Made available in DSpace on 2015-05-15T11:46:17Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1816441 bytes, checksum: 97d911fa4449e54f12d9f203b8b50c90 (MD5) Previous issue date: 2013-03-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work, we study qualitative properties of solutions of the semilinear elliptic equation class 8<: u + f(u) = 0, em , u = 0, em @ , defined in different kinds of unbounded domains of Rn, among them, infinite cylinders, half spaces and Lipschitz domains. We analyze properties like convergence, monotonocity and symmetry of solutions of the problem (1), when f satisfy certain conditions suitable. For this purpose, we will use various kinds of maximum principles, the moving planes method,elliptic estimates and compacity theorems. We also studied some results about Schrödinger operators and we prove the De Giorge conjecture in dimension n = 2. / Neste trabalho, estudamos propriedades qualitativas de soluções da seguinte classe de equações elípticas semilineares 8<:u + f(u) = 0, em u = 0, em @ , (1) definidas em vários tipos de domínios não limitados do Rn, dentre eles, cilíndros infinitos, semi espaços e domínios Lipschitzianos. Analisamos propriedades de convergência, monotonicidade e simetria de soluções de (1), quando f satisfaz certas condições adequadas. Para tanto, utilizaremos várias versões do princípio do máximo, o método dos planos móveis (moving planes), estimativas elípticas e teoremas de compacidade. Estudamos ainda resultados sobre operadores de Schrödinger e, como consequência, provamos a conjectura de De Giorgi em dimensão n = 2.

O conceito de taxa de variação é muito útil no estudo das funções reais de uma variável real e, por ser elementar, pode ser apresentado e discutido no Ensino Médio. A proposta deste trabalho é fazer um estudo detalhado e criterioso do uso da taxa de variação no estudo das funções afins, quadráticas e cúbicas. Com isso, pretende-se desenvolver resultados e apresentar diversas aplicações deste conceito a estas classes de funções que incluem, por exemplo, novas formas de obtenção dos seus gráficos, a interpretação geométrica de seus coeficientes, bem como resolver problemas simples e interessantes de otimização. Quanto às funções quadráticas, como aplicação do estudo feito sobre estas funções, apresentamos uma maneira matematicamente rigorosa de conceituar o Movimento Retilíneo Uniformemente Variado (MRUV), bem como, obtemos as equações horárias deste movimento. A definição que apresentamos é equivalente a usual, mas não faz uso dos Cálculos Infinitesimal e Integral. Como nosso objetivo é eminentemente matemático, não apresentamos qualquer interpretação Física deste conceito na forma que o introduzimos. Já o estudo das funções cúbicas pode ser visto como uma interessante aplicação das funções quadráticas, uma vez que, através do uso das taxas de variação, a descrição (traçado do gráfico) de uma função cúbica é reduzida ao estudo do sinal de uma função quadrática associada. / The concept of rate of change is very useful in the study of real functions of one real variable. Since this is a simple concept it can be presented and discussed in high school. The purpose of this work is to make a comprehensive study use of the rate of change in the study of the affine, quadratic and cubic functions. We prove results and present several applications of this concept to these classes of functions that include, for example, new ways of obtaining their graphs, geometric interpretation of their coefficients, and the solution of simple and interesting optimization problems. As for the quadratic functions, as an application of our study about these functions, we obtain in a mathematically rigorous definition of Uniformly Varied Rectilinear Motion and obtain the equations of the motion without using the Infinitesimal or Integral Calculus. Since we were essentially concerned with the mathematical aspect of this phenomenon, we did not present a discussion about the physical interpretation of our definition. The study of cubic functions can be seen as an interesting application of the quadratic functions, since through the use of rate changing, the description of the graph of a cubic function is reduced to the study of the sign of a quadratic function associated to the given cubic.

A maioria dos textos na literatura de testes de hipóteses trata de critérios de otimalidade para um determinado problema de decisão. No entanto, existem, em menor quantidade, alguns textos sobre os problemas de se realizar testes de hipóteses simultâneos e sobre a concordância lógica de suas soluções ótimas. Algo que se espera de testes de hipóteses simultâneos e que, se uma hipótese H1 implica uma hipótese H0, então é desejável que a rejeição da hipótese H0 necessariamente implique na rejeição da hipótese H1, para uma mesma amostra observada. Essa propriedade é chamada aqui de monotonicidade. A fim de estudar essa propriedade sob um ponto de vista mais geral, neste trabalho é definida a nocão de classe de testes de hipóteses, que estende a funcão de teste para uma sigma-álgebra de possíveis hipóteses nulas, e introduzida uma definição de monotonicidade. Também é mostrado, por meio de alguns exemplos simples, que, para um nível de signicância fixado, a classe de testes Razão de Verossimilhanças Generalizada (RVG) não apresenta monotonicidade, ao contrário de testes formulados sob a perspectiva bayesiana, como o teste de Bayes baseado em probabilidades a posteriori, o teste de Lindley e o FBST. Porém, são verificadas, sob a teoria da decisão, quando possível, quais as condições suficientes para que uma classe de testes de hipóteses tenha monotonicidade. / Most of the texts in the literature of hypothesis testing deal with optimality criteria for a single decision problem. However, there are, to a lesser extent, texts on the problem of simultaneous hypothesis testing and the logical consistency of the optimal solutions of such procedures. For instance, the following property should be observed in simultaneous hypothesis testing: if a hypothesis H implies a hypothesis H0, then, on the basis of the same sample observation, the rejection of the hypothesis H0 necessarily should imply the rejection of the hypothesis H. Here, this property is called monotonicity. To investigate this property under a more general point of view, in this work, it is dened rst the notion of a class of hypothesis testing, which extends the test function to a sigma-eld of possible null hypotheses, and then the concept of monotonicity is introduced properly. It is also shown, through some simple examples, that for a xed signicance level, the class of Generalized Likelihood Ratio tests (GLR) does not meet monotonicity, as opposed to tests developed under the Bayesian perspective, such as Bayes tests based on posterior probabilities, Lindleys tests and Full Bayesian Signicance Tests (FBST). Finally, sucient conditions for a class of hypothesis testing to have monotonicity are determined, when possible, under a decision-theoretic approach.

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This thesis contributes to the theoretical work on decision and game theory when decision makers or players perceive ambiguity. The first article introduces a new axiomatic framework for ambiguity aversion and provides axiomatic characterizations for important preference classes that thus far had lacked characterizations. The second article introduces a new axiom called Weak Monotonicity which is shown to play a crucial role in the multiple prior model. It is shown that for many important preference classes, the assumption of monotonic preferences is a consequence of the other axioms and does not have to be assumed. The third article introduces an intuitive definition of perceived ambiguity in the multiple prior model. It is shown that the approach allows an application to games where players perceive strategic ambiguity. A very general equilibrium existence result is given. The modelling capabilities of the approach are highlighted through the analysis of examples. The fourth article applies the model from the previous article to a specific class of games with a lattice-structure. We perform comparative statics on perceived ambiguity and ambiguity attitude. We show that more optimism does not necessarily lead to higher equilibria when players have Alpha-Maxmin preferences. We present necessary and sufficient conditions on the structure of the prior sets for this comparative statics result to hold. The introductory chapter provides the basis of the four articles in this thesis. An overview of axiomatic decision theory, decision-making under ambiguity and ambiguous games is given. It introduces and discusses the most relevant results from the literature.

Na Inferência Estatística, é comum, após a realização de um experimento, testar simultaneamente um conjunto de diferentes hipóteses de interesse acerca de um parâmetro desconhecido. Assim, para cada hipótese, realiza-se um teste de hipótese e, a partir disto, conclui-se algo sobre os parâmetros de interesse. O objetivo deste trabalho é avaliar a (falta de) concordância lógica entre as conclusões obtidas a partir dos testes realizados após a observação de um único experimento. Neste estudo, é apresentada uma definição de classe de testes de hipóteses, uma função que para cada hipótese de interesse associa uma função de teste. São então avaliadas algumas propriedades que refletem como gostaríamos que testes para diferentes hipóteses se comportassem em termos de coerência lógica. Tais propriedades são exemplificadas através de classes de testes que as satisfazem. A seguir, consideram-se conjuntos de axiomas para classes. Estes axiomas são baseados nas propriedades mencionadas. Classes de testes usuais são investigadas com relação aos conjuntos de axiomas propostos. São também estudadas propriedades advindas de tais conjuntos de axiomas. Por fim, estuda-se um resultado que estabelece uma espécie de conexão entre testes de hipóteses e estimação pontual. / In Statistical Inference, it is usual, after an experiment is performed, to test simultaneously a set of hypotheses of interest concerning an unknown parameter. Therefore, to each hypothesis, a statistical test is performed and a conclusion about the parameter is drawn based on it. The objective of this work is to evaluate the (lack of) logical coherence among conclusions obtained from tests conducted after the observation of a single experiment. In this study, a definition of class of hypotheses tests, a function that associates a test function to each hypothesis of interest, is presented. Some properties that reflect what one could expect (in terms of logical coherence) from tests to different hypotheses are then evaluated. These properties are exemplified by classes of hypotheses tests that respect them. Then, sets of axioms based on the properties studied are proposed to classes of hypotheses tests. Usual classes of hypotheses tests are investigated with respect to these sets of axioms. Some properties related to these sets of axioms are then analyzed. At last, a result which seems to connect hypotheses testing and point estimation is stated.

As we know the most popular inference methods for order restricted model are likelihood inference. In such models, the Maximum Likelihood Estimation (MLE) and Likelihood Ratio Testing (LRT) appear some suspect behaviour and unsatisfactory. In this thesis, I review the articles that focused in the behaviour of the Likelihood methods on Order restricted models. For those situations, an alternative method is preferred. However, likelihood inference is satisfactory for simple order cone restriction. But it is unsatisfactory when the restrictions are of the tree order, umbrella order, star-shaped and stochastic order types.

Recursion is a fundamental tool of mathematics used to define, construct, and analyze mathematical objects. This work employs induction, sieving, inversion, and other recursive methods to solve a variety of problems in the areas of algebraic number theory, topological and combinatorial graph theory, and analytic probability and statistics. A common theme of recursively defined functions, weighted sums, and cross-referencing sequences arises in all three contexts, and supplemented by sieving methods, generating functions, asymptotics, and heuristic algorithms. In the area of number theory, this work generalizes the sieve of Eratosthenes to a sequence of polynomial values called polynomial-value sieving. In the case of quadratics, the method of polynomial-value sieving may be characterized briefly as a product presentation of two binary quadratic forms. Polynomials for which the polynomial-value sieving yields all possible integer factorizations of the polynomial values are called recursively-factorable. The Euler and Legendre prime producing polynomials of the form n2+n+p and 2n2+p, respectively, and Landau's n2+1 are shown to be recursively-factorable. Integer factorizations realized by the polynomial-value sieving method, applied to quadratic functions, are in direct correspondence with the lattice point solutions (X,Y) of the conic sections aX2+bXY +cY2+X-nY=0. The factorization structure of the underlying quadratic polynomial is shown to have geometric properties in the space of the associated lattice point solutions of these conic sections. In the area of combinatorial graph theory, this work considers two topological structures that are used to model the process of homologous genetic recombination: assembly graphs and chord diagrams. The result of a homologous recombination can be recorded as a sequence of signed permutations called a micronuclear arrangement. In the assembly graph model, each micronuclear arrangement corresponds to a directed Hamiltonian polygonal path within a directed assembly graph. Starting from a given assembly graph, we construct all the associated micronuclear arrangements. Another way of modeling genetic rearrangement is to represent precursor and product genes as a sequence of blocks which form arcs of a circle. Associating matching blocks in the precursor and product gene with chords produces a chord diagram. The braid index of a chord diagram can be used to measure the scope of interaction between the crossings of the chords. We augment the brute force algorithm for computing the braid index to utilize a divide and conquer strategy. Both assembly graphs and chord diagrams are closely associated with double occurrence words, so we classify and enumerate the double occurrence words based on several notions of irreducibility. In the area of analytic probability, moments abstractly describe the shape of a probability distribution. Over the years, numerous varieties of moments such as central moments, factorial moments, and cumulants have been developed to assist in statistical analysis. We use inversion formulas to compute high order moments of various types for common probability distributions, and show how the successive ratios of moments can be used for distribution and parameter fitting. We consider examples for both simulated binomial data and the probability distribution affiliated with the braid index counting sequence. Finally we consider a sequence of multiparameter binomial sums which shares similar properties with the moment sequences generated by the binomial and beta-binomial distributions. This sequence of sums behaves asymptotically like the high order moments of the beta distribution, and has completely monotonic properties.