Table of contents

It is shown that the generating functions of modular graphs satisfy Burgers's equations, which enable one to obtain in a unified way the generating functions for the virtual Euler characteristic and the Poincaré polynomial of the moduli space of punctured curves and for the number (with weights ) of modular graphs of a definite type.

We consider symmetric flows of a viscous compressible barotropic fluid with free boundary driven by a general mass force  (depending on both the Eulerian and the Lagrangian coordinates) and an outer pressure , for a general monotone state function . The case of self-gravitation arising in astrophysics is covered. Studied first are the existence, the uniqueness, and the static stability of positive stationary solutions; a variational study of these solutions and their static stability in terms of potential energy is presented. In the astrophysical context it is proved that the stationary solution is unique and statically stable, provided that the first adiabatic exponent is at least  4/3. Next, in the case when the -limit set for the non-stationary density and free boundary contains a statically stable positive stationary solution a uniform stabilization to this solution is deduced and, as the main result, stabilization-rate bounds of exponential type as in   and  for the density and the velocity are established by constructing new non-trivial Lyapunov functionals for the problem. Moreover, it is proved that statically stable stationary solutions are exponentially asymptotically stable, and this non-linear dynamic stability is in addition stable with respect to small non-stationary perturbations of  and . A variational condition for the stationary solution is also introduced, which ensures global (with respect to the data) dynamic stability. The study is accomplished in the Eulerian coordinates and in the Lagrangian mass coordinates alike.

Approximation using linear combinations of exponentials with special constraints on the coefficients is investigated. A sufficient condition for such approximation is stated in simple geometric terms.

The Hermite-Padé approximants with common denominator are considered for a pair of Stieltjes functions with weights and , where , . On the basis of the method of the Riemann-Hilbert matrix problem the strong asymptotics of these approximants are found in the case . The limiting distribution of the zeros of the denominators of the Hermite-Padé approximants is shown to be equal to the equilibrium measure of a certain Nikishin system.

A trace formula is obtained for unbounded discrete operators perturbed by a Hilbert-Schmidt operator; this formula may be called the discrete analogue of M. Krein's formula for nuclear perturbations. A regularized trace formula of Krein's type is also proved for perturbations in the class , , for arbitrary compact and relatively compact perturbations depending on the behaviour at infinity of the distribution function of the spectrum of the unperturbed operator.