Almost all eigenfunctions of a rational polygon are uniformly distributed

Statistics of Wave Functions for a Point Scatterer on the Torus

Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-pote...

Periodic trajectories in the regular pentagon

The study of billiards in rational polygons and of directional flows on flat surfaces is a fast-growing and fascinating area of research. A classical construction reduces the billiard system in a rational polygon – a polygon whose angles are π-rational – to a constant flow on a flat surface with conical singularities, determined by the billiard polygon. In the most elementary case, the billiard...

Points at Rational Distance from the Vertices of a Unit Polygon

Network Location Problem with Stochastic and Uniformly Distributed Demands

This paper investigates the network location problem for single-server facilities that are subject to congestion. In each network edge, customers are uniformly distributed along the edge and their requests for service are assumed to be generated according to a Poisson process. A number of facilities are to be selected from a number of candidate sites and a single server is located at each facil...

Deviation of Ergodic Averages for Rational Polygonal Billiards

We prove a polynomial upper bound on the deviation of ergodic averages for almost all directional flows on every translation surface, in particular, for the generic directional flow of billiards in any Euclidean polygon with rational angles.