Table of contents

A solution of the local Pompeiu problem is obtained for functions with vanishing integrals over parallelepipeds lying in a fixed ball.

The problem of birational rigidity is investigated for smooth Mori fibre spaces representable as a pencil of del Pezzo surfaces of degree 1 or 2.

The present paper is devoted to the algorithmic construction of diffeomorphisms establishing the equivalence of geometric structures. For structures of finite type the problem reduces to integration of completely integrable distributions with a known symmetry algebra and further to integration of Maurer-Cartan forms. We develop algorithms that reduce this problem to integration of closed 1-forms and equations of Lie type that are characterized by the fact that they admit a non-linear superposition principle. As an application we consider the problem of constructive equivalence for the structures of absolute parallelism and for the transitive Lie algebras of vector fields on manifolds.

Ten series of matrix integrals (over non-compact Riemannian symmetric spaces) imitating the standard beta function are constructed. This is a broad generalization of Hua Loo Keng's integrals (1958) and Gindikin's B-integrals (1964). As a consequence Plancherel's formula for the Berezin kernel representations of classical groups is obtained in explicit form. Matrix models of non-compact Riemannian symmetric spaces are also discussed.

Let , , be a three-dimensional complex vector space. For the natural linear representation of the group in the space the orbits are classified and generators of the algebra of invariants are described.

A class of pseudodifferential operators in a subdomain of that is well adapted to the transfer to manifolds with (intersecting) edges of various dimensions is considered. A version of symbolic calculus is discussed. The operators in question act in Sobolev spaces with weighted norms. Stratified manifolds (with edges as strata) are introduced and pseudodifferential operators on such manifolds are defined.

Assume that , and let be an integer. Let , where the are points in the interval . The classes and are introduced. These consist of functions with absolutely continuous th derivative on such that their th and th derivatives satisfy certain conditions outside the set . It is proved that for the Fourier-Legendre sums realize the best approximation in the classes . Using the Fourier-Legendre expansions, polynomials of order are constructed that possess the following property: for the th derivative of the polynomial approximates on to within , and the accuracy is of order outside .