Differences Between Robin and Neumann Eigenvalues.

Let be a bounded planar domain, with piecewise smooth boundary . For , we consider the Robin boundary value problem where is the derivative in the direction of the outward pointing normal to . Let be the corresponding eigenvalues.

Volume distribution of nodal domains of random band-limited functions.

We study the volume distribution of nodal domains of families of naturally arising Gaussian random fields on generic manifolds, namely random band-limited functions. It is found that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and continuity of the cumulative probability function, are established.

On probability measures arising from lattice points on circles.

A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice , gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak limits, are said to be from lattice points on circles. We investigate the set of attainable measures and show that it contains all extreme points, in the sense of convex geometry, of the set of all probability measures that are invariant under some natural symmetries.