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Full-Text Articles in Physical Sciences and Mathematics
Generalized Branching In Circle Packing, James Russell Ashe
Generalized Branching In Circle Packing, James Russell Ashe
Doctoral Dissertations
Circle packings are configurations of circle with prescribed patterns of tangency. They relate to a surprisingly diverse array of topics. Connections to Riemann surfaces, Apollonian packings, random walks, Brownian motion, and many other topics have been discovered. Of these none has garnered more interest than circle packings' relationship to analytical functions. With a high degree of faithfulness, maps between circle packings exhibit essentially the same geometric properties as seen in classical analytical functions. With this as motivation, an entire theory of discrete analytic function theory has been developed. However limitations in this theory due to the discreteness of circle packings …
Discrete Geometric Homotopy Theory And Critical Values Of Metric Spaces, Leonard Duane Wilkins
Discrete Geometric Homotopy Theory And Critical Values Of Metric Spaces, Leonard Duane Wilkins
Doctoral Dissertations
Building on the work of Conrad Plaut and Valera Berestovskii regarding uniform spaces and the covering spectrum of Christina Sormani and Guofang Wei developed for geodesic spaces, the author defines and develops discrete homotopy theory for metric spaces, which can be thought of as a discrete analog of classical path-homotopy and covering space theory. Given a metric space, X, this leads to the construction of a collection of covering spaces of X - and corresponding covering groups - parameterized by the positive real numbers, which we call the [epsilon]-covers and the [epsilon]-groups. These covers and groups evolve dynamically as the …