Table of contents

A method for constructing the systems of hydrodynamic type having a number of Riemann invariants is indicated. The method is based on a new differential equation in a space of linear operators. The systems of hydrodynamic type naturally connected with the Volterra model and the Toda lattice are constructed. Their continuous limits are found, the equation (see [1] and [2]) of interaction between Riemann breaking waves and transversal long waves is among them. The equations for self-similar solutions of homogeneous systems of hydrodynamic type are derived.

The author constructs mappings between bundles on an elliptic surface and bundles on an elliptic surface that has been modified by a logarithmic transformation. These mappings are applied to the study of the moduli varieties of bundles with c₁=0 on elliptic surfaces.

Theorems are proved on the number of zeros of linear combinations of Dirichlet L-functions that lie on intervals of the critical line.

Effective conditions are found under which planar relaxation systems of a certain class have cycles with one δ-like component and another component close to discontinuous. A complete asymptotics of the cycles is constructed and the question on their stability is solved.

The asymptotics as is found for solutions of the Cauchy problem for a system of equations of surface waves with dissipation.

The author investigates an algorithm for the construction of a polynomial that realizes the simultaneous approximation of the solution of a boundary value problem and its derivatives with the same rapidity of decrease of the error as for the case of the best uniform polynomial approximation of the function on an interval.

The Morse-type index of a compact p-dimensional minimal submanifold is the index of the second variation of the p-dimensional volume functional. In this paper a definition is given for the index of a noncompact minimal submanifold, and the indices of some two-dimensional minimal surfaces in three-dimensional Euclidean space R³ and in three-dimensional Lobachevsky space H³ are computed. In particular, the indices of all the classic minimal surfaces in R³ are computed: the catenoid, Enneper surfaces, Scherk surfaces, Richmond surfaces, and others. The indices of spherical catenoids in H³ are computed, which completes the computation of the indices of catenoids in H³ (hyperbolic and parabolic catenoids have zero index, that is, they are stable). It is also proved that for a one-parameter family of helicoids in H³ the helicoids are stable for certain values of the parameter.

A twistor description of the Weil-Petersson metric on the moduli space of stable vector bundles on a K3-surface with hyper-Kähler structure is given, and this metric is extended to the compactification of the moduli space by torsion-free sheaves.

New classes of Heun's functions that are eigenfunctions of a two-parameter spectral problem with boundary conditions at singular points of Heun's equation are introduced. The asymptotics of the spectrum and of Heun's functions are studied.

It is proved that the extended matrix disk is a domain of holomorphy. This gives a positive answer to a conjecture that is a "compact" analogue of the conjecture on the extended future tube.

The author studies abelian varieties with infinite torsion in infinite extensions L of a number field K for which the Galois group Gal(L/K) is a compact l-adic Lie group.

An algorithm is found that decides, for an arbitrary system of semigroup identities, whether the orders of the finite semigroups satisfying the system are bounded by a function (or a recursive function) of the number of generators.

This paper is concerned with the question of the average value of the function τ(n), the number of divisors of n, for some special sequences of natural numbers.