Vol 211, No 1 (2020)

Billiards are considered in compact domains on a Minkowski plane whose boundary consists of arcs of confocal quadrics with angles at corner points $\le\pi/2$. A classification is obtained for these billiards, called simple billiards. The first integrals and trajectories of the motion of a ball in simple billiards are described. The Fomenko-Zieschang invariants are calculated for every simple billiard, and a theorem is proved which shows that only three different Liouville foliations of simple billiards exist on the Minkowski plane. Bibliography: 23 titles.

The problem of the distribution of the singular points of the sum of a series of exponential monomials on the boundary of the domain of convergence of the series is considered. Sufficient conditions are found for a singular point to exist on a prescribed arc on the boundary; these are stated in purely geometric terms. The singular point exists due to simple relations between the maximum density of the exponents of the series in an angle and the length of the arc on the boundary of the domain of convergence that corresponds to this angle. Necessary conditions for a singular point to exist on a prescribed arc on the boundary are also obtained. They are stated in terms of the minimum density of the exponents in an angle and the length of the arc. On this basis, for sequences with density, criteria are established for the existence of a singular point on a prescribed arc on the boundary of the domain of convergence. Bibliography: 27 titles.