15,133 research outputs found
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 in a quadrant, where the two half-axes represent
boundaries. We show that the mean cell area is less than when
the seed is located exactly at the boundary, and it can be larger than
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
Connectivity of confined 3D Networks with Anisotropically Radiating Nodes
Nodes in ad hoc networks with randomly oriented directional antenna patterns
typically have fewer short links and more long links which can bridge together
otherwise isolated subnetworks. This network feature is known to improve
overall connectivity in 2D random networks operating at low channel path loss.
To this end, we advance recently established results to obtain analytic
expressions for the mean degree of 3D networks for simple but practical
anisotropic gain profiles, including those of patch, dipole and end-fire array
antennas. Our analysis reveals that for homogeneous systems (i.e. neglecting
boundary effects) directional radiation patterns are superior to the isotropic
case only when the path loss exponent is less than the spatial dimension.
Moreover, we establish that ad hoc networks utilizing directional transmit and
isotropic receive antennas (or vice versa) are always sub-optimally connected
regardless of the environment path loss. We extend our analysis to investigate
boundary effects in inhomogeneous systems, and study the geometrical reasons
why directional radiating nodes are at a disadvantage to isotropic ones.
Finally, we discuss multi-directional gain patterns consisting of many equally
spaced lobes which could be used to mitigate boundary effects and improve
overall network connectivity.Comment: 12 pages, 10 figure
- …