In this thesis, we examined some Dirichlet type problems of the form: \begin{eqnarray*} \triangle u & = & 0\ {\rm in\ } \mathbb{R}^n \\ u & = & f\ {\rm on\ } \psi = 0, \end{eqnarray*} and we were particularly interested in finding entire solutions when entire data was prescribed. This is an extension of the work of D. Siegel, M. Mouratidis, and M. Chamberland, who were interested in finding polynomial solutions when polynomial data was prescribed. In the cases where they found that polynomial solutions always existed for any polynomial data, we tried to show that entire solutions always existed given any entire data. For half space problems we were successful, but when we compared this to the heat equation, we found that we needed to impose restrictions on the type of data allowed. For problems where data is prescribed on a pair of intersecting lines in the plane, we found a surprising dependence between the existence of an entire solution and the number theoretic properties of the angle between the lines. We were able to show that for numbers $\alpha$ with $\omega_1$ finite according to Mahler's classification of transcendental numbers, there will always be an entire solution given entire data for the angle $2\alpha\pi$ between the lines. We were also able to construct an uncountable, dense set of angles of measure 0, much in the spirit of Liouville's number, for which there will not always be an entire solution for all entire data. Finally, we investigated a problem where data is given on the boundary of an infinite strip in the plane. We were unable to settle this problem, but we were able to reduce it to other {\it a priori} more tractable problems.

<p>During the last decade, potential theory and p-harmonic functions have been developed in the setting of doubling metric measure spaces supporting a <em>p</em>-Poincar´e inequality. This theory unifies, and has applications in several areas of analysis, such as weighted Sobolev spaces, calculus on Riemannian manifolds and Carnot groups, subelliptic differential operators and potential theory on graphs.</p><p>In this thesis we investigate the double obstacle problem for p-harmonic functions on metric spaces. We show the existence and uniqueness of solutions and their continuity when the obstacles are continuous. Moreover the solution is p-harmonic in the open set where it does not touch the continuous obstacles. The boundary regularity of the solutions is also studied.</p><p>Furthermore we study two kinds of convergence problems for the solutions. First we let the obstacles vary and fix the boundary values and show the convergence of the solutions. Second we consider an increasing sequence of open sets, with union Ω, and fix the obstacles and the boundary values. We show that the solutions of the obstacle problems in these sets converge to the solution of the corresponding problem in Ω.</p> / <p>Låt oss börja med att betrakta följande situation: Vi vill förflytta oss från en plats vid ena sidan av en äng till en viss punkt på andra sidan ängen. På båda sidor om ängen finns skogsområden som vi inte får gå in i. Ängen är tyvärr inte homogen utan består av olika sorters mark som vi har noggrant beskrivet på en karta. Vi vill göra förflyttningen på smidigast sätt, men då ängen inte är homogen ska vi förmodligen inte gå rakaste vägen utan ska anpassa vägen optimalt efter terrängen. Detta är ett exempel på ett dubbelhinderproblem där hindren är skogsområdena på sidorna som vi måste hålla oss utanför.</p><p>Mer abstrakt vill man minimiera energin hos funktioner som tar vissa givna randvärden (de givna start- och slutpunkterna i exemplet ovan) och som håller sig mellan ett undre och ett övre hinder. I denna avhandling studeras detta dubbelhinderproblem i väldigt allmänna situationer.</p><p>För att kunna lösa hinderproblemet krävs det att vi tillåter ickekontinuerliga lösningar och då visas i avhandlingen att hinderproblemet är entydigt lösbart. Ett huvudresultat i avhandlingen är att om våra hinder är kontinuerliga så blir även lösningen kontinuerlig. Vidare visas diverse konvergenssatser som visar hur lösningarna varierar när hindren eller området i vilket problemet löses varierar.</p><p>Hinderproblem har utöver eget intresse viktiga tillämpningar i potentialteorin, bland annat för att studera motsvarande energiminimeringsproblem utan hinder.</p>

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 85-86).

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (p. 115-123).

Thesis--Lund. / Extra t.p., with thesis note inserted. Imprint on cover: Lund, C.W.K. Gleerup. "A study of problems connected with Riesz' generalization of the Riemann-Liouville integral."--Introd. "References": p. [98].

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1996. / Includes bibliographical references (p. 147-149) and index.

Thesis (Ph. D. in Mathematics)--Vanderbilt University, Aug. 2006. / Title from title screen. Includes bibliographical references.

Made available in DSpace on 2014-06-11T19:26:17Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-08-27Bitstream added on 2014-06-13T20:54:29Z : No. of bitstreams: 1 paulino_wb_me_bauru.pdf: 510809 bytes, checksum: 5b329ee9792382dc555572a1524fe865 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Com a privatização da malha ferroviária no meio dos anos de 1990, a produtividade do setor vem crescendo em decorrência do aumento de investimentos por parte das concessionárias, tanto na modernização das vias quanto na melhoria das práticas de manutenção. Dentro deste contexto, o uso de modelos probabilísticos é de fundamental importância para se determinar o comportamento de um dado sistema, não sendo com as ferrovias. Ao se adequar os tempos de reparo ao modelo de distribuição de Lei de Potência, espera-se aproveitar uma de suas principais propriedades, que é a invariância de escala, que permite a generalização do modelo empregado em um trecho para toda malha. Os dados se adequaram e o modelo pôde ser utilizado como mais uma ferramenta de auxílio à resolução de problemas e antecipação de indisponibilidades da malha / The privatization of the railroad in the middle of the years of 1990 increases the segment productivity by the concessionary investments in modernization and maintenance practices improvement. Inside the context of the maintenance, the use of probability models has a fundamental importance for determine system behavior, not being different with the maintenance of the railroads. Determine the times of repair in to the Power Law distribution means takes advantages of their main properties like a free scale. That in the build model can be generalized for the entire railroad. The data were adapted at the model and can be used as one more tool of aid for the resolution of problems and antecipation of unavailability of the railroad

Orientador: Sueli Irene Rodrigues Costa / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-11T17:00:37Z (GMT). No. of bitstreams: 1 Costa_FelixSilva_M.pdf: 655518 bytes, checksum: d511c33452fb4b2c113deef060ff88eb (MD5) Previous issue date: 2008 / Resumo: Neste trabalho são descritos métodos para o cálculo da área de regiões planas delimitadas por curvas simples e algumas propriedades de transformações do plano no plano que preservam áreas. No Capítulo 1, a área de polígonos é introduzida como uma soma de determinantes e utilizada para discutir o cálculo da área de regiões planas contornadas por curvas simples quando estas são aproximadas por polígonos com vértices ajustados por parâmetros geométricos. A fundamentação, baseada no Teorema de Green, de processos mecânicos (planímetros) para o cálculo destas áreas é descrita no Capítulo 2. Propriedades e famílias especiais de aplicações do plano no plano que preservam áreas são apresentadas no Capítulo 3. / Abstract: We describe here methods for the area estimation of plane regions bounded by simple curves and also some properties of plane transformations which preserve area. In Chapter 1 the area of polygons, described as a sum of determinants, is used to discuss the calculus of the area of plane regions bounded by simple curves approached by polygons adjusted through geometric parameters. Mechanical processes ( planimeters) based on the Green's Theorem are described in Chapter 2. Properties and special families of area preserving mappings are presented in Chapter 3. / Mestrado / Geometria / Mestre em Matemática

Orientador: Serguei Popov / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T09:42:00Z (GMT). No. of bitstreams: 1 Costa_MarceloRocha_M.pdf: 576974 bytes, checksum: 3b01a9f15e6e0f9fdd98631dc69cd202 (MD5) Previous issue date: 2014 / Resumo: O problema considerado no presente trabalho cumpre o papel de reforçar a eficácia dos métodos apresentados nos capítulos introdutórios, bem como investiga a resposta de um problema até então não publicado na literatura especializada. Introduzimos uma partícula realizando um passeio aleatório simples no espaço, ou seja, uma partícula que a cada passo escolhe uniformemente um de seus vizinhos para onde irá saltar. Fixando sua posição inicial ao longo da fronteira do cubo, pergunta-se: qual é a probabilidade de que a trajetória de tal partícula nunca mais retorne ao cubo? Em outras palavras, se T é o tempo de primeiro retorno ao cubo, estamos interessados em descrever o comportamento assintótico da probabilidade de que T seja infinito / Abstract: It has been considered in this work a problem which play a role of showing the effectiveness of the content covered in the introductory chapters, as well as it is a unsolved problem across the specialized literature. We introduce a particle performing a simple random walk in space, i.e., a particle which at each step choose uniformly one of its neighbourhood sites to which it then jumps into. Fixed its initial position along the boundary of a cube, we are interested in answering the following question: what is the probability that such particle's trajectory will never reach the cube again. In other words, if T is the first return time to the cube, we aim to analyse the asymptotic behaviour of the probability that T is infinite / Mestrado / Estatistica / Mestre em Estatística