Dissertations / Theses on the topic 'Calculus Mathematics Mathematical analysis'

This thesis consists of two main chapters along with an introduction andconclusion. In the introduction, we address the inspiration for the thesis, whichoriginates in a common calculus problem wherein travel time is minimized across two media separated by a single, straight boundary line. We then discuss the correlation of this problem with physics via Snells Law. The first core chapter takes this idea and develops it to include the concept of two media with a circular border. To make the problem easier to discuss, we talk about it in terms of running and swimming speeds. We first address the case where the starting and ending points for the passage are both on the boundary. We find the possible optimal paths, and also determine the conditions under which we travel along each path. Next we move the starting point to a location outside the boundary. While we are not able to determine the exact optimal path, we do arrive at some conclusions about what does not constitute the optimal path. In the second chapter, we alter this problem to address a rectangular enclosed boundary, which we refer to as a swimming pool. The variations in this scenario prove complex enough that we focus on the case where both starting and ending points are on the boundary. We start by considering starting and ending points on adjacent sides of the rectangle. We identify three possibilities for the fastest path, and are able to identify the conditions that will make each path optimal. We then address the case where the points are on opposite sides of the pool. We identify the possible paths for a minimum time and once again ascertain the conditions that make each path optimal. We conclude by briefly designating some other scenarios that we began to investigate, but were not able to explore in depth. They promise insightful results, and we hope to be able to address them in the future.

In this thesis we focus on high school students who graduated from a B.C. high school in 1985 and then proceeded directly to the University of British Columbia (UBC) and registering in a first year calculus course in the 1985 fall term. From this data, we want to determine the best predictor of success (the high school assigned grade for Algebra 12, or the provincial grade for Algebra 12, or the average of the high school and the provincial grade for Algebra 12) in first year calculus at UBC. We first analyze the data using simple descriptive statistics and continuous methods such as regression and analysis of variance techniques. In subsequent chapters, the categorical approach is taken and we use scaling techniques as well as loglinear models. Finally, we summarize our analysis and give conclusions in the final chapter.
Science, Faculty of
Statistics, Department of
Graduate

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
The present study aims to analyze the development of early discipline of Differential and Integral Calculus in Mathematics graduation course of Universidade de São Paulo (USP) since 1934, when this institution was founded and in which the first Mathematics graduation course was implemented, until 1994, when the subject Calculus I of Teaching Course became officially different from that offered in the Bachelor. Through interviews, conducted in accordance with the methodology of Thematic Oral History, with involved people, in different years, in the process of teaching and learning of Calculus at USP, by the analysis of the textbooks contents used at different times and by the obtained data in official documents or in the researches done by other scholars, it is noticed that initially there was not in the curriculum of Mathematics course of the investigated institution a discipline called Calculus. It was implanted at USP the European model, in which the studied concepts in this discipline had been worked which formal an high level of rigor in the course of Mathematical Analysis, introduced by Italian mathematician Luigi Fantappiè in 1934 and taught to the students since they attended the higher education, by mean that followed against the historic development of Calculus and Analysis as fields of knowledge. As time passing, by didactic reasons, some professors, especially Elza Furtado Gomide, Omar Catunda e Carlos Benjamin de Lyra, were led to defend that before studying Analysis, students should go attend on initial course of Calculus, in which the concepts would be discussed with a lower level of rigor and more manipulative way, an idea that culminated in the introduction to the discipline called Differential and Integral Calculus in the Mathematics course of the institution in 1964. This discipline had been conducted essentially with analytical orientation for many years. The process of transition from Analysis to Calculus often seen as a pre-Analysis - was slow, gradual, full of comings and goings and its detail is one of the aim to this research, which focuses its attention also in the didactic concerns and levels of rigor present in different years in the discipline of Calculus I and in the textbooks used as reference by professors. For the presented analyzes it has not recourse to a general theory that establishes the study; in it chapter specific theoretical considerations referring to studied theme are searched. In the beginning, it is noticed that the professors did not accept the existence of different levels of rigor then they did not consider necessary to adequate to the target public of the discipline that was being taught. Gradually, however, the adaptation of the way how the concepts were presented became to be defended, considering the mathematic maturation of the students, the course in which the discipline was inserted and the professional profile that would like to be formed. It is noticed that the most part of manifested concern by the authors of textbooks and by the professors of analyzed discipline was strictly related with the intention of these professors and/or the authors in giving conditions to the students in order to get, in fact, comprehension of the Calculus study done in high levels of rigor and the formalism. Furthermore, it may observed that the distinction between the discipline of Calculus I for Teaching Course and Bachelor´s Degree also was given by didactical reasons: it was searched to offer to the future teachers an initial course that would enable them to review, with a way that was more suitable to the goals of higher education, concepts even studied in the Basic Education that students do not usually dominate them when they enter the university, and at the same time, introducing the specific contends of Calculus in a appropriated way to the future teacher
O presente estudo tem como objetivo analisar o desenvolvimento da disciplina inicial de Cálculo Diferencial e Integral do curso de graduação em Matemática da Universidade de São Paulo, desde 1934, ano em que tal instituição foi fundada e nela foi implantado o primeiro curso superior de Matemática do país, até 1994, momento em que a disciplina de Cálculo I do curso de Licenciatura passou a ser oficialmente diferente daquela oferecida no Bacharelado. Por meio de entrevistas, realizadas de acordo com a metodologia da História Oral Temática, com pessoas envolvidas, em diferentes épocas, no processo de ensino e aprendizagem do Cálculo na USP, pela análise de livros didáticos deste conteúdo adotados em diferentes momentos e pelos dados obtidos em documentos oficiais ou em pesquisas realizadas por outros estudiosos, verifica-se que, inicialmente, não havia no currículo do curso de Matemática da instituição investigada uma disciplina chamada Cálculo Diferencial e Integral. Implantou-se na USP o modelo europeu, no qual os conceitos usualmente estudados nesta disciplina eram trabalhados, já totalmente sistematizados, de maneira bastante formal e com alto nível de rigor, no curso de Análise Matemática, introduzido pelo matemático italiano Luigi Fantappiè em 1934 e ministrado aos alunos desde o momento em que estes ingressavam no ensino superior, em uma abordagem que seguia na contramão da história da constituição do Cálculo e da Análise como campos de conhecimento. Com o passar do tempo, razões de caráter didático, levaram alguns professores, em especial Elza Furtado Gomide, Omar Catunda e Carlos Benjamin de Lyra, a defenderem que, antes de estudar Análise, os alunos deveriam passar por um curso inicial de Cálculo, no qual os conceitos fossem abordados com um nível menos elevado de rigor e de forma mais manipulativa, idéia que culminou na introdução no curso de Matemática da instituição, em 1964, de uma disciplina chamada Cálculo Diferencial e Integral que, na prática, foi conduzida ainda durante anos com uma orientação essencialmente analítica. O processo de transição de uma disciplina inicialmente de Análise para outra efetivamente de Cálculo vista com frequência como uma pré-Análise foi lento, gradual e repleto de idas e vindas, e seu detalhamento é um dos pontos-chaves desta investigação, que foca sua atenção também nas preocupações didáticas e nos níveis de rigor presentes, em diferentes épocas, nos cursos de Cálculo I da Matemática e nos manuais utilizados como referência pelos docentes dos mesmos. Para as análises apresentadas, não se recorre a uma teoria geral que embasa o estudo; em cada capítulo buscam-se considerações teóricas específicas referentes ao tema nele abordado. Percebe-se que, inicialmente, os professores não concebiam a existência de diferentes níveis de rigor e, portanto, não consideravam necessário adequá-lo ao público-alvo da disciplina que estava sendo ministrada. Paulatinamente, no entanto, passou-se a defender a adequação da forma como os conceitos eram apresentados, levando em conta a maturidade matemática dos estudantes, o curso no qual a disciplina estava inserida e o perfil do profissional que se desejava formar. Observa-se que grande parte das preocupações manifestadas pelos autores de livros-didáticos e pelos docentes da disciplina analisada esteve estreitamente relacionada com a intenção destes professores e/ou autores em dar condições aos alunos para que estes pudessem, de fato, compreender uma abordagem do Cálculo feita com níveis elevados de rigor e de formalismo. Além disso, verifica-se que a distinção entre a disciplina de Cálculo I da Licenciatura e do Bacharelado também se deu por razões didáticas: buscou-se oferecer aos licenciandos um primeiro curso que os possibilitassem rever, com uma abordagem que fosse mais adequada aos objetivos do ensino superior, conceitos já trabalhados na Educação Básica e que estes usualmente não dominam ao ingressar na universidade, e, ao mesmo tempo, introduzir os conteúdos específicos do Cálculo de forma mais apropriada ao futuro professor

This thesis is devoted to the mathematical analysis of models describing the energy of defects in crystalline solids via variational methods. The first part of this work studies a discrete model describing the energy of a point defect in a one dimensional chain of atoms. We derive an expansion of the ground state energy using Gamma-convergence, following previous work on similar models [BDMG99,BC07,SSZ11]. The main novelty here is an explicit characterisation of the first order limit as the solution of a variational problem in an infinite lattice. Analysing this variational problem, we prove a regularity result for the perturbation caused by the defect, and demonstrate the order of the next term in the expansion. The second main topic is a discrete model describing screw dislocations in body centred cubic crystals. We formulate an anti plane lattice model which describes the energy difference between deformations and, using the framework defined in [AO05], provide a kinematic description of the Burgers vector, which is a key geometric quantity used to describe dislocations. Apart from the anti plane restriction, this model is invariant under all the natural symmetries of the lattice and in particular allows for the creation and annihilation of dislocations. The energy difference formulation enables us to provide a clear definition of what it means to be a stable deformation. The main results of the analysis of this model are then first, a proof that deformations with unit net Burgers vector exist as globally stable states in an infinite body, and second, that deformations containing multiple screw dislocations exist as locally stable states in both infinite bodies and finite convex bodies. To prove the former result, we establish coercivity with respect to the elastic strain, and exploit a concentration compactness principle. In the latter case, we use a form of the inverse function theorem, proving careful estimates on the residual and stability of an ansatz which combines continuum linear elasticity theory with an atomistic core correction.

This multiple-case study examines the explicit and implicit assumptions of six veteran calculus instructors from three types of educational institutions, comparing and contrasting their views on the iteration of conceptual understanding and procedural fluency of pre-calculus topics. There were three components to the research data recording process. The first component was a written survey, the second component was a “think-aloud” activity of the instructors analyzing the results of a function diagnostic instrument administered to a calculus class, and for the third component, the instructors responded to two quotations. As a result of this activity, themes were found between and among instructors at the three types of educational institutions related to their expectations of their incoming students' prior knowledge of pre-calculus topics related to functions. Differences between instructors of the three types of educational institutions included two identifiable areas: (1) the teachers' expectations of their incoming students and (2) the methods for planning instruction. In spite of these differences, the veteran instructors were in agreement with other studies' findings that an iterative approach to conceptual understanding and procedural fluency are necessary for student understanding of pre-calculus concepts.
Ph.D.
Doctorate
Dean's Office, Education
Education and Human Performance
Education; Math Education

The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by pn^(λ)(x) instead of the standard notation pn^(λ) (x/2, π/2), where λ > 0. The limiting case of these sequences of polynomials pn⁽⁰⁾ (x) =lim_λ→0 pn^(λ)(x), is obtained, and is shown to be an orthogonal sequence in the strip, S = {z ∈ ℂ : −1≤ℭ (z)≤1}.

From the point of view of Umbral Calculus, this sequence has a special property that makes it unique in the Symmetric Meixner-Pollaczek class of polynomials: it is of convolution type. A convolution type sequence of polynomials has a unique associated operator called a delta operator. Such an operator is found for pn⁽⁰⁾ (x), and its integral representation is developed. A convolution type sequence of polynomials may have associated Sheffer sequences of polynomials. The set of associated Sheffer sequences of the sequence pn⁽⁰⁾(x) is obtained, and is found

to be ℙ = {{pn^(λ) (x)} =0 : λ ∈ R}. The major properties of these sequences of polynomials are studied.

The polynomials {pn^(λ) (x)}^∞n₌₀, λ < 0, are not orthogonal polynomials on the real line with respect to any positive real measure for failing to satisfy Favard’s three term recurrence relation condition. For every λ ≤ 0, an associated nonstandard inner product is defined with respect to which pn^(λ)(x) is orthogonal.

Finally, the connection and linearization problems for the Symmetric Meixner-Pollaczek polynomials are solved. In solving the connection problem the convolution property of the polynomials is exploited, which in turn helps to solve the general linearization problem.

The work in this thesis is inspired by the fabrication of Zn₄₅Au₃₀Cu₂₅. This is the first alloy undergoing ultra-reversible martensitic transformations and closely satisfying the cofactor conditions, particular conditions of geometric compatibility between phases, which were conjectured to influence reversibility. With the aim of better understanding reversibility, in this thesis we study the martensitic microstructures arising during thermal cycling in Zn₄₅Au₃₀Cu₂₅, which are complex and different in every phase transformation cycle. Our study is developed in the context of continuum mechanics and nonlinear elasticity, and we use tools from nonlinear analysis. The first aim of this thesis is to advance our understanding of conditions of geometric compatibility between phases. To this end, first, we further investigate cofactor conditions and introduce a physically-based metric to measure how closely these are satisfied in real materials. Secondly, we introduce further conditions of compatibility and show that these are nearly satisfied by some twins in Zn₄₅Au₃₀Cu₂₅. These might influence reversibility as they improve compatibility between high and low temperature phases. Martensitic phase transitions in Zn₄₅Au₃₀Cu₂₅ are a complex phenomenon, especially because the crystalline structure of the material changes from a cubic to a monoclinic symmetry, and hence the energy of the system has twelve wells. There exist infinitely many energy-minimising microstructures, limiting our understanding of the phenomenon as well as our ability to predict it. Therefore, the second aim of this thesis is to find criteria to select physically-relevant energy minimisers. We introduce two criteria or selection mechanisms. The first involves a moving mask approximation, which allows one to describe some experimental observations on the dynamics, while the second is based on using vanishing interface energy. The moving mask approximation reflects the idea of a moving curtain covering and uncovering microstructures during the phase transition, as appears to be the case for Zn₄₅Au₃₀Cu₂₅, and many other materials during thermally induced transformations. We show that the moving mask approximation can be framed in the context of a model for the dynamics of nonlinear elastic bodies. We prove that every macroscopic deformation gradient satisfying the moving mask approximation must be of the form 1 + a(x) ⊗ n(x), for a.e. x. With regards to vanishing interface energy, we consider a one-dimensional energy functional with three wells, which simplifies the physically relevant model for martensitic transformations, but at the same time highlights some key issues. Our energy functional admits infinitely many minimising gradient Young measures, representing energy-minimising microstructures. In order to select the physically relevant ones, we show that minimisers of a regularised energy, where the second derivatives are penalised, generate a unique minimising gradient Young measure as the perturbation vanishes. The results developed in this thesis are motivated by the study of Zn₄₅Au₃₀Cu₂₅, but their relevance is not limited to this material. The results on the cofactor conditions developed here can help for the understanding of new alloys undergoing ultra-reversible transformations, and as a guideline for the fabrication of future materials. Furthermore, the selection mechanisms studied in this work can be useful in selecting physically relevant microstructures not only in Zn₄₅Au₃₀Cu₂₅, but also in other materials undergoing martensitic transformations, and other phenomena where pattern formation is observed.

Cette thèse développe une nouvelle méthodologie permettant d'établir des approximations analytiques pour les prix des options européennes. Notre approche combine astucieusement des développements stochastiques et le calcul de Malliavin afin d'obtenir des formules explicites et des évaluations d'erreur précises. L'intérêt de ces formules réside dans leur temps de calcul qui est aussi rapide que celui de la formule de Black et Scholes. Notre motivation vient du besoin croissant de calculs et de procédures de calibration en temps réel, tout en contrôlant les erreurs numériques reliées aux paramètres du modèle. On traite ainsi quatre catégories de modèles, en réalisant des paramétrisations spécifiques pour chaque modèle afin de mieux cibler le bon modèle proxy et obtenir ainsi des termes correctifs faciles à évaluer. Les quatre parties traitées sont : les diffusions avec sauts, les volatilités locales ou modèles à la Dupire, les volatilités stochastiques et finalement les modèles hybrides (taux-action). Il faut signaler aussi que notre erreur d'approximation est exprimée en fonction de tous les paramètres du modèle en question et est analysée aussi en fonction de la régularité du payoff.

Problems involving the minimization of functionals date back to antiquity. The mathematics of the calculus of variations has provided a framework for the analytical solution of a limited class of such problems. This paper describes a numerical approximation technique for obtaining machine solutions to minimal path problems. It is shown that this technique is applicable not only to the common case of finding geodesics on parameterized surfaces in R3, but also to the general case of finding minimal functionals on hypersurfaces in Rn associated with an arbitrary metric.

We look at three distinct topics in analysis. In the first we give a direct and easy proof that the usual Newton-Leibniz rule implies the fundamental theorem of algebra that any nonconstant complex polynomial of one complex variable has a complex root. Next, we look at the Riesz representation theorem and show that the Riesz representing measure often can be given in the form of mini sums just like in the case of the usual Lebesgue measure on a cube. Lastly, we look at the idea of holomorphic domination and use it to define a class of complex Banach manifolds that is similar in nature and definition to the class of Stein manifolds.

Cette thèse s’intéresse à l’élaboration d’alternatives pour l’enseignement du calcul algébrique au collège et en particulier de la propriété de distributivité qui joue un rôle central.En appui sur des recherches antérieures en didactique de l’algèbre, nous analysons les spécificités des savoirs à enseigner et enseignés sur le calcul algébrique, au regard de difficultés protomathématiques (Chevallard 1985) prégnantes du côté des élèves. Ceci conduit à appréhender de nouvelles formes de savoirs à enseigner accompagnant les savoirs mathématiques liés aux aspects sémantiques et syntaxiques des écritures symboliques algébriques. La notion de transformation de mouvement (Drouhard 1992) et l’exploration des caractères formalisateur, unificateur et généralisateur (ou FUG, Robert 1998) amène à envisager la distributivité au regard d’un domaine d’étude plus large, à la fois numérique et algébrique. L’étude d’une transposition possible des savoirs à enseigner permet de dégager des conditions et des contraintes pour élaborer une ingénierie didactique. Les résultats d’une expérimentation en classe de 5e (élèves de 12-13 ans) à partir d’analyses a priori et a posteriori, concernent les discours dont les élèves parviennent à s’emparer, justifiant et soutenant leurs techniques de calcul, ainsi que les organisations de connaissances qui émergent. Une nouvelle étude didactique et épistémologique relative à la notion de substitution vient clore la thèse afin de déterminer en quoi elle pourrait fonder un prolongement possible aux enjeux FUG pour l’enseignement de la distributivité et poursuivre l’ingénierie didactique amorcée visant à enseigner le calcul algébrique tout au long du collège
This thesis seeks to explore alternatives for the teaching of algabraic calculus in second grade, and more specifically of the distributive law that plays a central role.Drawing on prior researches on didactic of algebra, characteristics of the knowledge to be taught and the knowledge taught about algebraic calculus are analyzed towards protomathematics difficulties (Chevallard 1985) constantly arising in students’work. This leads to consider new forms of knowledge, along with mathematical knowledge, that would be linked to semantic and syntactic aspects of symbolic algebraic expressions.Exploring the notion of movement transformation (Drouhard 1992) and the potential of formalizing, unifying, and generalizing (or FUG, Robert 1998), brings out the distributive law in a larger study field both numerical and algebraic.The study of a possible transposition of the knowledge to be taught yields a set of conditions and constraints to design a didactic situation. The results from a first experiment in a 5th grade class (12-13 year-olds) are based on a priori and a posteriori analysis. They focus on the discourses built and used by the students, justifying and supporting their manipulations, along with the knowledge organizations arising out.The last chapter addresses a new didactic and epistemological study of the notion of substitution aiming at discussing its potential to extend the FUG point of view on the teaching of the distributive property, and further on to provide a new perspective of research to carry on with our didactic design to teach algebraic calculus all along secondary school

Orientador: Marcos Lutz Müller
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciências Humanas
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Resumo: Foi o objetivo do presente trabalho apresentar os pressupostos histórico-filosóficos da crítica de Hegel ao cálculo infinitesimal, bem como acompanhar de perto praticamente todos os aspectos dessa crítica, tal como ela se apresenta no capítulo intermediário da Doutrina do Ser da Ciência da Lógica. A primeira tarefa, pois, foi levada a cabo através da análise interpretativa de três capítulos da Fenomenologia do Espírito (Força e Entendimento, Consciência de Si e a primeira subdivisão do capítulo da Razão), bem como através de um confrontamento com alguns aspectos da filosofia de Kant (a doutrina das grandezas negativas), Fichte e Schelling. Buscou-se mostrar como faltava à filosofia transcendental um conceito não quantitativo da qualidade em decorrência de uma proximidade fundamental à maneira tal como Leibniz inventara o cálculo infinitesimal. Além disso, coube observar como já na Fenomenologia do Espírito o tema da crítica ao cálculo infinitesimal se faz notar de maneira notável, preparando (no Força e Entendimento) e concluindo (na Observação da Natureza) o conceito dialético da consciência de si. A segunda tarefa foi, por sua vez, levada a cabo através de uma leitura minunciosa dos capítulos da Qualidade e da Quantidade da Ciência da Lógica, onde se pode mostrar como os temas trazidos à tona de maneira introdutória na Fen. do Espírito eram então consumados nas duas versões da obra máxima do método dialético especulativo hegeliano. Um confrontamento radical com a filosofia de Leibniz foi, portanto, uma das principais linhas de força do presente esforço. Nisso, mostrou-se igualmente necessário elaborar, a partir de Hegel, uma reconstrução dos contornos históricos que guiaram as práticas matemáticas infinitesimais desde Pitágoras até Cauchy, bem como propor, para além de Hegel, baseando-se porém, em seu diagnóstico, alguns prognósticos a respeito do desenvolvimento da análise matemática nos sécs. XIX e XX
Abstract: It was the goal of the present work to elucidate the historic-philosophical presuppositions of Hegel¿s critique of infinitesimal calculus and to follow very closely the way in which this critique was effectively carried forth in the intermediary chapter of the Doctrine of Being of the Science of Logic. The first of these two tasks was approached by an interpretative analysis of three chapters of the Phenomenology of Spirit (Force and Understanding, Self-consciousness and the first of the subdivisions of Reason: Observation of Nature), just as by a confrontation with some aspects of Kant¿s philosophy (the doctrine of the negative magnitudes) and the further developments of this conceptual starting point in the philosophies of Fichte and Schelling. By doing so it was attempted to show how the transcendental philosophy, due to a fundamental binding to the manner with which Leibniz had invented infinitesimal calculus, lacked a non-quantitative concept of quality; furthermore, it was aimed at showing how the theme of infinitesimal calculus critique can be observed, already in the Phenomenology of Spirit, as noticeably preparing (in Force and Understanding) and essentially resolving (in Observation of Nature) the dialectical concept of self-consciousness. The second task was, on its turn, carried forth by a detailed reading of the chapters Quality and Quantity of the Science of Logic, through which it became possible to show how the themes brought to light in an introductory manner in the Phenomenology of Spirit were then resolved in the two versions of the first volume of the most important work of Hegel¿s speculative dialectics. A radical confrontation with the philosophy of Leibniz was, therefore, one of the main red-lines of the present enterprise. In this regard, it became equally necessary to elaborate, departing from Hegel, a reconstruction of the outlines of the historical development of mathematical infinitesimal praxis from Pythagoras to Cauchy, just as propounding ¿ now beyond the scope of Hegel¿s diagnosis, but essentially based upon it ¿ some observations regarding the development of mathematical analysis in the 19th and 20th centuries
Doutorado
Filosofia
Doutor em Filosofia

In this thesis, we develop preconditioned iterative methods for the solution of matrix systems arising from PDE-constrained optimization problems. In order to do this, we exploit saddle point theory, as this is the form of the matrix systems we wish to solve. We utilize well-known results on saddle point systems to motivate preconditioners based on effective approximations of the (1,1)-block and Schur complement of the matrices involved. These preconditioners are used in conjunction with suitable iterative solvers, which include MINRES, non-standard Conjugate Gradients, GMRES and BiCG. The solvers we use are selected based on the particular problem and preconditioning strategy employed. We consider the numerical solution of a range of PDE-constrained optimization problems, namely the distributed control, Neumann boundary control and subdomain control of Poisson's equation, convection-diffusion control, Stokes and Navier-Stokes control, the optimal control of the heat equation, and the optimal control of reaction-diffusion problems arising in chemical processes. Each of these problems has a special structure which we make use of when developing our preconditioners, and specific techniques and approximations are required for each problem. In each case, we motivate and derive our preconditioners, obtain eigenvalue bounds for the preconditioners where relevant, and demonstrate the effectiveness of our strategies through numerical experiments. The goal throughout this work is for our iterative solvers to be feasible and reliable, but also robust with respect to the parameters involved in the problems we consider.

Ma recherche se situe dans le cadre de l'analyse harmonique et fonctionnelle avec des applications en théorie du contrôle. Le fil conducteur de mes travaux est le calcul fonctionnel ainsi que les estimations de fonctions carrées associées. Mes travaux concernent les thèmes ci-dessous : a) calcul fonctionnel H1 et estimations de fonctions carrées, b) applications des estimations de fonctions carrées au probl eme de Cauchy stochastique, c) résultats de perturbation pour des opérateurs (R) sectoriels, d) admissibilité et observabilité d'opérateurs de contrôle et d'observation, e) applications aux equations non-autonomes ou non-linéaires, en particulier aux équations de type Volterra et aux équations de Navier-Stokes, f) liens entre la théorie du contrôle et les mesures de Carleson.

Stochastic Calculus has been applied to the problem of pricing financial derivatives since 1973 when Black and Scholes published their famous paper "The Pricing of Options and Corporate Liabilities" in the Joumal of Political Economy. The purpose of this thesis is to show the mathematical principles underlying the methods applied to finance and to present a new model of the stock price process. As part of this paper, we present proofs of Ito's Formula and Girsanov's Theorem which are frequently used in financial applications. We demonstrate the application of these theorems to calculating the fair price of a European call option. There are two methods that result in the same price: the risk neutral valuation and the Black-Scholes partial differential equation. A new model of the stock price process is presented in Section 4. This model was inspired by the model of Cox and Ross published in 1976. We develop the model such that a martingale measure will exist for the present value of the stock price. We fit data to the traditional geometric Brownian motion model and the new model and compare the resulting prices. The data fit some stocks well, but in some cases the new model provided a better fit. The price of a European call is calculated for both models for several different stocks.

Since the introduction of the Riemann integral in the middle of the nineteenth century, integration theory has been subject to significant breakthroughs on a relatively frequent basis. We have now reached a point where integration theory has been thoroughly researched to a point where one has to delve quite deep into a particular subject in order to encounter open conjectures. In education the Riemann integral has for quite some time been the standard integral in elementary analysis courses and as the complexity of these courses incrementally increase the more general Lebesgue integral eventually becomes the standard integral.  Unfortunately, in the transition from the Riemann integral to the Lebesgue integral there are certain topics of pure theoretical interest which to a certain extent are neglected. This is particularly the case for topics regarding the inverse relationship between differential and integral calculus and the integration of exceedingly complicated functions which for example might be of a highly oscillatory nature. From an applied mathematician's point of view, the partial neglection of these topics in the case of highly problematic functions might be justified in the sense that this theory is unnecessary for modeling most problems that appear in nature. From a theoretician's point of view however this negligence is unacceptable. Consequently, there are alternative integrals which give rise to theories which one can use in an attempt to study these aforementioned topics. An example of such an integral is the Henstock–Kurzweil integral, which can be developed in a rather similar manner to that of the Riemann integral.  In this thesis we will develop the Henstock–Kurzweil integral in order to answer some of the questions which to a certain extent are beyond the scope of the Lebesgue integral while using rather basic proof techniques from complex analysis and measure theory. In addition to that we extended various properties of the Lebesgue integral to the Henstock–Kurzweil integral, in particular when it comes to Lebesgue's fundamental theorem of calculus and the basic convergence theorems of the Lebesgue integral.