We study the survival probability for long times in an open spherical billiard, extending previous work on the circular billiard. We provide details of calculations regarding two billiard configurations, specifically a sphere with a circular hole and a sphere with a square hole. The constant terms of the long-term survival probability expansions have been derived analytically. Terms that vanish in the long time limit are investigated analytically and numerically, leading to connections with the Riemann hypothesis. I IntroductionA mathematical billiard is a dynamical system within which a particle is in motion via alternating straight line movements in its interior and mirror-like reflections with its boundary without losing speed [18]. There are many dynamical properties that are possibly present within such systems (regular, chaotic, etc.) which are obtained depending on their shapes [18]. Important applications include microwave experiments [16] and microlasers [15].The circular billiard is a simple but important example of regular dynamics. Orbits in the circular billiard are related to the study of mushroom billiards, since circular orbits are present in the caps of such billiards' configuration, which are a prominent example of sharply divided phase space [2], and widely studied both classically and quantum mechanically [19]. There, typical values of a control parameter allow the existence of marginally unstable periodic orbits (MUPOs) that exhibit stickiness, specifically that unstable orbits approach regular regions in phase space [3]. In addition, MUPOs are present in an annular billiard [7], within which orbits resemble those from the circular billiard. MUPOs have been realised in the context of directional emission in dielectric microcavities [3]. The drive-belt stadium billiard has similar properties to its straight counterpart including hyperbolicity and mixing, as well as intermittency due to MUPOs whereas the big distinction between the straight and drivebelt cases is the presence of multiple MUPO families in the drivebelt [2]. In each of these examples, the MUPOs correspond to periodic orbits of a corresponding circular billiard.Perturbations of the class of such closed systems by the introduction of a small hole, referred to as open systems [4], allows us to probe their internal dynamical nature. We will denote the probability of survival for time t in the circular billiard by P c (t). The density of orbits implies that P c (t) → 0 as t → ∞ since unperturbed periodic orbits constitute a zero-measure set in phase space. Furthermore, the leading coefficient of P c (t) is related to the Riemann hypothesis [13], perhaps the greatest unsolved problem in number theory [12].Here, we study the survival probability for the spherical billiard, showing that this is also related to the Riemann hypothesis. The spherical billiard is of particular interest for applications, e.g. whispering gallery mode emission from a spherical microcavity [20], while simple enough as a starting point for the study of o...

Recent research has demonstrated the importance of boundary effects on the overall connection probability of wireless networks but has largely focused on convex deployment regions. We consider here a scenario of practical importance to wireless communications, in which one or more nodes are located outside the convex space where the remaining nodes reside. We call these 'external nodes', and assume that they play some essential role in the macro network functionality e.g. a gateway to a dense self-contained mesh network cloud. Conventional approaches with the underlying assumption of only line-of-sight (LOS) or direct connections between nodes, fail to provide the correct analysis for such a network setup. To this end we present a novel analytical framework that accommodates for the non-convexity of the domain and explicitly considers the effects of non-LOS nodes through reflections from the domain boundaries. We obtain analytical expressions in 2D and 3D which are confirmed numerically for Rician channel fading statistics and discuss possible extensions and applications.

A dynamical billiard consists of a point particle moving uniformly except for mirror-like collisions with the boundary. Recent work has described the escape of the particle through a hole in the boundary of a circular or spherical billiard, making connections with the Riemann Hypothesis. Unlike the circular case, the sphere with a single hole leads to a non-zero probability of never escaping. Here, we study variants in which almost all initial conditions escape, with multiple small holes or a thin strip. We show that equal spacing of holes around the equator is an efficient means of ensuring almost complete escape and study the long time survival probability for small holes analytically and numerically. We find that it approaches a universal function of a single parameter, hole area multiplied by time.