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

Egyptian fractions are what we know as unit fractions that are of the form 1/n - with the exception, by the Egyptians, of 2/3. Egyptian fractions have actually played an important part in mathematics history with its primary roots in number theory. This paper will trace the history of Egyptian fractions by starting at the time of the Egyptians, working our way to Fibonacci, a geologist named Farey, continued fractions, Diophantine equations, and unsolved problems in number theory.

Radar systems have been very effective in gathering information in a battlefield, so that the tactical actions can be decided. On the contrary, self-protection systems have been developed to break this activity of radars, for which radar signals must be intercepted to be able to take counter measures on time. Ideally, interception should be done in a certain time with a 100% probability, but in reality this is not the case. To intercept radar signals in shortest time with the highest probability, a search strategy should be developed for the receiver. This thesis studies the conditions under which the intercept time increases and the probability of intercept decreases. Moreover, it investigates the performance of the search strategy of Clarkson with respect to these conditions, which assumes that a priori knowledge about the radars that will be intercepted is available. Then, the study identifies the cases where the search strategy of Clarkson may be not desirable according to tactical necessities, and proposes a probabilistic search strategy, in which it is possible to intercept radar signals with a specified probability in a certain time.