The purpose of this paper is to review and summarize the topics involved in the study of real analysis. Real analysis is a branch of mathematics that studies the field of real numbers including the calculus of real numbers, analytical properties of real functions and sequences. This includes limits of sequences of real numbers, continuity, completeness, and related properties of real functions. While all topics in the course were important and vital to understanding analysis, the goal of this paper is to review, research, and report on a few of the more interesting topics covered in the class. / text

<p>The paper begins with a short survey of monotone functions. The functions of bounded variation are introduced and some basic properties of these functions are given. Finally the jump function of a function of bounded variation is defined.</p>

The paper begins with a short survey of monotone functions. The functions of bounded variation are introduced and some basic properties of these functions are given. Finally the jump function of a function of bounded variation is defined.

The Probabilistic Method was primarily used in Combinatorics and pioneered by Erdös Pai, better known to Westerners as Paul Erdos in the 1950s. The probabilistic method is a powerful tool for solving many problems in discrete mathematics, combinatorics and also in graph .theory. It is also very useful to solve problems in number theory, combinatorial geometry, linear algebra and real analysis. More recently, it has been applied in the development of efficient algorithms and in the study of various computational problems.Broadly, the probabilistic method is somewhat opposite of the extremal graph theory. Instead of considering how a graph can behave in the extreme, we consider how a collection of graphs behave on 'average' where by we can formulate a probability space. The method allows one to prove the existence of a structure with particular properties by defining an appropriate probability space of structures and show that the desired properties hold in the space with positive probability.(please see PDF for complete abstract)

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In this paper we will present two different approaches to the development of product measures. In the second chapter we follow the lead of H. L. Royden in his book Real Analysis and develop product measure in the context of outer measure. The approach in the third and fourth chapters will be the one taken by N. Dunford and J. Schwartz in their book Linear Operators Part I. Specifically, in the fourth chapter, product measures arise almost entirely as a consequence of integration theory. Both developments culminate with proofs of well known theorems due to Fubini and Tonelli.

L'analyse réelle a de nombreuses applications car c'est un outil approprié pour modéliser de nombreux phénomènes physiques et socio-économiques. En tant que tel, sa formalisation dans des systèmes de preuve formelle est justifié pour permettre aux utilisateurs de vérifier formellement des théorèmes mathématiques et l'exactitude de systèmes critiques. La bibliothèque standard de Coq dispose d'une axiomatisation des nombres réels et d'une bibliothèque de théorèmes d'analyse réelle. Malheureusement, cette bibliothèque souffre de nombreuses lacunes. Par exemple, les définitions des intégrales et des dérivées sont basées sur les types dépendants, ce qui les rend difficiles à utiliser dans la pratique. Cette thèse décrit d'abord l'état de l'art des différentes bibliothèques d'analyse réelle disponibles dans les assistants de preuve. Pour pallier les insuffisances de la bibliothèque standard de Coq, nous avons conçu une bibliothèque facile à utiliser : Coquelicot. Une façon plus facile d'écrire les formules et les théorèmes a été mise en place en utilisant des fonctions totales à la place des types dépendants pour écrire les limites, dérivées, intégrales et séries entières. Pour faciliter l'utilisation, la bibliothèque dispose d'un ensemble complet de théorèmes couvrant ces notions, mais aussi quelques extensions comme les intégrales à paramètres et les comportements asymptotiques. En plus, une hiérarchie algébrique permet d'appliquer certains théorèmes dans un cadre plus générique comme les nombres complexes pour les matrices. Coquelicot est une extension conservative de l'analyse classique de la bibliothèque standard de Coq et nous avons démontré les théorèmes de correspondance entre les deux formalisations. Nous avons testé la bibliothèque sur plusieurs cas d'utilisation : sur une épreuve du Baccalauréat, pour les définitions et les propriétés des fonctions de Bessel ainsi que pour la solution de l'équation des ondes en dimension 1. / Real analysis is pervasive to many applications, if only because it is a suitable tool for modeling physical or socio-economical systems. As such, its support is warranted in proof assistants, so that the users have a way to formally verify mathematical theorems and correctness of critical systems. The Coq system comes with an axiomatization of standard real numbers and a library of theorems on real analysis. Unfortunately, this standard library is lacking some widely used results. For instance, the definitions of integrals and derivatives are based on dependent types, which make them cumbersome to use in practice. This thesis first describes various state-of-the-art libraries available in proof assistants. To palliate the inadequacies of the Coq standard library, we have designed a user-friendly formalization of real analysis: Coquelicot. An easier way of writing formulas and theorem statements is achieved by relying on total functions in place of dependent types for limits, derivatives, integrals, power series, and so on. To help with the proof process, the library comes with a comprehensive set of theorems that cover not only these notions, but also some extensions such as parametric integrals and asymptotic behaviors. Moreover, an algebraic hierarchy makes it possible to apply some of the theorems in a more generic setting, such as complex numbers or matrices. Coquelicot is a conservative extension of the classical analysis of Coq's standard library and we provide correspondence theorems between the two formalizations. We have exercised the library on several use cases: in an exam at university entry level, for the definitions and properties of Bessel functions, and for the solution of the one-dimensional wave equation.

Made available in DSpace on 2015-04-22T22:11:27Z (GMT). No. of bitstreams: 1 Adilson Alves.pdf: 1168831 bytes, checksum: b0b3c601292bebc707ce420d4f776ce6 (MD5) Previous issue date: 2013-10-13 / The search for sophisticated methods more of evaluation of investments that are capable to deal with uncertainty, irreversibilidade, risk and that they guarantee flexibility, has been one exausto exercise on the part of researchers, mainly in projects that involve investments in products with high aggregate value. From a bibliographical survey and of laboratory, the present dissertation analyzes the viability of the industry gas-chemistry in the Industrial Polo region of Manaus. It considers an economic-financial, established analysis in use of Real Options (OR), applied Simulation for Monte Carlo (SMC). Thus, the prices of the entrances and exits are shaped through one determined random process, in this in case that Geometric Movement Browniano (MGB), is calculated the parameters (drift and volatileness) of each series of prices and after that are made its SMC. The variable fluctuation of prices, variations of market, delays or advancings in the execution of the project, estimates of cost and forecasts for the attainment of the necessary approvals, are observed to Real Options (OR). Next, considers the combination of entrances and exits that the function maximizes profit in one determined period and calculates the liquid present value (VPL). Finally, it currently presents the used methods in this precificacion, giving emphasis to the method of the Monte Simulation Carlo (SMC), it points the advantages in relation to the classic analysis and identifies the main existing OR in a project of great transport / A busca por métodos mais sofisticados de avaliação de investimentos que sejam capazes de lidar com incerteza, irreversibilidade, risco e que garantam flexibilidade, têm sido um exausto exercício por parte de pesquisadores, principalmente em projetos que envolvam investimentos em produtos com alto valor agregado. A partir de um levantamento bibliográfico e de laboratório, a presente dissertação analisa a viabilidade da indústria gás-química no Pólo Industrial de Manaus. Propõe uma análise econômico-financeiro , baseado em uso de Opções Reais (OR), aplicado por Simulação de Monte Carlo (SMC). Desta forma, os preços das entradas e saídas são modelados através de um determinado processo estocástico, neste caso o Movimento Geométrico Browniano (MGB), são calculados os parâmetros (drift e volatilidade) de cada série de preços e em seguida são feitos sua Simulação por Monte Carlo (SMC). As variáveis flutuação de preços, variações de mercado, atrasos ou adiantamentos na execução do projeto, estimativas de custo e previsões para a obtenção das aprovações necessárias, são observadas para se precificar Opções Reais (OR). Seguidamente, propõe a combinação de entradas e saídas que maximiza a função lucro num determinado período e calcula o valor presente líquido (VPL). Finalmente, apresenta os métodos usados atualmente nessa precificação, dando ênfase ao método da Simulação de Monte Carlo (SMC), aponta as vantagens em relação à análise clássica e identifica as principais OR existentes em um projeto de grande porte

Orientador: Roger Miarka / Resumo: Esta pesquisa, como uma espécie de barco, devém composição que se lança num mar em busca de uma dobra. Uma prega na linha do fora na qual seja possível inventar um dentro habitável e dar passagem a afetos disso que é conhecido por rigor. Um rigor que se relaciona à disciplina de Análise Real, a qual é contemplada numa grande maioria de cursos de licenciatura em Matemática em nosso país. Tal rigor mostra-se discursivamente atrelado a muitas práticas que ocorrem na formação de professores de Matemática, as quais influenciam sobremaneira a forma como licenciandos lidam com a própria graduação e com suas práticas profissionais. Em nossa composição, adotamos uma estética de escrita-fluxo, em que narrativas ficcionais foram compostas tendo como inspiração e suporte vivências experienciadas pelo próprio autor e por toda uma multidão de sujeitos e obras que o circundam e o atravessam, no intuito de construir uma Tese sem Órgãos, ou seja, uma composição desprovida de uma organização prévia. Para tal, estabelecemos alianças teóricas mais estreitas com o pensamento arqueológico de Michel Foucault e com o conceito de Corpo sem Órgãos, de Gilles Deleuze e Félix Guattari, bem como consonâncias junto a constelações teóricas que orbitam cada um desses pensadores. Desta maneira, nas linhas rizomáticas compostas a partir de acoplamentos, desacoplamentos e interrupções de fluxo causadas nas e pelas narrativas construídas, foi possível operar nas fissuras e pensar o conceito de Rigor sem Órgãos,... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: This research, as a kind of boat, becomes composition that is launched in a sea in search for a fold. A crease in the line of the outside in which it is possible to invent a habitable inside and and give passage to affections of that which is known as rigor. A rigor that is related to the discipline of Real Analysis, which is included in a large majority of undergraduate courses in Mathematics in our country. Such rigor is shown to be discursively linked to many practices that occur in the formation of mathematics teachers, which greatly influence the way they deal with their own graduation and their professional practices. In our composition, we adopted a flow-writing aesthetic, in which fictional narratives were composed having as inspiration and support experiences experienced by the author himself and by a whole multitude of subjects and works that surround and cross him, in order to build a Thesis without Organs, that is, a composition without prior organization. To this goal, we established closer theoretical alliances with the archaeological thought by Michel Foucault and with the concept of Body without Organs, by Gilles Deleuze and Félix Guattari, as well as consonances with theoretical constellations that orbit each of these thinkers. In this way, in the rhizomatic lines composed from couplings, decouplings and interruptions of flow caused in and by the constructed narratives, it was possible to operate in the cracks and think about the concept of Rigor without Orga... (Complete abstract click electronic access below) / Doutor

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-02-23T14:01:44Z No. of bitstreams: 1 willianjosedacruz.pdf: 5165704 bytes, checksum: 494d07f969a44ae41c762d7617ed44a5 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-02-23T14:34:27Z (GMT) No. of bitstreams: 1 willianjosedacruz.pdf: 5165704 bytes, checksum: 494d07f969a44ae41c762d7617ed44a5 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-02-23T14:34:49Z (GMT) No. of bitstreams: 1 willianjosedacruz.pdf: 5165704 bytes, checksum: 494d07f969a44ae41c762d7617ed44a5 (MD5) / Made available in DSpace on 2017-02-23T14:34:49Z (GMT). No. of bitstreams: 1 willianjosedacruz.pdf: 5165704 bytes, checksum: 494d07f969a44ae41c762d7617ed44a5 (MD5) Previous issue date: 2011-04-29 / Com a percepção que distingue a matemática escolar em vários aspectos da matemática científica, esta pesquisa propõe uma aproximação entre essas duas formas conceituais, no que discute o entender e fazer matemática, perpassando pelas ideias que diferenciam cada uma delas. Ainda que num tratamento formal na apresentação dos Números Reais, serão apontadas possíveis consequências e aplicações desta apresentação nas séries finais do ensino fundamental e do ensino médio. Esta pesquisa também é uma tentativa de iniciar uma reflexão, a partir dos Números Reais, que possa permitir uma mudança na forma de trabalho com a disciplina Análise Real, nos cursos de Licenciatura em Matemática, diminuindo a dicotomia entre a formação matemática do professor e sua prática docente. / With the perception that distinguishes school mathematics in various scientific aspects of mathematics, this research proposes a rapprochement between these two conceptual ways in which discusses the understanding and doing mathematics, passing by the ideas that distinguish each one. Although a formal treatment in the presentation of real numbers, will be pointed out the possible consequences and applications of this presentation in the final grades of elementary school and high school. This research is also an attempt to initiate a reflection from the real numbers, that would enable a shift in the discipline of working with Real Analysis courses in Mathematics, reducing the dichotomy between mathematics teacher education and practice teacher.