For a certain class of Z²-actions, we provide a proof of a conjecture that the ratio of the Perron eigenvalues of the transfer matrices of the free boundary restrictions converge to the entropy of that action. Also, a novel method for computing the entropy of Z²-actions is conjectured. / Graduation date: 2009 / Presentation date: 2008-07-23

Not available / text

Thesis (Ph.D.)--McMaster University, 2005. / Supervisor: Jean-Pierre Gabardo. Includes bibliographical references (leaves 138-145).

Thesis (Ph. D.)--University of Washington, 1992. / Vita. Includes bibliographical references (leaves [104]-105).

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Minimum Cost Polygon Overlay (MCPO) is a unique two-dimensional optimization problem that involves the task of covering a polygon shaped area with a series of rectangular shaped panels. The challenges in solving MCPO problems are related to the interdependencies that exist among the parameters and constraints that may be applied to the solution.This thesis examines the MCPO problem to construct a model that captures essential parameters to be solved using optimization algorithms. The purpose of the model is to make it possible that a solution for an MCPO problem can be generated automatically. A software application has been developed to provide a framework for validating the model.The development of the software has uncovered a host of geometric operations that are required to enable optimization to take place. Many of these operations are non-trivial, demanding novel, well-constructed algorithms based on careful appreciation of the nature of the problem.For the actual optimization task, three algorithms have been implemented: a greedy search, a Monte Carlo method, and a Genetic Algorithm. The behavior of the completed software is observed through its application on a series of test data. The results are presented to show the effectiveness of the software under various settings. This is followed by critical analysis of various findings of the research.Conclusions are drawn to summarize lessons learned from the research. Important issues about which no satisfactory explanation exists are given as material to be studied by future research.

Minimum Cost Polygon Overlay (MCPO) is a unique two-dimensional optimization problem that involves the task of covering a polygon shaped area with a series of rectangular shaped panels. The challenges in solving MCPO problems are related to the interdependencies that exist among the parameters and constraints that may be applied to the solution.This thesis examines the MCPO problem to construct a model that captures essential parameters to be solved using optimization algorithms. The purpose of the model is to make it possible that a solution for an MCPO problem can be generated automatically. A software application has been developed to provide a framework for validating the model.The development of the software has uncovered a host of geometric operations that are required to enable optimization to take place. Many of these operations are non-trivial, demanding novel, well-constructed algorithms based on careful appreciation of the nature of the problem.For the actual optimization task, three algorithms have been implemented: a greedy search, a Monte Carlo method, and a Genetic Algorithm. The behavior of the completed software is observed through its application on a series of test data. The results are presented to show the effectiveness of the software under various settings. This is followed by critical analysis of various findings of the research.Conclusions are drawn to summarize lessons learned from the research. Important issues about which no satisfactory explanation exists are given as material to be studied by future research.

Typescript. Includes bibliographical references.

Orientador: Eduardo Garibaldi / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T00:20:51Z (GMT). No. of bitstreams: 1 Colle_CleberFernando_M.pdf: 1068657 bytes, checksum: 78c9700800b05194ffcf66838581b081 (MD5) Previous issue date: 2011 / Resumo: Focaremos sobre aspectos dinâmicos e ergódicos de shifts multidimensionais, atentando especialmente para suas relações com estados fundamentais e quase-cristais em reticulados. Por exemplo, em mecânica estatística, dado um potencial invariante por translação, seus estados fundamentais são medidas de probabilidade invariantes por translação suportadas no conjunto de suas configurações fundamentais, isto é, das configurações com energia específica mínima. Estados fundamentais são naturalmente associados com o bordo de certos polítopos convexos dimensionalmente finitos. Esse bordo se torna drasticamente diferente se a dimensão do modelo em questão passa de d = 1 para d > 1, pois no caso multidimensional existe shift de tipo finito unicamente ergódico sem configurações periódicas / Abstract: We will focus on dynamic and ergodic aspects of multidimensional shifts, with particular care to their relations with ground states and quasicrystals in lattices. For example, in statistical mechanics, given a translation-invariant potential, its ground states are translation-invariant probability measures supported on the set of its ground configurations, i.e., of configurations with minimal specific energy. Ground states are naturally associated with the boundary of certain finite-dimensional convex polytopes. This boundary becomes drastically different if the dimension of the model in question changes from d = 1 to d > 1, because in the multidimensional case there exists uniquely ergodic shift of finite type with no periodic configurations / Mestrado / Matematica / Mestre em Matemática

Orientador: Reginaldo Palazzo Junior / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-18T16:51:18Z (GMT). No. of bitstreams: 1 Leskow_LucilaHelenaAllan_D.pdf: 2583405 bytes, checksum: 3161d9deabaa60a8965a9e3d20ff36aa (MD5) Previous issue date: 2011 / Resumo: Neste trabalho apresentamos como contribuição um novo conjunto de tesselações do plano hiperbólico construídas a partir de uma tesselação bem conhecida, a tesselação de Farey. Nestas tesselações a região de Dirichlet é formada por polígonos hiperbólicos de n lados, com n > 3. Explorando as características dessas tesselações, apresentamos alguns tipos possíveis de aplicações. Inicialmente, estudando a relação existente entre a teoria das frações contínuas e a tesselação de Farey, propomos um novo método de codificação de geodésicas. A inovação deste método está no fato de ser possível realizar a codificação de uma geodésica pertencente a PSL(2,Z) em qualquer uma das tesselações ou seja, para qualquer valor de n com n > 3. Neste método mostramos como é possível associar as sequências cortantes de uma geodésica em cada tesselação à decomposição em frações contínuas do ponto atrator desta. Ainda explorando as características dessas novas tesselações, propomos dois tipos de aplicação em teoria de codificação de fontes discretas. Desenvolvendo dois novos códigos para compactação de fontes (um código de árvore e um código de bloco), estes dois métodos podem ser vistos como a generalização dos métodos de Elias e Tunstall para o caso hiperbólico / Abstract: In this work we present as contribution a new set of tessellations of the hyperbolic plane, built from a well known tessellation, the Farey tessellation. In this set of tessellations the Dirichlet region is made of hyperbolic polygons with n sides where n > 3. While studying these tessellations and theirs properties, we found some possible applications. In the first one, while exploring the relationship between the continued fractions theory and the Farey tessellation we propose a new method for coding geodesics. Using this method, it is possible to obtain a relationship between the cutting sequence of a geodesic belonging to PSL(2,Z) in each tessellation and the continued fraction decomposition of its attractor point. Exploring the characteristics of these tessellations we also propose two types of applications regarding the discrete memoryless source coding theory, a fixed-to-variable code and a variable length-to-fixed code. These methods can be seen as a generalized version of the Elias and Tunstall methods for the hyperbolic case / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica