Table of contents

Ensembles of random Hermitian matrices with a distribution measure defined by an anharmonic potential perturbed by an external source are considered. The limiting characteristics of the eigenvalue distribution of the matrices in these ensembles are related to the asymptotic behaviour of a certain system of multiple orthogonal polynomials. Strong asymptotic formulae are derived for this system. As a consequence, for matrices in this ensemble the limit mean eigenvalue density is found, and a variational principle is proposed to characterize this density.

Bibliography: 35 titles.

We study Fano threefolds birationally equivalent to a quartic containing a plane. We prove that linear systems that have no maximal singularities at a singular point of the variety can have maximal singularities only along curves of degree one. We construct corresponding birational automorphisms.

Bibliography: 10 titles.

This paper looks at ordinary differential equations (DE) containing the derivative of the unknown functions with respect to a measure  which is continuous with respect to the Lebesgue measure. It is shown that the Cauchy problem for a linear normal system of DE with a -derivative is uniquely solvable. A necessary and sufficient condition is obtained for the solvability of an equation of Riccati type with a -derivative. It is related to a boundary-value problem for a linear system of DE. Using this condition a necessary and sufficient condition is obtained for a Volterra factorization to exist for linear operators that differ from the identity by an integral operator that is completely continuous in the space , .

Bibliography: 12 titles.

All complete Ricci-flat Kähler -invariant metrics on the cotangent bundle of a compact rank-one symmetric space , (with the fixed Kähler form, the canonical symplectic structure ), are classified. It is proved that the set of equivalence classes of such metrics can be parametrized by positive numbers. The representative of each class is constructed by using explicit expressions. An alternative description of these structures based on the Kähler reduction procedure is proposed. We show also that the complete Ricci-flat Kähler metrics, constructed by Stenzel, are diffeomorphic to these ones.

Bibliography: 26 titles.

Estimates of the best -approximation of functions by polynomials in an affine system (system of dilations and translations), which are similar to well-known estimates due to Ul'yanov and Golubov for approximations in the Haar system, are obtained. An analogue of A.F. Timan and M.F. Timan's inequality is shown to hold under certain conditions on the generating function of the affine system; this analogue fails for the Haar system for .

Bibliography: 10 titles.