Nesta dissertação apresentamos e desenvolvemos o Método de Perron, fazendo uma aplicação ao ploblema de Dirichlet para a equação das superfícies de curvatura média constante em R3. Apresentamos também uma extensão deste método dentro de EDP's e, por fim, obtemos uma extensão geométrica que se aplica a superfícies ao invés de gráficos. Comentamos a aplicação deste método geométrico á existência de superfícies mínimas tendo como bordo duas curvas convexas em planos paralelos do R3. / In this work we explain Perron's method and obtain an application of it to the Dirichlet Problem for the constant mean curvature surface equation in R3. We also obtain an extension of this method within the P.D.E theory and, finally, we obtain a geometric extension which applies to surfaces instead of graphs. This geometric extension can be used to prove the existence of a minimal compact surface having as boundary two convex curves in palallel plane of R3. We discuss this result at the final part of the work.

Nesta dissertação apresentamos e desenvolvemos o Método de Perron, fazendo uma aplicação ao ploblema de Dirichlet para a equação das superfícies de curvatura média constante em R3. Apresentamos também uma extensão deste método dentro de EDP's e, por fim, obtemos uma extensão geométrica que se aplica a superfícies ao invés de gráficos. Comentamos a aplicação deste método geométrico á existência de superfícies mínimas tendo como bordo duas curvas convexas em planos paralelos do R3. / In this work we explain Perron's method and obtain an application of it to the Dirichlet Problem for the constant mean curvature surface equation in R3. We also obtain an extension of this method within the P.D.E theory and, finally, we obtain a geometric extension which applies to surfaces instead of graphs. This geometric extension can be used to prove the existence of a minimal compact surface having as boundary two convex curves in palallel plane of R3. We discuss this result at the final part of the work.

A proposta principal desta. dissertação é provar a existência de medidas invariante absolutamente contínuas para uma clas$e de funções monótonas por partes com um número finito de descontinuidade mas o resultado pode ser estedido para funções monótonas por partes com um número infini to de descontinuidades. O método de prova explora a existência de pontos fixos para o operador de Perron- Frobenius e utiliza o Teorema de Helly e o Teorema Ergódico de Kakutani-Yosida. / The main purpose of this dissertation is to prove the existence of invariant absolutely continuous measures for a class of piecewise monotonic functions with a finite number of descontinuities but it can be extended to piecewise monotonic functions with infinite numbers of descontinuities. The method of the proof explores the existence of fixe·d points to Perron-Frobenius operator and employs the Helly's Theorem and the Kakutani - Yosida ergodic Theorem.

A proposta principal desta. dissertação é provar a existência de medidas invariante absolutamente contínuas para uma clas$e de funções monótonas por partes com um número finito de descontinuidade mas o resultado pode ser estedido para funções monótonas por partes com um número infini to de descontinuidades. O método de prova explora a existência de pontos fixos para o operador de Perron- Frobenius e utiliza o Teorema de Helly e o Teorema Ergódico de Kakutani-Yosida. / The main purpose of this dissertation is to prove the existence of invariant absolutely continuous measures for a class of piecewise monotonic functions with a finite number of descontinuities but it can be extended to piecewise monotonic functions with infinite numbers of descontinuities. The method of the proof explores the existence of fixe·d points to Perron-Frobenius operator and employs the Helly's Theorem and the Kakutani - Yosida ergodic Theorem.

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[pt] Estudaremos dinâmicas simbólicas associadas a alfabetos finitos. Consideraremos seqüências bi-infinitas e espaços com memória finita. Estudaremos propriedades invariantes por conjugação. Analisaremos a relação entre os espaços de seqüências e propriedades de matrizes não negativas. O principal exemplo desta correlação é o Teorema de Perron- Frobenius que relaciona a entropia de um espaço de seqüências e os autovalores de uma matriz não negativa associada ao espaço. Neste contexto, certos grafos e suas propriedades aparecem de forma natural. / [en] We study symbolic dynamics associated to finite alphabets. We consider bi-infinite sequences and spaces with finite memory. We pay attention to properties which are invariant by conjugations. We analyze the relation between spaces of sequences and properties of non-negative matrices. The main example is given by the Perron-Frobenius theorem relating the entropy of a space of sequences and the eigerrvalues of a non-negative matrix associated to the space. In this setting, certain graphs and their properties appear in a natural way.

In this work, we study existence and non existence for the Dirichlet problem for the minimal graph equation in non convex domains of the plane. We search for conditions on the boundary data which be the less restricted possible for the solubility of the Dirichlet problem. / Neste trabalho estudamos existência e não existência do problema de Dirichlet para a equação dos gráficos mínimos em domínios não convexos do plano. Procuramos por condições sobre o dado no bordo que sejam as menos restritivas possíveis para que o problema de Dirichlet em questão tenha solução

近年來鉅額貿易出超，台幣持續升值，貨幣數量急速擴張以及股票與房地產交易異常 熱絡下，反映出台灣經濟與金融情勢的變動。在此變遷下，中央銀行是否可藉貨幣政 策來影響所得，為學術與實務上重要之研究課題。 另一方面，貨幣理論發展至今，仍存在許多爭論，其根源在於各學說所強調的重點與 影響途徑不同，使分析結果產生差異，因此國內學者對有關之研究，或因模型設定之 不同與選擇期間之差異，所得結果頗為分岐。 貨幣與所得或物價間單向或雙向影響之爭論由來已久，Granger 與Sims等人提出因果 關係檢定方法來解決，為實證研究之一項重大突破，然而由於方法上若干缺失，故實 證結果常有出入。本論文除針對傳統模型實證方法上之缺失，設法修正外，更進一步 分析總體數列非恆定以及共異積情況下之因果關係檢定，希望藉有系統而且較可靠之 方法探討貨幣與所得之基本關係，以供政策採行、修改。 為確實檢討與修正實證方法，以及獲得穩健、可靠台灣實證結果，本研究計劃結構如 下： 第一章：前言：研究動機以前研究的檢討。 第二章：單根檢定方法與結果：兼採Perron-Phillips 與Stock-Watson 方法。 第三章：Cointegration 檢定方法與結果。 第四章：非恆定下因果關係之檢定。 第五章：實證結果。 第六章：結論。 在不同模型設定下，系統性因果檢定，以期充分且正確地發現過去貨幣與所得之關連 ，以作未來經濟預測以及政策采行之參考。

Systems whose time evolutions are entirely deterministic can nevertheless be studied probabilistically, <em>i. e. </em> in terms of the evolution of probability distributions rather than individual trajectories. This approach is central to the dynamics of ensembles (statistical mechanics) and systems with uncertainty in the initial conditions. It is also the basis of ergodic theory--the study of probabilistic invariants of dynamical systems--which provides one framework for understanding chaotic systems whose time evolutions are erratic and for practical purposes unpredictable. Delay differential equations (DDEs) are a particular class of deterministic systems, distinguished by an explicit dependence of the dynamics on past states. DDEs arise in diverse applications including mathematics, biology and economics. A probabilistic approach to DDEs is lacking. The main problems we consider in developing such an approach are (1) to characterize the evolution of probability distributions for DDEs, <em>i. e. </em> develop an analog of the Perron-Frobenius operator; (2) to characterize invariant probability distributions for DDEs; and (3) to develop a framework for the application of ergodic theory to delay equations, with a view to a probabilistic understanding of DDEs whose time evolutions are chaotic. We develop a variety of approaches to each of these problems, employing both analytical and numerical methods. In transient chaos, a system evolves erratically during a transient period that is followed by asymptotically regular behavior. Transient chaos in delay equations has not been reported or investigated before. We find numerical evidence of transient chaos (fractal basins of attraction and long chaotic transients) in some DDEs, including the Mackey-Glass equation. Transient chaos in DDEs can be analyzed numerically using a modification of the "stagger-and-step" algorithm applied to a discretized version of the DDE.

Objects such as documents, people, words or utilities, that are related in some way, for instance by citations, friendship, appearance in definitions or physical connections, may be conveniently represented using graphs or networks. An increasing number of such relational databases, as for instance the World Wide Web, digital libraries, social networking web sites or phone calls logs, are available. Relevant information may be hidden in these networks. A user may for instance need to get authority web pages on a particular topic or a list of similar documents from a digital library, or to determine communities of friends from a social networking site or a phone calls log. Unfortunately, extracting this information may not be easy. This thesis is devoted to the study of problems related to information extraction in large graphs with the help of dominant vectors of nonnegative matrices. The graph structure is indeed very useful to retrieve information from a relational database. The correspondence between nonnegative matrices and graphs makes Perron--Frobenius methods a powerful tool for the analysis of networks. In a first part, we analyze the fixed points of a normalized affine iteration used by a database matching algorithm. Then, we consider questions related to PageRank, a ranking method of the web pages based on a random surfer model and used by the well known web search engine Google. In a second part, we study optimal linkage strategies for a web master who wants to maximize the average PageRank score of a web site. Finally, the third part is devoted to the study of a nonlinear variant of PageRank. The simple model that we propose takes into account the mutual influence between web ranking and web surfing.