京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19470号 / 理博第4130号 / 新制||理||1594(附属図書館) / 32506 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 熊谷 隆, 教授 岡本 久, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

This thesis studies effective resistances of finite and infinite weighted graphs. Classical results state that it is a metric on the set of vertices of the graph and that it can be expressed completely in terms of the graph’s random walk. The first goal of this thesis is to provide a concise and accessible starting point for new scholars interested in the topic. In that spirit, we reproduce existing results and review different approaches to effective resistances using tools from several fields such as linear algebra, probability theory, geometry and functional analysis. The second goal is to characterize which metric spaces are given by the effective resistance of a graph. For the finite case, we begin by reconstructing the associated graph from the effective resistance. This leads to a complete algebraic characterization in terms of triangle inequality defects. A more geometric condition is given by showing that a metric space can only be an effective resistance if its minimal graph realization contains no incomplete cycles. We also show that our algebraic characterization can be applied to the more general theory of resistance forms as defined by Kigami. The third goal of this thesis is to investigate probabilistic representations of effective resistances. Building on the work of Tetali and Barlow, we characterize under which conditions known representations for finite graphs can be extended to infinite graphs.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-04-24T15:04:21Z No. of bitstreams: 1 luizfernandoalvesguedes.pdf: 8533376 bytes, checksum: 1fb75359e16225fb5b36b42d374e4b7a (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-04-24T16:59:50Z (GMT) No. of bitstreams: 1 luizfernandoalvesguedes.pdf: 8533376 bytes, checksum: 1fb75359e16225fb5b36b42d374e4b7a (MD5) / Made available in DSpace on 2017-04-24T16:59:50Z (GMT). No. of bitstreams: 1 luizfernandoalvesguedes.pdf: 8533376 bytes, checksum: 1fb75359e16225fb5b36b42d374e4b7a (MD5) Previous issue date: 2012-12-17 / Este trabalho tem o objetivo de estudar, projetar, simular e obter resultados experimentais, a partir de um conversor flyback operando no modo descontínuo (DCM), para acionar uma luminária de diodos emissores de luz (LEDs). Além disso, deseja-se que o conversor possua elevado fator de potência, alta eficiência, baixa distorção harmônica da corrente de entrada, vida útil elevada (através da possibilidade de substituir de capacitores eletrolíticos por capacitores de filme metálico) e baixo custo. Inicialmente, o trabalho apresenta um breve estudo da iluminação a LEDs, o modelo elétrico simplificado dos LEDs, bem como uma revisão de alguns conversores aplicados ao acionamento deste tipo de fonte luminosa. Em seguida é analisada a operação do conversor flyback em modo DCM alimentado a partir da retificação (em ponte) da tensão da rede elétrica. Além do estudo teórico, a simulação do conversor é apresentada de modo a validar o projeto do conversor e servir de base de comparação com os resultados experimentais. O projeto do transformador (indutores acoplados) do conversor flyback tem destaque especial neste trabalho, pois, este elemento é fundamental para elevar a eficiência do conversor. Portanto, parte deste trabalho é dedicada ao estudo de perdas do transformador. Isto é feito através de uma adaptação de técnicas consagradas nesta área por meio de uma compilação didática visando a repetitividade do estudo. A partir do estudo teórico sobre as perdas no transformador, programas elaborados no MATLAB e no MICROSOFT EXCEL auxiliam o projeto do elemento magnético do conversor flyback. Estes dois programas, visam facilitar o exaustivo equacionamento das perdas e auxiliam no desenvolvimento de um projeto com elevada eficiência do transformador flyback elevando, consequentemente, a eficiência global do conversor. Empregando os programas elaborados, o projeto é calculado e resultados experimentais são obtidos a partir de medições realizadas em um protótipo em escala. Estes resultados são analisados e comparados com os resultados teóricos e de simulação. Ao final, é realizada uma análise simplificada do custo dos componentes do conversor e também são apresentadas as principais conclusões deste trabalho. / This work aims to study, design, simulate and obtain experimental results from a flyback converter operating in discontinuous mode (DCM), to trigger a fixture of light emitting diodes (LEDs). Furthermore, it is desired that the converter has high power factor, high efficiency, low harmonic distortion of input current, high life (through the possibility of replacing the electrolytic capacitors for metal film capacitors) and low cost. Initially, the paper presents a brief study of lighting LEDs, LEDs for simplified electrical model, as well as a review of some converters used to drive this type of light source. Then it analyzed the operation of the flyback converter in DCM mode fed from the rectification (bridged) the utility voltage. Besides the theoretical study, the simulation of the converter is presented in order to validate the design of the converter as a basis for comparison with experimental results. The transformer design (coupled inductors) flyback converter has special emphasis in this work, because this element is essential to raise the efficiency of the converter. So part of this work is devoted to the study of transformer losses. This is done by an adaptation of standard techniques in this area via a teaching order to build repeatability study. From the theoretical study on the losses in the transformer, programs developed in MATLAB and MICROSOFT EXCEL aid the design of the magnetic element of the flyback converter. These two programs are designed to facilitate the comprehensive equating of losses and assist in developing a project with high efficiency flyback transformer raising therefore the overall efficiency of the converter. Employing the programs drawn up, the project is calculated and experimental results are obtained from measurements performed on a prototype scale. These results are analyzed and compared with the theoretical results and simulation. At the end we provide a simplified analysis of the cost of the converter components and also presents the main conclusions of this work.