Table of contents

We obtain necessary and sufficient conditions for the invertibility of the difference operator , , , whose domain is given by the condition , where , , is the Banach space of sequences (of vectors in a Banach space ) summable with weight for and bounded with respect to  for ,  is a bounded linear operator, and is a closed -invariant subspace of . We give applications to the invertibility of differential operators with an unbounded operator coefficient (the generator of a strongly continuous operator semigroup) in weight spaces of functions.

For an equation of hyperbolic type in the plane or one of its third-order analogues in three-dimensional space, we solve boundary-value problems in special classes introduced by the authors on domains containing two lines (planes) of singularity of the coefficients of these equations under conjugation conditions on a characteristic line (plane).

We consider an equation of internal gravitational waves in a stratified fluid in which non-linear sources of a general form are taken into account. In the case of the corresponding initial-boundary-value problem in a bounded domain with homogeneous Dirichlet conditions on the boundary, we prove the local-in-time solubility and also obtain sufficient conditions for the blow-up of a solution in finite time.

We introduce a natural structure of a semigroup (isomorphic to the factorization semigroup of the identity in the symmetric group) on the set of irreducible components of the Hurwitz space of coverings of marked degree  of  of fixed ramification types. We shall prove that this semigroup is finitely presented. We study the problem of when collections of ramification types uniquely determine the corresponding irreducible components of the Hurwitz space. In particular, we give a complete description of the set of irreducible components of the Hurwitz space of three-sheeted coverings of the projective line.

We develop harmonic analysis in certain categories of filtered Abelian groups and vector spaces. The objects of these categories include local fields and adelic spaces arising from arithmetic surfaces. We prove some structure theorems for quotients of the adèle groups of algebraic and arithmetic surfaces.

We study the question of whether the topological quotient of a real linear representation of a simple three-dimensional compact Lie group is a manifold. We obtain an upper bound for the dimension of a representation whose quotient is a manifold, and examine most of the remaining cases.

We prove results on exact asymptotics as  for the expectations and probabilities , where is a sequence of independent identically Laplace-distributed random variables, , , is the corresponding random walk on , is a positive continuous function satisfying certain conditions, and , , are fixed numbers. Our results are obtained using a new method which is developed in this paper: the Laplace method for the occupation time of discrete-time Markov chains. For  one can take , , , , or , , , for example. We give a detailed treatment of the case when using Bessel functions to make explicit calculations.

Errata to the paper by M A Korolev 'On large distances between consecutive zeros of the Riemann zeta-function' Izv. Ross. Akad. Nauk Ser. Mat. 72:2, 2008, 91-104. English transl.: Izv. Math., 72, 2008, 291-304.