Orientador: Samuel Conceição de Oliveira / Resumo: A determinação da estratégia de controle adequada para bioprocessos batelada e batelada alimentada é uma questão prática importante devido ao alto valor agregado de alguns bioprodutos. Desde que é altamente desejável otimizar a produção de bioprodutos, vários métodos têm sido propostos para esse objetivo. Uma vez dispondo de um modelo matemático adequado para o bioprocesso, o problema de otimização pode ser formulado no âmbito do princípio do máximo de Pontryagin e da teoria de controle ótimo para determinar a melhor trajetória de controle para certas variáveis manipuladas, como temperatura, pH e taxa de alimentação do substrato. Neste estudo, duas aplicações dessas técnicas baseadas em modelos matemáticos para otimizar e controlar bioprocessos de produção de antibióticos são revisadas e novos aspectos são enfatizados. Os casos analisados incluem a otimização da taxa de alimentação de substrato em um reator batelada alimentada e da temperatura em um reator batelada durante fermentações penicilínicas. Os principais resultados obtidos neste estudo foram: (i) a constatação de que métodos numéricos simples (Runge-Kutta, Newton-Raphson) podem ser aplicados para resolver satisfatoriamente os problemas de valor no contorno propostos; (ii) a demonstração de que a operação não isotérmica é mais produtiva em antibiótico do que a operação sob temperatura constante; (iii) a necessidade de adoção de um modelo matemático apropriado para o bioprocesso visando à resolução do problema de cont... (Resumo completo, clicar acesso eletrônico abaixo) / Mestre

In this dissertation work, I present a broad historical account of how the calculus of variations was applied in economics in the 1920s up until the 1940s. In the interwar period, mathematical economics was a vibrant and plural community of authors. Previous historical works on this period have focused on specific points of these authors. The present dissertation focuses on the mathematical technique, i.e., the calculus of variations and how it was used in economics. This history also encompasses the early mathematization of economics, the early history of econometrics, and the struggles to devise a dynamic theory of economics in a general equilibrium framework. I follow mainly the works of American mathematician Griffith C. Evans (18871973) whom I argue is a seminal author in this literature. In 1924, Evans used the calculus of variations to put forward a dynamic version of A. Cournot\'s classic analysis of monopoly. In the following decades, a handful of authors followed Evans\'s approach and used the calculus of variations to research depreciation, business cycles, optimal savings, and general equilibrium. In the late 1960s, similar mathematical formulations became common place in the form of optimal control and dynamic programming. These new mathematical techniques shared intimate relations with the calculus of variations. / Neste trabalho de dissertação, apresento uma história geral de como o cálculo variacional foi aplicado na economia no período dos anos 1920 até 1940. Durante o período do entreguerras, havia uma comunidade plural e vibrante de autores trabalhando com economia matemática. Trabalhos históricos sobre esse período se debruçaram sobre pontos específicos desses autores. O presente trabalho tem como foco a técnica matemática, i.e., o cálculo variacional e como ele foi utilizado na economia. Minha história também abarca o início da matematização da economia, os primeiros anos da econometria, e os desenvolvimentos de uma teoria dinâmica de economia dentro de um modelo de equilíbrio geral. Este trabalho segue de perto a obra do matemático estadunidense Griffith C. Evans (1887-1973), um autor seminal nesta literatura. Em 1924, Evans usa o cálculo variacional para dinamizar a análise clássica de monopólio de A. Cournot. Nas próximas décadas, a maior parte dos autores que usaram o cálculo variacional em economia seguiram a abordagem de Evans: eles encontraram aplicações para o cálculo de variações em teorias de depreciação, ciclos de negócios e equilíbrio geral. No final da década de 60, modelos matemáticos usando controle ótimo e programação dinâmica se popularizaram em economia. Estas novas técnicas matemáticas têm íntima relação com o cálculo variacional.

O cálculo funcional de operadores em espaços de Banach tem uma longa história, sendo inicialmente desenvolvido por F. Riesz, N. Dunford entre outros. Em 1986, uma importante contribuição foi feita por Alan McIntosh, que definiu um cálculo funcional holomorfo de operadores setoriais e destacou uma importante classe de operadores setoriais desses operadores: a dos operadores com cálculo funcional holomorfo limitado (CFHL). Do ponto de vista de operadores diferenciais e pseudodiferenciais, alguns elementos envolvidos neste cálculo já estavam presentes nos trabalhos de R. T. Seeley sobre potências complexas de operadores diferenciais elípticos. Mais tarde mostrou-se que diversos operadores possuem CFHL. Um artigo recente nesta direção e base para esta dissertação foi publicado por Bilyj, Schrohe e Seiler. Neste trabalho mostraremos que certos operadores pseudodiferenciais, agindo em espaços de Banach apropriados, são setoriais e possuem CFHL. Para isso faremos o estudo da álgebra dos símbolos de ordem zero e utilizaremos uma construção para a parametriz do resolvente. A apresentação procura ser uma versão mais didática do artigo de Bilyj, Schrohe e Seiler. Além disso, fazemos certas adaptações nas demonstrações com o propósito de facilitar a compreensão dos argumentos. Também vamos apresentar aplicações do resultado obtido. / Functional calculus for operators acting on Banach Spaces has a long history. It was initially developed by F. Riesz, N. Dunford among others. In 1986, an important contribution was made by Alan McIntosh who defined a holomorphic functional calculus for sectorial operators and put on the scene an important class of sectorial operators, namely, operators with a bounded holomorphic functional calculus (BHFC). From the point of view of differential and pseudodifferential operators, some elements treated in this calculus were already in the works of R. T. Seeley about complex powers of elliptic differential operators. Later it was shown that several operators have BHFC. A recent paper in this direction, and the one on which this dissertation is based, was published by Bilyj, Schrohe and Seiler. In this work we show that certain pseudodifferential operators, acting on appropriate Banach spaces, are sectorial and have BHFC. For this we will study the algebra of symbols of order zero and use a construction for the parametrix. This presentation aims to explore and detail the paper of Bilyj, Schrohe and Seiler. Furthermore, we make adaptations in the proofs in order to clarify the argument. We also show applications of the obtained results.

Let G be a compact Lie group and let π be an irreducible representation of G of highest weight λ. We study the operator-valued Fourier transform of the product of the j-function and the pull-back of ?? by the exponential mapping. We show that the set of extremal points of the convex hull of the support of this distribution is the coadjoint orbit through ?? + ??. The singular support is furthermore the union of the coadjoint orbits through ?? + w??, as w runs through the Weyl group. Our methods involve the Weyl functional calculus for noncommuting operators, the Nelson algebra of operants and the geometry of the moment set for a Lie group representation. In particular, we re-obtain the Kirillov-Duflo correspondence for compact Lie groups, independently of character formulae. We also develop a "noncommutative" version of the Kirillov character formula, valid for noncentral trigonometric polynomials. This generalises work of Cazzaniga, 1992.

In this thesis we propose the use of Dijkstra's concept of a predicate transformer [Dij75] for the determination of database transaction safety [SS89] and the generation of simple conditions to check that a transaction will not violate the integrity constraints in the case that it is not safe. The generation of this simple condition is something that can be done statically, thus providing a mechanism for generating safe transactions. Our approach treats a database as state, a database transaction as a program, and the database's integrity constraints as a postcondition in order to use a predicate transformer [Dij75] to generate a weakest precondition. We begin by introducing a set-oriented update language for relational databases for which a predicate transformer is then defined. Subsequently, we introduce a more powerful update language for deductive databases and define a new predicate transformer to deal with this language and the more powerful integrity constraints that can be expressed using recursive rules. Next we introduce a data model with object-oriented features including methods, inheritance and dynamic overriding. We then extend the predicate transformer to handle these new features. For each of the predicate transformers, we prove that they do indeed generate a weakest precondition for a transaction and the database integrity constraints. However, the weakest precondition generated by a predicate transformer still involves much redundant checking. For several general classes of integrity constraint, including referential integrity and functional dependencies, we prove that the weakest precondition can be substantially further simplified to avoid checking things we already know to be true under the assumption that the database currently satisfies its integrity con-straints. In addition, we propose the use of the predicate transformer in combination with meta-rules that capture the exact incremental change to the database of a particular transaction. This provides a more general approach to generating simple checks for enforcing transaction safety. We show that this approach is superior to known existing previous approaches to the problem of efficient integrity constraint checking and transaction safety for relational, deductive, and deductive object-oriented databases. Finally we demonstrate several further applications of the predicate transformer to the problems of schema constraints, dynamic integrity constraints, and determining the correctness of methods for view updates. We also show how to support transactions embedded in procedural languages such as C.

In this thesis, results will be presented that pertain to the global regularity of solutions to boundary value problems having the general form \begin{align} F\left[D^2u-A(\,\cdot\,,u,Du)\right] &= B(\,\cdot\,,u,Du),\quad\text{in}\ \Omega^-,\notag\\ T_u(\Omega^-) &= \Omega^+, \end{align} where $A$, $B$, $T_u$ are all prescribed; and $\Omega^-$ along with $\Omega^+$ are bounded in $\mathbb{R}^n$, smooth and satisfying notions of c-convexity and c^*-convexity relative to one another (see [MTW05] for definitions). In particular, the case where $F$ is a quotient of symmetric functions of the eigenvalues of its argument matrix will be investigated. Ultimately, analogies to the global regularity result presented in [TW06] for the Optimal Transportation Problem to this new fully-nonlinear elliptic boundary value problem will be presented and proven. It will also be shown that the A3w condition (first presented in [MTW05]) is also necessary for global regularity in the case of (1). The core part of this research lies in proving various a priori estimates so that a method of continuity argument can be applied to get the existence of globally smooth solutions. The a priori estimates vary from those presented in [TW06], due to the structure of F, introducing some complications that are not present in the Optimal Transportation case.¶ In the final chapter of this thesis, the A3 condition will be reformulated and analysed on round spheres. The example cost-functions subsequently analysed have already been studied in the Euclidean case within [MTW05] and [TW06]. In this research, a stereographic projection is utilised to reformulate the A3 condition on round spheres for a general class of cost-functions, which are general functions of the geodesic distance as defined relative to the underlying round sphere. With this general expression, the A3 condition can be readily verified for a large class of cost-functions that depend on the metrics of round spheres, which is tantamount (combined with some geometric assumptions on the source and target domains) to the classical regularity for solutions of the Optimal Transportation Problem on round spheres.

Les travaux autour de la conception, de la vérification et de la surveillance de compositions de services Web forment un domaine de recherche très actif actuellement. Cependant, peu de ces travaux s'intéressent à la prise en compte globale des problématiques de composition, vérification et surveillance grâce à un formalisme unifié. Dans la thèse, nous proposons l'approche DISC qui est une approche déclarative unifiée utilisant un formalisme à base d'événements, et qui permet grâce à un formalisme unique de concevoir, de vérifier et de surveiller des compositions de services, réduisant ainsi les transformations nécessaires pour passer d'un formalisme à un autre. De plus, le formalisme utilisé permet de prendre en compte des aspects non fonctionnels tels que les données, les aspects temporels, ou certains aspects liés à la sécurité. L'approche permet en outre d'instancier et de vérifier des compositions de services, et d'exécuter et surveiller ces compositions lors de l'exécution. Enfin, les effets de violations lors de l'exécution de la composition peuvent être calculés, et un ensemble d'actions de recouvrement sont proposées, permettant aux compositions d'être en partie autonomes vis-à-vis des problèmes pouvant survenir à l'exécution.

The key to understanding a program is recognizing familiar algorithmic fragments and data structures in it. Automating this recognition process will make it easier to perform many tasks which require program understanding, e.g., maintenance, modification, and debugging. This report describes a recognition system, called the Recognizer, which automatically identifies occurrences of stereotyped computational fragments and data structures in programs. The Recognizer is able to identify these familiar fragments and structures, even though they may be expressed in a wide range of syntactic forms. It does so systematically and efficiently by using a parsing technique. Two important advances have made this possible. The first is a language-independent graphical representation for programs and programming structures which canonicalizes many syntactic features of programs. The second is an efficient graph parsing algorithm.

Real-time system refers to the computing, communication, and information system with deadline requirements. To meet these deadline requirements, most systems use a mechanism known as the schedulability test which determines whether each of the admitted tasks can meet its deadline. A new task will not be admitted unless it passes the schedulability test. Schedulability tests can be either direct or indirect. The utilization based schedulability test is the most common schedulability test approach, in which a task can be admitted only if the total system utilization is lower than a pre-derived bound. While the utilization bound based schedulability test is simple and effective, it is often difficult to derive the bound. For its analytical complexity, utilization bound results are usually obtained on a case-by-case basis. In this dissertation, we develop a general framework that allows effective derivation of schedulability bounds for different workload patterns and schedulers. We introduce an analytical model that is capable of describing a wide range of tasks' and schedulers'€™ behaviors. We propose a new definition of utilization, called workload rate. While similar to utilization, workload rate enables flexible representation of different scheduling and workload scenarios and leads to uniform proof of schedulability bounds. We introduce two types of workload constraint functions, s-shaped and r-shaped, for flexible and accurate characterization of the task workloads. We derive parameterized schedulability bounds for arbitrary static priority schedulers, weighted round robin schedulers, and timed token ring schedulers. Existing utilization bounds for these schedulers are obtained from the closed-form formula by direct assignment of proper parameters. Some of these results are applied to a cluster computing environment. The results developed in this dissertation will help future schedulability bound analysis by supplying a unified modeling framework and will ease the implementation practical real-time systems by providing a set of ready to use bound results.

In this thesis, we discuss the formulation of variational problems on supermanifolds. Supermanifolds incorporate bosonic as well as fermionic degrees of freedom. Fermionic fields take values in the odd part of an appropriate Grassmann algebra and are thus showing an anticommutative behaviour. However, a systematic treatment of these Grassmann parameters requires a description of spaces as functors, e.g. from the category of Grassmann algberas into the category of sets (or topological spaces, manifolds). After an introduction to the general ideas of this approach, we use it to give a description of the resulting supermanifolds of fields/maps. We show that each map is uniquely characterized by a family of differential operators of appropriate order. Moreover, we demonstrate that each of this maps is uniquely characterized by its component fields, i.e. by the coefficients in a Taylor expansion w.r.t. the odd coordinates. In general, the component fields are only locally defined. We present a way how to circumvent this limitation. In fact, by enlarging the supermanifold in question, we show that it is possible to work with globally defined components. We eventually use this formalism to study variational problems. More precisely, we study a super version of the geodesic and a generalization of harmonic maps to supermanifolds. Equations of motion are derived from an energy functional and we show how to decompose them into components. Finally, in special cases, we can prove the existence of critical points by reducing the problem to equations from ordinary geometric analysis. After solving these component equations, it is possible to show that their solutions give rise to critical points in the functor spaces of fields. / In dieser Dissertation wird die Formulierung von Variationsproblemen auf Supermannigfaltigkeiten diskutiert. Supermannigfaltigkeiten enthalten sowohl bosonische als auch fermionische Freiheitsgrade. Fermionische Felder nehmen Werte im ungeraden Teil einer Grassmannalgebra an, sie antikommutieren deshalb untereinander. Eine systematische Behandlung dieser Grassmann-Parameter erfordert jedoch die Beschreibung von Räumen durch Funktoren, z.B. von der Kategorie der Grassmannalgebren in diejenige der Mengen (der topologischen Räume, Mannigfaltigkeiten, ...). Nach einer Einführung in das allgemeine Konzept dieses Zugangs verwenden wir es um eine Beschreibung der resultierenden Supermannigfaltigkeit der Felder bzw. Abbildungen anzugeben. Wir zeigen, dass jede Abbildung eindeutig durch eine Familie von Differentialoperatoren geeigneter Ordnung charakterisiert wird. Darüber hinaus beweisen wir, dass jede solche Abbildung eineindeutig durch ihre Komponentenfelder, d.h. durch die Koeffizienten einer Taylorentwickelung bzgl. von ungeraden Koordinaten bestimmt ist. Im Allgemeinen sind Komponentenfelder nur lokal definiert. Wir stellen einen Weg vor, der diese Einschränkung umgeht: Durch das Vergrößern der betreffenden Supermannigfaltigkeit ist es immer möglich, mit globalen Koordinaten zu arbeiten. Schließlich wenden wir diesen Formalismus an, um Variationsprobleme zu untersuchen, genauer betrachten wir eine super-Version der Geodäte und eine Verallgemeinerung von harmonischen Abbildungen auf Supermannigfaltigkeiten. Bewegungsgleichungen werden von Energiefunktionalen abgeleitet und wir zeigen, wie sie sich in Komponenten zerlegen lassen. Schließlich kann in Spezialfällen die Existenz von kritischen Punkten gezeigt werden, indem das Problem auf Gleichungen der gewöhnlichen geometrischen Analysis reduziert wird. Es kann dann gezeigt werden, dass die Lösungen dieser Gleichungen sich zu kritischen Punkten im betreffenden Funktor-Raum der Felder zusammensetzt.