Usando teoria de perturbação, fizemos uma análise detalhada da divergência infravermelha em 4ª ordem do processo de aniquilação do par elétron-pósitron. Mostramos que o teorema de Bloch-Nordsieck garante o cancelamento destas divergências em QED. Em seguida, consideramos um processo em QCD com duas partículas coloridas no estado inicial, somando sobre as cores destas duas partículas. O processo escolhido foi o de aniquilação do par quark-anti-quark. O processo em 4ª ordem de perturbação e mostramos neste caso que o Teorema de Block-Nordsieck não se verifica. Este resultado é consequência do fato desta teoria ter um caráter não-abeliano. O não-cancelamento das divergências infravermelhas significa que a natureza do estado ligado é fundamental para o uso consistente da teoria de perturbação na QCD. / Using pertubation theory, we made a detailed analysis of the infrared divergencies to the 4th order for the phocess of annihilation of the pair electron-positron. We showed that the Bloch-Nordsieck theorem guarantees the cancellation of these divergencies in QED. Then we considered a process in QED involving two coloured particles in the initial state, summing over the colours of these particles. The process chosen was the annihilation of the pair quark-anti-quark. We calculated the process to the 4ª order in perturbation theory, and showed that in this case the Bloch-Nordsieck theorem in not verified. This result is a consequence of the fact that this theory has a non-abelian caracter. The non-cancellation of the infrared divergencies signifies that the nature of the bound-state is fundamental for the consistent use of perturbation theory in QCD.

We employ some of the modern tools of economic theory and time series econometrics to consider a number of economic problems. The communication and coordination problems we study are relevant in accounting, business, economics and finance. The thesis begins by examining the behaviour of people and organisations, who are supposed to share a common goal. Then it considers the equilibriating mechanisms of behaviour by groups of economic agents, who usually have conflicting interests. We apply the tools of non-cooperative game theory, which constitutes a large part of modern economic theory. In the sequel, we address the question of why people behave the way they do in their economic a↵airs. Peoples' economic behaviour is mirrored in the aggregates of macroeconomics. We propose a Time Varying Autoregressive model to study the movements in the five main macroeconomic variables. The methods come from standard Time Series Analysis, but we do introduce some innovative time series techniques. Finally, we conduct an empirical investigation of the movements in one of the five main macroeconomic variables, the rate of inflation. Among the econometric tools employed are standard Autoregressive models (AR), Autoregressive Distributed Lag models (ADL) and the more recent Generalised Autoregressive Conditional Heteroskedasticity (GARCH) methodology.

O principal objetivo desta dissertação é obter um Teorema do Envelope que permita mecanismos não diferenciáveis, preferências arbitrárias e que possa ser aplicado em modelos com múltiplos agentes. Nós alcançamos isto ao expandir a análise de Milgrom e Segal (2002), generalizando seus resultados para espaços de tipos multidimensionais. Dessa forma, continuamos permitindo que a regra de escolha (mecanismo) seja descontínua. Para obter nosso resultado, é necessário o uso do Teorema do Máximo de Berge e, consequentemente, devemos impor compacidade no conjunto de escolha. Inicialmente esta hipótese pode parecer forte, porém argumentamos que em aplicações _e muito improvável termos um conjunto de escolha aberto ou, principalmente, não limitado. Nós também identificamos condições para que a função valor seja absolutamente contínua e mostramos que sua representação integral também é válida para espaços de tipos multidimensionais. Inicialmente propomos uma generalização direta do resultado de Milgrom e Segal (2002), utilizando a hipótese de continuidade absoluta da função de utilidade do agente. Entretanto, esta exigência não possui muito significado econômico e é considerada pouco elegante por parte da literatura. Neste sentido, incorporamos uma hipótese adicional de diferenciabilidade da utilidade em todo o domínio que gera a mesma representação integral e possui uma maior interpretação econômica. Nossos resultados são, em geral, aplicados a modelos com múltiplos agentes, em especial Economia do Setor Público (provisão de bens públicos e taxação ótima) e teoria dos leilões. / The main objective of this dissertation is to obtain an Envelope Theorem that allows non-di erentiable mechanisms, arbitrary preferences, and that can be applied to models with multiple agents. We achieve that by expanding the analysis of Milgrom and Segal (2002) and generalizing their results to multidimensional type spaces. Thus, we continue allowing that the choice rule (mechanism) is discontinuous. For our result, it is necessary to use the Berge's Maximum Theorem and therefore we must impose compactness in the choice set. Initially this assumption may seem strong, but we argue that in applications there is an open or unbounded choice set is very unlikely. We also identify conditions for the value function is absolutely continuous and show that its integral representation is also valid for multidimensional type spaces. Firstly we propose a direct generalization of the Milgrom and Segal (2002)'s result, using the assumption of absolute continuity of the agent's utility function. However, this requirement does not have much economic interpretation and it is considered not very elegant in the literature. In this sense, we incorporate an additional assumption of di erentiability of the utility in all range that generates the same integral representation and it possesses a greater economic interpretation. Our results are generally applied to models with multiple agents, in particular Public Economics (public goods supply and optimal taxation) and auction theory.

In this thesis, we provide constructive proofs of serveral generalizations of Sperner's Lemma, a combinatorial result which is equivalent to the Brouwer Fixed Point Theorem. This lemma makes a statement about the number of a certain type of simplices in the triangulation of a simplex with a special labeling. We prove generalizations for polytopes with simplicial facets, for arbitrary 3-polytopes, and for polygons. We introduce a labeled graph which we call a nerve graph to prove these results. We also suggest a possible non-constructive proof for a polytopal generalization.

We give an overview of Hahn-Banch Theorems. We present the statements of these theorems along with some definitions that are required to understand these statements and work out some applications.

Interactive theorem provers are software tools that help users create machine-checked proofs. Although difficult to use, they have been playing an important role in the effort to create highly reliable software. I present several novel user interface ideas for interactive theorem provers, generalizable to other mathematics and programming tools. Prototypes tailored to the Coq interactive theorem prover were developed and tested in an experiment with human participants. The results show promising directions for making interactive theorem provers easier to use.

In the late 19th and early 20th centuries famous scientists like Boltzmann, Loschmidt, Maxwell and Einstein tried, unsuccessfully, to find the link between the time-reversible equations of motion of individual molecules and irreversible thermodynamics. The solution to this puzzle was found in 1993, and the link is now known as the Fluctuation Theorem (FT). In the decade that followed theory and computer simulation tested the FT and, in 2002, an experiment indirectly demonstrated the FT.¶ This thesis describes original experiments that demonstrate the FT directly using Optical Tweezers. A related expression, known as the Kawasaki Identity, is also experimentally demonstrated. These experimental results provide a rigorous demonstration that irreversible dynamics can be obtained from a system with time-reversible dynamics.

A well known result in probability is that convergence almost surely (a.s.) of a sequence of random elements implies weak convergence of their laws. The Ukrainian mathematician Anatoliy Volodymyrovych Skorohod proved the lemma known as Skorohods a.s. representation Theorem, a partial converse of this result. In this work we discuss the notion of continuous representations, which allows us to provide generalizations of Skorohods Theorem. In Chapter 2, we explore Blackwell and Dubinss extension [3] and Ferniques extension [10]. In Chapter 3 we present Cortissozs result [5], a variant of Skorokhods Theorem. It is shown that given a continuous path inM(S) it can be associated a continuous path with fixed endpoints in the space of S-valued random elements on a nonatomic probability space, endowed with the topology of convergence in probability. In Chapter 4 we modify Blackwell and Dubins representation for particular cases of S, such as certain subsets of R or R^n. / Mathematics

Thesis (M.S.)--Brigham Young University. Dept. of Physics and Astronomy, 2006. / Includes bibliographical references (p. 113-115).

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