Table of contents

We construct a mirror-type correspondence that assigns variations (that is, local systems, -modules or -adic sheaves) to pairs , where  is a variety and  is a complex of densely filtered vector bundles over . We consider Calabi-Yau complete intersections in projective spaces. In the particular case when the complex is quasi-isomorphic to the tangent bundle on a generic Calabi-Yau complete intersection, this construction yields the variation that arises in the relative cohomology of the mirror-dual pencil. We call it the Riemann-Roch variation. The Riemann-Roch data of the divisorial sublattice in the -group can be read off the Riemann-Roch local system since it encodes the information about the Euler characteristics of all  sheaves (in an essentially non-commutative way).

We consider a two-dimensional model Schrödinger equation with logarithmic integral non-linearity. We find asymptotic expansions for its solutions (Airy polarons) that decay exponentially at the "semi-infinity" and oscillate along one direction. These solutions may be regarded as new special functions, which are somewhat similar to the Airy function. We use them to construct global asymptotic solutions of Schrödinger equations with a small parameter and with integral non-linearity of Hartree type.

For functions in the Hardy-Sobolev class , which is defined as the set of functions analytic in the unit disc and satisfying , we construct best quadrature formulae that use the values of the functions and their derivatives on a given system of points in the interval . For the periodic Hardy-Sobolev class , which is defined as the set of -periodic functions analytic in the strip and satisfying , we prove that the rectangle rule is the best for an equidistant system of points, and we calculate the error in this formula. We construct best quadrature formulae on the class , which is defined similarly to , except that the boundary values of functions are taken in the -norm. We also construct an optimal method for recovering functions in  from the Taylor information .

We construct new acyclic resolutions of quasicoherent sheaves. These resolutions are connected with multidimensional local fields. The resolutions obtained are applied to construct a generalization of the Krichever map to algebraic varieties of any dimension.

This map canonically produces two -subspaces and from the following data: an arbitrary algebraic -dimensional Cohen-Macaulay projective integral scheme  over a field , a flag of closed integral subschemes such that  is an ample Cartier divisor on  and  is a smooth point on all , formal local parameters of this flag at the point , a rank  vector bundle  on , and a trivialization of  in the formal neighbourhood of the point  where the -dimensional local field is associated with the flag . In addition, the map constructed is injective, that is, one can uniquely reconstruct all the original geometric data. Moreover, given the subspace , we can explicitly write down a complex which calculates the cohomology of the sheaf  on  and, given the subspace , we can explicitly write down a complex which calculates the cohomology of  on .

We show that one-dimensional semilinear second-order parabolic equations have finite-dimensional dynamics on attractors. In particular, this is true for reaction-diffusion equations with convection on (0,1).

We obtain new topological criteria for a class of dissipative equations of parabolic type in Banach spaces to have finite-dimensional dynamics on invariant compact sets. The dynamics of these equations on an attractor  is finite-dimensional (can be described by an ordinary differential equation) if  can be embedded in a finite-dimensional -submanifold of the phase space.

We consider the asymptotic behavior of the real spectrum of the indefinite Sturm-Liouville problem for large values of the spectral parameter and prove the existence of infinitely many asymptotic terms under the condition that the coefficients of the equation are analytic.

The results of the first part of this work (see [1]) are used only in §7 of this paper, from which subsequent results follow. We pose new dual problems for weight spaces of holomorphic functions of one and several variables defined on a domain in , namely, the problem of non-triviality of a given space, description of zero sets, description of sets of (non-)uniqueness, existence of holomorphic functions of certain classes that "suppress" the growth of a given holomorphic function, and representation of meromorphic functions as quotients of holomorphic functions contained in a given space.

In this paper we prove that any T-space over a field of characteristic zero has a finite basis. This result generalizes Kemer's theorem on the existence of a finite basis for any system of associative identities over a field of characteristic zero.