O objetivo deste trabalho é caracterizar os polinômios cujas raízes têm todas parte real negativa, chamados de polinômios estáveis ou de Hurwitz. Para este fim, apresentaremos e provaremos o critério de Routh-Hurwitz. Também estenderemos este resultado para obter uma caracterização da estabilidade para uma família de polinômios com seus coeficientes variando independentemente num intervalo limitado. Aplicaremos os resultados para obter um critério de estabilidade robusta para um sistema de equações diferenciais que descreve um sistema mecânico. / The objective of this work is to determine when all of zeros of a given polynomial have negative real parts, called stable or Hurwitz polynomials. We will present and prove the Routh-Hurwitz criterion. Furthermore we will extend the result for classes of polynomials defined by letting their coeficients vary independently in an arbitrary finite interval. Then we will apply them to derive a robust stability condition for a mechanical system.

O objetivo deste trabalho é caracterizar os polinômios cujas raízes têm todas parte real negativa, chamados de polinômios estáveis ou de Hurwitz. Para este fim, apresentaremos e provaremos o critério de Routh-Hurwitz. Também estenderemos este resultado para obter uma caracterização da estabilidade para uma família de polinômios com seus coeficientes variando independentemente num intervalo limitado. Aplicaremos os resultados para obter um critério de estabilidade robusta para um sistema de equações diferenciais que descreve um sistema mecânico. / The objective of this work is to determine when all of zeros of a given polynomial have negative real parts, called stable or Hurwitz polynomials. We will present and prove the Routh-Hurwitz criterion. Furthermore we will extend the result for classes of polynomials defined by letting their coeficients vary independently in an arbitrary finite interval. Then we will apply them to derive a robust stability condition for a mechanical system.

The goal of this paper is to explain the mathematics involved in modeling biochemical oscillations. We first discuss several important biochemical concepts fundamental to the construction of descriptive mathematical models. We review the basic theory of differential equations and stability analysis as it relates to two-variable models exhibiting oscillatory behavior. The importance of the Hopf Bifurcation will be discussed in detail for the central role it plays in limit cycle behavior and instability. Once we have exposed the necessary mathematical framework, we consider several specific models of biochemical oscillators in three or more variables. This will include a detailed analysis of Goodwin's equations and their modification first studied by Painter. Additionally, we consider the consequences of introducing both distributed and discrete time delay into Goodwin's model. We will show that the presence of distributed time lag modifies Goodwin's model in no significant way. The final section of the paper will discuss discrete time lag in the context of a minimal model of the circadian rhythm. In the main, this paper will address mathematical, as opposed to biochemical, issues. Nevertheless, the significance of the mathematics to the biochemistry will be considered throughout. / Master of Science

O objetivo deste trabalho é estudar a ocorrência de mudanças de regime típicas de comportamentos em sistemas complexos, em particular no contexto de sistemas dinâmicos aplicados. Para isso, desenvolvemos um modelo matemático que representa a interação entre uma cultura de ostras utilizadas para consumo humano e os processos de eutrofização e biorremediação do ecossistema que as contém. As interações entre as populações de ostras e do fitoplâncton entre si e com a matéria suspensa, subproduto das relações entre os componentes do meio e seu processo de eutrofização, alteram os níveis de oxigenação e a consequente qualidade da ´agua devido `a realização de maior ou menor quantidade de fotossíntese pelas vegetações mais profundas do meio. Neste trabalho propomos um sistema dinâmico de três variáveis para modelar esse sistema e analisamos seus pontos de equilíbrio usando duas técnicas, método de Quirk-Ruppert e os critérios de Routh-Hurwitz, além de resolvê-lo numericamente para um conjunto de parâmetros realísticos (fenomenológicos) obtidos a partir da literatura especializada. Nossos resultados indicam que o limite de extração diária de ostras que pode ser realizado sem levar a cultura ao colapso gira em torno de 4.8% da população / The objective of this work is to study the occurrence of regime shifts that are typical in the behavior of complex systems, in particular in the context of applied dynamical systems. Accordingly, we have developed a mathematical model that represents the interaction between a culture of oysters used for human consumption and the eutrophication and bioremediation processes of the ecosystem containing the culture. The interactions between the oyster populations and the phytoplankton between themselves and with the suspended matter, that appears as a by-product of the relationship between the components of the medium and its eutrophication process, change the oxygenation levels and the resulting water quality due to the realization of a greater or lesser amount of photosynthesis by the vegetation of the deeper levels. In this paper we propose a dynamical system of three variables to model the system and analyze its points of equilibrium using two techniques, the Quirk-Ruppert method and the Routh-Hurwitz criteria, besides solving the equations numerically for a realistic phenomenological) set of parameters obtained from the literature. Our results indicate that the daily extraction threshold that can be achieved without collapsing the culture of oysters amounts to approximately 4.8% of the total population

O objetivo deste trabalho é estudar a ocorrência de mudanças de regime típicas de comportamentos em sistemas complexos, em particular no contexto de sistemas dinâmicos aplicados. Para isso, desenvolvemos um modelo matemático que representa a interação entre uma cultura de ostras utilizadas para consumo humano e os processos de eutrofização e biorremediação do ecossistema que as contém. As interações entre as populações de ostras e do fitoplâncton entre si e com a matéria suspensa, subproduto das relações entre os componentes do meio e seu processo de eutrofização, alteram os níveis de oxigenação e a consequente qualidade da ´agua devido `a realização de maior ou menor quantidade de fotossíntese pelas vegetações mais profundas do meio. Neste trabalho propomos um sistema dinâmico de três variáveis para modelar esse sistema e analisamos seus pontos de equilíbrio usando duas técnicas, método de Quirk-Ruppert e os critérios de Routh-Hurwitz, além de resolvê-lo numericamente para um conjunto de parâmetros realísticos (fenomenológicos) obtidos a partir da literatura especializada. Nossos resultados indicam que o limite de extração diária de ostras que pode ser realizado sem levar a cultura ao colapso gira em torno de 4.8% da população / The objective of this work is to study the occurrence of regime shifts that are typical in the behavior of complex systems, in particular in the context of applied dynamical systems. Accordingly, we have developed a mathematical model that represents the interaction between a culture of oysters used for human consumption and the eutrophication and bioremediation processes of the ecosystem containing the culture. The interactions between the oyster populations and the phytoplankton between themselves and with the suspended matter, that appears as a by-product of the relationship between the components of the medium and its eutrophication process, change the oxygenation levels and the resulting water quality due to the realization of a greater or lesser amount of photosynthesis by the vegetation of the deeper levels. In this paper we propose a dynamical system of three variables to model the system and analyze its points of equilibrium using two techniques, the Quirk-Ruppert method and the Routh-Hurwitz criteria, besides solving the equations numerically for a realistic phenomenological) set of parameters obtained from the literature. Our results indicate that the daily extraction threshold that can be achieved without collapsing the culture of oysters amounts to approximately 4.8% of the total population

Résumé : La principale caractéristique des marchés OTC (Over-The-Counter) est l’absence d’un mécanisme de négociation centralisée (comme des ventes aux enchères, des spécialistes ou des limit-order books). Les acheteurs et les vendeurs sont donc souvent dans l'ignorance des prix actuellement disponibles auprès d'autres contreparties potentielles et ont une connaissance limitée de l’amplitude des transactions récemment négociées ailleurs sur le marché. C'est la raison pour laquelle les marchés OTC sont qualifiés de relativement opaques et nommés «Dark Markets» par Duffie (2012) dans sa récente monographie afin de refléter le fait que les investisseurs sont en quelque sorte dans le noir au sujet du meilleur prix disponible et de la personne à contacter pour faire la meilleure transaction. Dans ce travail, nous sommes particulièrement intéressés à l’évolution temporelle de la transmission de l’information au cours des séances de négociation. Plus précisément, nous cherchons à établir la stabilité asymptotique de la dynamique de partage de l'information au sein d’une large population d’investisseurs caractérisés par la fréquence/intensité des rencontres entre investisseurs. L’effort optimal déployé par un agent en recherche d’informations dépend de son niveau actuel d'information et de la distribution transversale des efforts de recherche des autres agents. Dans le cadre défini par Duffie-Malamud-Manso (2009), à l’équilibre, les agents recherchent au maximum jusqu'à ce que la qualité de leur information atteigne un certain niveau, déclenchant une nouvelle phase de recherche minimale. Dans le contexte de percolation d'information entre agents, l'information peut être transmise parfaitement ou imparfaitement. La première étude de ce problème de percolation a été faite par Duffie-Manso (2007), puis par Duffie-Giroux-Manso (2010). Dans cette deuxième étude, le cas de la percolation de l'information par des groupes de plus de deux investisseurs a été abordé et résolu. Cette dernière étude a conduit au problème de l'extension des sommes de Wild dans Bélanger-Giroux (2013). D'autre part, dans Duffie-Malamud-Manso (2009), chaque agent est doté de signaux quant à l'issue probable d'une variable aléatoire d'intérêt commun dans l’optique de transmission d’informations dans une large population d'agents. Un tel contexte conduit à des systèmes d'équations non linéaires d’évolution. Leur objectif est d'obtenir une politique d'équilibre déterminée par un ensemble de paramètres d'une politique de cible traduisant le fait que l’effort de recherche qui doit être minimal lorsqu’un agent possède suffisamment d’information. Dans ce travail, nous sommes en mesure d'obtenir l'existence de l’état d’équilibre, même lorsque la fonction d'intensité n'est pas un produit. De plus, nous sommes également en mesure de montrer la stabilité asymptotique pour toute loi initiale par un changement de noyaux. Enfin, nous élargissons les hypothèses de Bélanger-Giroux (2012) pour montrer la stabilité exponentielle par le critère de Routh-Hurwitz pour un autre exemple de système à un nombre fini d’équations. // Abstract : Over-the-counter (OTC) markets have the main characteristic that they do not use a centralized trading mechanism (such as auctions, specialist, or limit-order book) to aggregate bids and offers and to allocate trades. The buyers and sellers have often a limited knowledge of trades recently negotiated elsewhere in the market. They are also negotiating in potential ignorance of the prices currently available from other counterparties. This is the reason why OTC markets are said to be relatively opaque and are qualified as «Dark Markets» by Duffie (2012) in his recent monograph to reflect the fact that investors are somewhat in the dark about the most attractive available deals and about whom to contact. In this work, we are particularly interested in the evolution over time of the distribution across investors of information learned from private trade negotiations. Specifically, we aim to establish the asymptotic stability of equilibrium dynamics of information sharing in a large interaction set. An agent’s optimal current effort to search for information sharing opportunities depends on that agent’s current level of information and on the cross-sectional distribution of information quality and search efforts of other agents. Under the Duffie-Malamud-Manso (2009) framework, in equilibrium, agents search maximally until their information quality reaches a trigger level and then search minimally. In the context of percolation of information between agents, the information can be transmitted directly or indirectly. The first studies of such a problem were made by Duffie-Manso (2007) and then by Duffie-Giroux-Manso (2010). In that second study the case of the percolation of information by groups of more than 2 investors was addressed and solved for a perfect information transmission kernel. That last study has led Bélanger-Giroux (2013) to the problem of extending the Wild sums for a general interacting kernel (not only for the kernel which adds the information). On the other hand, in Duffie-Malamud-Manso (2009), the authors explain that, for the information sharing in a large population, each agent is endowed with signals regarding the likely outcome of a random variable of common concern, like the price of an asset of common interest. Such a setting leads to nonlinear systems of evolution equations. The agents’ goal is to obtain an equilibrium policy specified by a set of parameters of a trigger policy; more specifically the minimal search effort trigger policies. We concentrate our study on those trigger policies in order to provide more intuitive and practical results. Doing so, we are able to obtain the existence of the steady state even when the intensity function is not a product. And in our framework, we are even able to show the asymptotic stability starting with any initial law. This can be done because we are able to show that, by a change of kernels, the systems of ODE’s, which are expressed by a set of kernels (one 1-airy and one 2-airy) are equivalent to systems expressed with a single 2-airy kernel even with a constant intensity equal to one (by a change of time). We show also that starting from any distribution, the solution converges to the limit proportions. Furthermore, we are able to show the exponential stability using the Routh-Hurwitz criterion for an example of a finite system of differential equations. The solution of such a system of equations describes the cross distribution of types in the market.

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A presente pesquisa tem por objetivo a análise das contribuições e limitações oriundas da cooperação acadêmica estabelecida entre universidades de países distintos. Com o emprego da triangulação metodológica, buscou-se recuperar as diferentes concepções de universidade e configurações tomadas ao longo do percurso histórico para responder a seguinte questão: faz-se possível que países com temporalidade e características distintas possam construir ações conjuntas e colaborar entre si? A partir do estudo da história do Ensino Superior no Brasil e em Angola e da análise do convênio de cooperação acadêmica entre a Faculdade de Educação da Universidade de São Paulo (BR) e a Universidade Lueji ANkonde (AO), a pesquisa busca responder esta questão levando em consideração as possibilidades e dificuldades originadas deste processo no desenvolvimento de projetos e ações que têm em comum o objetivo da produção de um conhecimento sem fronteiras. / This research aims to analyze the contributions and limitations arising from the academic cooperation established between universities in different countries. With the use of methodological triangulation, it sought to recover the different university concepts and configurations taken along the historic route to answer the question: is it possible that countries with temporality and distinct characteristics can build joint actions and cooperate with each other? From the study of the history of higher education in Brazil and Angola and analysis of academic cooperation agreement between the Faculty of Education, University of São Paulo (BR) and the University Lueji A\'Nkonde (AO), the research seeks to answer this issue taking into account the possibilities and difficulties arising from this process in the development of projects and actions that have in common the goal of producing a borderless knowledge.

A presente pesquisa tem por objetivo a análise das contribuições e limitações oriundas da cooperação acadêmica estabelecida entre universidades de países distintos. Com o emprego da triangulação metodológica, buscou-se recuperar as diferentes concepções de universidade e configurações tomadas ao longo do percurso histórico para responder a seguinte questão: faz-se possível que países com temporalidade e características distintas possam construir ações conjuntas e colaborar entre si? A partir do estudo da história do Ensino Superior no Brasil e em Angola e da análise do convênio de cooperação acadêmica entre a Faculdade de Educação da Universidade de São Paulo (BR) e a Universidade Lueji ANkonde (AO), a pesquisa busca responder esta questão levando em consideração as possibilidades e dificuldades originadas deste processo no desenvolvimento de projetos e ações que têm em comum o objetivo da produção de um conhecimento sem fronteiras. / This research aims to analyze the contributions and limitations arising from the academic cooperation established between universities in different countries. With the use of methodological triangulation, it sought to recover the different university concepts and configurations taken along the historic route to answer the question: is it possible that countries with temporality and distinct characteristics can build joint actions and cooperate with each other? From the study of the history of higher education in Brazil and Angola and analysis of academic cooperation agreement between the Faculty of Education, University of São Paulo (BR) and the University Lueji A\'Nkonde (AO), the research seeks to answer this issue taking into account the possibilities and difficulties arising from this process in the development of projects and actions that have in common the goal of producing a borderless knowledge.

Made available in DSpace on 2015-01-26T13:21:16Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-04-15Bitstream added on 2015-01-26T13:30:54Z : No. of bitstreams: 1 000801690.pdf: 625904 bytes, checksum: 6c904be347933eff6bc28f2d0cf3ad4f (MD5) / Neste trabalho, ser a apresentado um estudo detalhado da estabilidade dos pontos de equilíbrio de alguns modelos matemáticos que representam o funcionamento de um ciruito elétrico que possui o memristor em sua composição, além dos outros componentes elétricos, formados por sistemas de equações diferenciais ordinárias de terceira e quarta ordens, envolvendo funções lineares por partes. Em tal processo e de fundamental importância o conhecimento de resultados relacionados a zeros de polinômios, pois a análise da estabilidade de tais sistemas está relacionado a determina ção dos autovalores da matriz dos coeficientes do sistema. Em tal estudo ser a utilizado o Critério de Routh-Hurwitz. / In this work, a detailed study of the stability of the equilibrium points of some mathematical models that represent the that represent the behavior of an electrical circuit with a memristor in your composition, consisting, consisting of ordinary di erential equations of third and fourth order systems, involving piecewise linear functions. In this theory is very important the study of results related to the zeros of polynomials, because the stability analysis of these systems is related to the eigenvalues of the coe cient matrix of the system. The Routh-Hurwitz criterion will be used.