Paul Castle | Searcharxiv

arXiv2018

Taylor series for generalized Lambert W functions

The Lambert W function gives the solutions of a simple exponential polynomial. The generalized Lambert W function was defined by Mez\"{o} and Baricz, and has found applications in delay differential equations and physics. In this article we describe an even more general function, the inverse of a product of powers of linear functions and one exponential term. We show that the coefficients of the Taylor series for these functions can be described by multivariable hypergeometric functions of the parameters. We also present a surprising conjecture for the radius of convergence of the Taylor series, with a rough proof.

math.CA↗

arXiv2017

Unilluminable rooms, billiards with hidden sets, and Bunimovich mushrooms

The illumination problem is a popular topic in recreational mathematics: In a mirrored room, is every region illuminable from every point in the region? So-called \enquote{unilluminable rooms} are related to \enquote{trapped sets} in inverse scattering, and to billiards with divided phase space in dynamical systems. In each case, a billiard with a semi-ellipse has always been put forward as the standard counterexample: namely the Penrose room, the Livshits billiard, and the Bunimovich mushroom respectively. In this paper, we construct a large class of planar billiard obstacles, not necessarily featuring ellipses, that have dark regions, hidden sets, or a divided phase space. The main result is that for any convex set $\mathcal{H}$, we can construct a convex, everywhere differentiable billiard table $K$ (at any distance from $\mathcal{H}$) such that trajectories leaving $\mathcal{H}$ always return to $\mathcal{H}$ after one reflection. This billiard generalises the Bunimovich mushroom. As corollaries, we give more general answers to the illumination problem and the trapped set problem. We use recent results from nonsmooth analysis and convex function theory, to ensure that the result applies to all convex sets.

math.DS↗