Table of contents

In this paper, we study the behaviour of harmonic variation in a neighbourhood of a regular point for functions belonging to some multi-dimensional Waterman classes. As a consequence, some new results are obtained on the convergence of multiple trigonometric Fourier series for functions in these classes.

We study the generic fibre functor for finite group schemes over the rings of integers of complete discrete valuation fields. We prove that it is "almost full". Whence we deduce a "finite wild" criterion for good reduction of Abelian varieties.

We study rectifiable curves given by mutually singular coordinate functions in finite-dimensional normed spaces. We describe these curves in terms of the behaviour of approximative tangents and find a simple formula for their lengths. We deduce from these results new necessary and sufficient conditions for the mutual singularity of finitely many functions of bounded variation.

We investigate the properties of stable (and unstable) hypersurfaces with prescribed mean curvature in Euclidean space and establish some necessary and sufficient tests for stability stated in terms of the external geometric structure of the surface. We prove an analogue of a well-known theorem of A. D. Aleksandrov that generalizes the variational property of the sphere and find an exact estimate for the extent of a stable tubular surface of constant mean curvature. Our method is based on an analysis of the first and second variations of area-type functionals for the surfaces under consideration.

We prove that the space of non-singular real three-dimensional cubics has precisely nine connected components. We also study the space of real canonical curves of genus 4 and prove, in particular, that it consists of eight connected components.

We study real Campedelli surfaces up to real deformations and exhibit examples of such surfaces which are equivariantly diffeomorphic but not real deformation equivalent (DIFDEF).

We consider approximation by convex sets in the space of continuous maps from a compact topological space to a locally convex space with respect to certain asymmetric seminorms. We suggest new criteria for elements of least deviation, make a definition of strongly unique elements of least deviation and study the problems of characterization and existence of such elements. The most detailed study concerns the approximation with a sign-sensitive weight of real-valued continuous functions defined on a compact metric space or on a line segment by elements of the Chebyshev space.

In the class , , of all functions that are analytic in the unit disc  and such that in , we obtain asymptotic estimates for the coefficients for small and sufficiently large . We suggest an algorithm for determining those for which the canonical functions provide the local maximum of  in . We describe the set of functionals for which the canonical functions provide the maximum of  in  for small and large values of . The proofs are based on optimization methods for solutions of control systems of differential equations.