Table of contents

We extend the definition of Hurwitz numbers to the case of seamed surfaces, which arise in new models of mathematical physics, and prove that they form a system of correlators for a Klein topological field theory in the sense defined in [1]. We find the corresponding Cardy-Frobenius algebras, which yield a method for calculating the Hurwitz numbers. As a by-product, we prove that the vector space generated by the bipartite graphs with edges possesses a natural binary operation that makes this space into a non-commutative Frobenius algebra isomorphic to the algebra of intertwining operators for a representation of the symmetric group on the space generated by the set of all partitions of a set of elements.

Necessary and sufficient conditions are found for the solubility of difference inclusions in the space of vector sequences determined by a linear relation (a multi-valued linear operator) on a Banach space.

The paper deals with the spectrum of a periodic self-adjoint differential operator on the real axis perturbed by a small localized non-self-adjoint operator. We show that the continuous spectrum does not depend on the perturbation, the residual spectrum is empty, and the point spectrum has no finite accumulation points. We study the problem of the existence of eigenvalues embedded in the continuous spectrum, obtain necessary and sufficient conditions for the existence of eigenvalues, construct asymptotic expansions of the eigenvalues and corresponding eigenfunctions and consider some examples.

The bijections of associativity and commutativity arise from symmetries of the Littlewood-Richardson coefficients. We define these bijections in terms of arrays and show that they coincide with analogous bijections defined in terms of discretely concave functions using the octahedron recurrence as well as with bijections defined in terms of Young tableaux. The main ingredient in the proof of their coincidence is a functional version of the Robinson-Schensted-Knuth correspondence.

This paper deals with the operator inclusion , where is a multi-valued map of monotonic type from a reflexive space to its conjugate and is the cone normal to the closed set , which, generally speaking, is not convex. To estimate the number of solutions of this inclusion we introduce topological characteristics of multi-valued maps and Lipschitzian functionals that have the properties of additivity and homotopy invariance. We prove some infinite-dimensional versions of the Poincaré-Hopf theorem.

We obtain a joint universality theorem of Voronin type for systems of periodic Hurwitz zeta-functions with parameters such that the system , is linearly independent over the field of rational numbers.

We use the `graph' method to obtain estimates for the derivatives of any order of inverse functions in terms of those of the original functions. We construct explicit asymptotics of the estimates obtained as the order of the derivative tends to infinity. For analytic functions and functions in Gevrey's class, we obtain explicit estimates for all derivatives of the inverse functions.

We consider the problem of constructing asymptotically exact (for ) uniform (with respect to parameters ) estimates for oscillatory integrals containing a large parameter . We suggest a possible multidimensional analogue of Vinogradov's well-known estimate for one-dimensional integrals. Based on this suggestion, we estimate the integrals with singularities of type , (in Arnold's classification) and use the special case of to discuss the possibility of generalizing our results.

If the Hodge conjecture holds for some generic (in the sense of Weil) geometric fibre of an Abelian scheme over a smooth projective curve , then numerical equivalence of algebraic cycles on coincides with homological equivalence. The Hodge conjecture for all complex Abelian varieties is equivalent to the standard conjecture of Lefschetz type on the algebraicity of the Hodge operator for all Abelian schemes over smooth projective curves. We investigate some properties of the Gauss-Manin connection and Hodge bundles associated with Abelian schemes over smooth projective curves, with applications to the conjectures of Hodge and Tate.