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Survival and connection probabilities of open three dimensional billiards

The purpose of the work presented in this thesis was carrying out survival and connection probability analyses in open three dimensional billiards. Results obtained are based on and extended upon previous work on analogous two dimensional counterparts. Typical analyses of such systems involve using the law of reflection in studying ranges of paths travelled by particles and/or waves that have constant velocity unless they make contact with the boundary or escape the system. The two forms of systems whose results will be presented are billiards (probabilities that some enclosed particle does not escape the region in question for a minimum measure of time), in our cases spherical in shape (hence the spherical billiard problem) and wireless communication channels (connection probabilities within a nodal network). We will let A and B denote the constant and 0(1/t) terms in the long-time expansions of the survival probability in the billiard system in question for time t. Results found regarding the spherical billiard work, analytically as well as numerically for a circular hole configuration and at best numerically for a square hole configuration include a long-time trend of the form A +B/t. In both hole type configurations, analytic expressions for the constant terms in the long-time survival probability expansions have been computed. For the case of the circular-shaped hole, the 0(1/t) coefficient has a small ε (angular displacement from a vector pointing to the center of the hole to a vector pointing to its boundary) expansion with one of its terms resulting from the Riemann hypothesis, perhaps the greatest unsolved number-theoretical problem. A key extension to the single-hole problem is the multi-hole problem. Results found regarding the wireless communications work include the connection probability within a network of nodes being approximately exponentially dependent on absolute distances between nodal pairs and being dependent on reflections with the channel boundary. In particular, results for communication channels with convex geometries (in past work) are presented and it has been justified that the corresponding analysis can be used in the study of non-convex geometries (in this work).

Identiferoai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:685438
Date January 2015
CreatorsRahman, Mohammed Rizwanur
PublisherUniversity of Bristol
Source SetsEthos UK
Detected LanguageEnglish
TypeElectronic Thesis or Dissertation

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