Hausdorff -content of the set are obtained, where is a measuring function and  is a fixed positive number. These estimates are shown to be sharp up to the values of the absolute constants involved. A similar problem is also considered for potentials with arbitrary real non-increasing kernels of positive measure in , . As an application of the so-developed machinery, results on connections between the analytic capacity and the Hausdorff measure are obtained (for instance, an analogue of Frostman's theorem on classical capacities).

Bibliography: 37 titles.