Recent advances in open billiards with some open problems

Much recent interest has focused on "open" dynamical systems, in which a classical map or flow is considered only until the trajectory reaches a "hole", at which the dynamics is no longer considered. Here we consider questions pertaining to the survival probability as a function of time, given an initial measure on phase space. We focus on the case of billiard dynamics, namely that of a point particle moving with constant velocity except for mirror-like reflections at the boundary, and give a number of recent results, physical applications and open problems.Comment: 16 pages, 1 figure in six parts. To appear in Frontiers in the study of chaotic dynamical systems with open problems (Ed. Z. Elhadj and J. C. Sprott, World Scientific

Symmetric motifs in random geometric graphs

We study symmetric motifs in random geometric graphs. Symmetric motifs are subsets of nodes which have the same adjacencies. These subgraphs are particularly prevalent in random geometric graphs and appear in the Laplacian and adjacency spectrum as sharp, distinct peaks, a feature often found in real-world networks. We look at the probabilities of their appearance and compare these across parameter space and dimension. We then use the Chen-Stein method to derive the minimum separation distance in random geometric graphs which we apply to study symmetric motifs in both the intensive and thermodynamic limits. In the thermodynamic limit the probability that the closest nodes are symmetric approaches one, whilst in the intensive limit this probability depends upon the dimension.Comment: 11 page

Distribution of Cell Area in Bounded Poisson Voronoi Tessellations with Application to Secure Local Connectivity

Poisson Voronoi tessellations have been used in modeling many types of systems across different sciences, from geography and astronomy to telecommunications. The existing literature on the statistical properties of Poisson Voronoi cells is vast, however, little is known about the properties of Voronoi cells located close to the boundaries of a compact domain. In a domain with boundaries, some Voronoi cells would be naturally clipped by the boundary, and the cell area falling inside the deployment domain would have different statistical properties as compared to those of non-clipped Voronoi cells located in the bulk of the domain. In this paper, we consider the planar Voronoi tessellation induced by a homogeneous Poisson point process of intensity $\lambda\!>\!0$ in a quadrant, where the two half-axes represent boundaries. We show that the mean cell area is less than $\lambda^{-1}$ when the seed is located exactly at the boundary, and it can be larger than $\lambda^{-1}$ when the seed lies close to the boundary. In addition, we calculate the second moment of cell area at two locations for the seed: (i) at the corner of a quadrant, and (ii) at the boundary of the half-plane. We illustrate that the two-parameter Gamma distribution, with location-dependent parameters calculated using the method of moments, can be of use in approximating the distribution of cell area. As a potential application, we use the Gamma approximations to study the degree distribution for secure connectivity in wireless sensor networks deployed over a domain with boundaries.Comment: to be publishe

Quantization for uniform distributions on equilateral triangles

We approximate the uniform measure on an equilateral triangle by a measure supported on $n$ points. We find the optimal sets of points ($n$-means) and corresponding approximation (quantization) error for $n\leq4$, give numerical optimization results for $n\leq 21$, and a bound on the quantization error for $n\to\infty$. The equilateral triangle has particularly efficient quantizations due to its connection with the triangular lattice. Our methods can be applied to the uniform distributions on general sets with piecewise smooth boundaries

Periodic compression of an adiabatic gas: Intermittency enhanced Fermi acceleration

A gas of noninteracting particles diffuses in a lattice of pulsating scatterers. In the finite horizon case with bounded distance between collisions and strongly chaotic dynamics, the velocity growth (Fermi acceleration) is well described by a master equation, leading to an asymptotic universal non-Maxwellian velocity distribution scaling as v ~ t. The infinite horizon case has intermittent dynamics which enhances the acceleration, leading to v ~ t ln t and a non-universal distribution.Comment: 6 pages, 4 figures, to appear in EPL (http://epljournal.edpsciences.org/