In this thesis, we propose an ε-Stokes problem connecting the Stokes problem and the corresponding pressure-Poisson equation using one pa- rameter ε > 0. We prove that the solution to the ε-Stokes problem, converges as ε tends to 0 or ∞ to the Stokes and pressure-Poisson prob- lem, respectively. Most of these results are new. The precise statements of the new results are given in Proposition 3.5, Theorem 4.1, Theorem 5.2, and Theorem 5.3. Numerical results illustrating our mathematical results are also presented. / STINT (DD2017-6936) "Mathematics Bachelor Program for Efficient Computations"

The use of fractal-based methods in imaging was first popularized with fractal image compression in the early 1990s. In this application, one seeks to approximate a given target image by the fixed point of a contractive operator called the fractal transform. Typically, one uses Local Iterated Function Systems with Grey-Level Maps (LIFSM), where the involved functions map a parent (domain) block in an image to a smaller child (range) block and the grey-level maps adjust the shading of the shrunken block. The fractal transform is defined by the collection of optimal parent-child pairings and parameters defining the grey-level maps. Iteration of the fractal transform on any initial image produces an approximation of the fixed point and, hence, an approximation of the target image. Since the parameters defining the LIFSM take less space to store than the target image does, image compression is achieved.This thesis extends the theoretical and practical frameworks of fractal imaging to one involving a particular type of multifunction that captures the idea that there are typically many near-optimal parent-child pairings. Using this extended machinery, we treat three application areas. After discussing established edge detection methods, we present a fractal-based approach to edge detection with results that compare favourably to the Sobel edge detector. Next, we discuss two methods of information hiding: first, we explore compositions of fractal transforms and cycles of images and apply these concepts to image-hiding; second, we propose and demonstrate an algorithm that allows us to securely embed with redundancy a binary string within an image. Finally, we discuss some theory of certain random fractal transforms with potential applications to texturing. / The Natural Sciences and Engineering Research Council and the University of Guelph helped to provide financial support for this research.

In this thesis, using rigorous methods, we determine the low-lying energy-momentum spectrum of a lattice Quantum Chromodynamics model (QCD) in dimension 3 + 1, imaginary time and in the strong coupling regime. We consider a QCD model with the Wilson action, three quark avors and 4 X 4 Dirac spin matrices. Under these conditions, we reanalyze que question about the existence of baryon particles in the context of the Gell-Mann and Ne´eman Eightfold Way. Completing previous works, where the octet baryons where shown to exist, here we prove the existence of the decuplet baryons detected by showing the existence of isolated dispersion curves in the energy-momentum espectrum restricted to the subspace of the physical quantum-mechanical Hilbert H of the model, associated with states with an odd number of fermions. Besides, we obtain some smoothness properties veried by these curves, as well as analytical properties of the spectral measure for the two-baryon correlations. Also, the octet and the decuplet baryons are shown to be the only states in the odd subspace up to near the energy threshold of the meson-baryon bound state. Using the results obtained here, we can go up in the spectrum and validate some previously obtained results on the two-baryon bound state spectrum beyond the so called ladder approximation, for the complete model. / Nessa tese, usando métodos rigorosos, determinamos a parte inicial do espectro de energia-momento para o modelo de Cromodinâmica Quântica (QCD) na rede, em dimensão 3 + 1, com tempo imaginário e no regime de acoplamento forte. Consideramos o modelo de QCD com a ação de Wilson, três sabores de quarks e matrizes de spin de Dirac 4 X 4. Sob estas condições, reconsideramos a questão da demonstração da existência de bárions, no contexto do Eightfold Way de Gell-Mann e Ne ´eman. Complementando trabalhos anteriores onde a existência dos octetos de bárions foi demonstrada, aqui demonstramos a existência dos decupletos de bárions. Estas partículas são detectadas através de curvas de dispersão isoladas no espectro de energia-momento do modelo, considerando-se o subespaço H0, do espaço de Hilbert (quântico) físico H, com vetores com um número ímpar de férmions. Além disso, obtemos propriedades de suavidade destas curvas, assim como propriedade analíticas da medida espectral da correlação de dois bárions. Estes bárions são demonstrados ser os únicos estados espectrais no modelo, no subespaço ímpar, até o limiar de energia próximo do estado méson-bárion. Com os resultados obtidos nesta tese, podemos subir no espectro e validar, para o modelo completo, resultados espectrais envolvendo estados ligados de dois-bárion obtidos anteriormente na chamada aproximação escada (dominante).

In this thesis, using rigorous methods, we determine the low-lying energy-momentum spectrum of a lattice Quantum Chromodynamics model (QCD) in dimension 3 + 1, imaginary time and in the strong coupling regime. We consider a QCD model with the Wilson action, three quark avors and 4 X 4 Dirac spin matrices. Under these conditions, we reanalyze que question about the existence of baryon particles in the context of the Gell-Mann and Ne´eman Eightfold Way. Completing previous works, where the octet baryons where shown to exist, here we prove the existence of the decuplet baryons detected by showing the existence of isolated dispersion curves in the energy-momentum espectrum restricted to the subspace of the physical quantum-mechanical Hilbert H of the model, associated with states with an odd number of fermions. Besides, we obtain some smoothness properties veried by these curves, as well as analytical properties of the spectral measure for the two-baryon correlations. Also, the octet and the decuplet baryons are shown to be the only states in the odd subspace up to near the energy threshold of the meson-baryon bound state. Using the results obtained here, we can go up in the spectrum and validate some previously obtained results on the two-baryon bound state spectrum beyond the so called ladder approximation, for the complete model. / Nessa tese, usando métodos rigorosos, determinamos a parte inicial do espectro de energia-momento para o modelo de Cromodinâmica Quântica (QCD) na rede, em dimensão 3 + 1, com tempo imaginário e no regime de acoplamento forte. Consideramos o modelo de QCD com a ação de Wilson, três sabores de quarks e matrizes de spin de Dirac 4 X 4. Sob estas condições, reconsideramos a questão da demonstração da existência de bárions, no contexto do Eightfold Way de Gell-Mann e Ne ´eman. Complementando trabalhos anteriores onde a existência dos octetos de bárions foi demonstrada, aqui demonstramos a existência dos decupletos de bárions. Estas partículas são detectadas através de curvas de dispersão isoladas no espectro de energia-momento do modelo, considerando-se o subespaço H0, do espaço de Hilbert (quântico) físico H, com vetores com um número ímpar de férmions. Além disso, obtemos propriedades de suavidade destas curvas, assim como propriedade analíticas da medida espectral da correlação de dois bárions. Estes bárions são demonstrados ser os únicos estados espectrais no modelo, no subespaço ímpar, até o limiar de energia próximo do estado méson-bárion. Com os resultados obtidos nesta tese, podemos subir no espectro e validar, para o modelo completo, resultados espectrais envolvendo estados ligados de dois-bárion obtidos anteriormente na chamada aproximação escada (dominante).

Die vorliegende Dissertation befasst sich mit der mathematischen Modellierung semi-klassischer Licht-Materie-Interaktion. Im semiklassischen Bild wird Materie durch eine Dichtematrix "rho" beschrieben. Das Konzept der Dichtematrizen ist quantenmechanischer Natur. Auf der anderen Seite wird Licht durch ein klassisches elektromagnetisches Feld "(E,H)" beschrieben. Wir stellen einen mathematischen Rahmen vor, in dem wir systematisch dissipative Effekte in die Liouville-von-Neumann-Gleichung inkludieren. Bei unserem Ansatz sticht ins Auge, dass Lösungen der resultierenden Gleichung eine intrinsische Liapunov-Funktion besitzen. Anschließend koppeln wir die resultierende Gleichung mit den Maxwell-Gleichungen und erhalten ein neues selbstkonsistentes, dissipatives Modell vom Maxwell-Bloch-Typ. Der Fokus dieser Arbeit liegt auf der intensiven mathematischen Studie des dissipativen Modells vom Maxwell-Bloch-Typ. Da das Modell Lipschitz-Stetigkeit vermissen lassen, kreieren wir eine regularisierte Version des Modells, das Lipschitz-stetig ist. Wir beschränken unsere Analyse im Wesentlichen auf die Lipschitz-stetige Regularisierung. Für regularisierte Versionen des dissipativen Modells zeigen wir die Existenz von Lösungen des zugehörigen Anfangswertproblems. Der Kern des Existenzbeweises besteht aus einem Resultat von ``compensated compactness'''', das von P. Gérard bewiesen wurde, sowie aus einem Lemma vom Rellich-Typ. In Teilen folgt dieser Beweis dem Vorgehen einer älteren Arbeit von J.-L. Joly, G. Métivier und J. Rauch. / This thesis deals with the mathematical modeling of semi-classical matter-light interaction. In the semi-classical picture, matter is described by a density matrix "rho", a quantum mechanical concept. Light on the other hand, is described by a classical electromagnetic field "(E,H)". We give a short overview of the physical background, introduce the usual coupling mechanism and derive the classical Maxwell-Bloch equations which have intensively been studied in the literature. Moreover, We introduce a mathematical framework in which we state a systematic approach to include dissipative effects in the Liouville-von-Neumann equation. The striking advantage of our approach is the intrinsic existence of a Liapunov function for solutions to the resulting evolution equation. Next, we couple the resulting equation to the Maxwell equations and arrive at a new self-consistent dissipative Maxwell-Bloch type model for semi-classical matter-light interaction. The main focus of this work lies on the intensive mathematical study of the dissipative Maxwell-Bloch type model. Since our model lacks Lipschitz continuity, we create a regularized version of the model that is Lipschitz continuous. We mostly restrict our analysis to the Lipschitz continuous regularization. For regularized versions of the dissipative Maxwell-Bloch type model, we prove existence of solutions to the corresponding Cauchy problem. The core of the proof is based on results from compensated compactness due to P. Gérard and a Rellich type lemma. In parts, this proof closely follows the lines of an earlier work due to J.-L. Joly, G. Métivier and J. Rauch.

Inverse problems occur in a wide variety of applications and are an active area of research in many disciplines. We consider inverse problems for a broad class of nonlinear systems of partial differential equations (PDEs). We develop collage-based approaches for solving inverse problems for nonlinear PDEs of elliptic, parabolic and hyperbolic type. The original collage method for solving inverse problems was developed in [29] with broad application, in particular to ordinary differential equations (ODEs). Using a consequence of Banach’s fixed point theorem, the collage theorem, one can bound the approximation error above by the so-called collage distance, which is more readily minimizable. By minimizing the collage distance the approximation error can be controlled. In the case of nonlinear PDEs we consider the weak formulation of the PDE and make use of the nonlinear Lax-Milgram representation theorem and Galerkin approximation theory in order to develop a similar upper-bound on the approximation error. Supporting background theory, including weak solution theory,is presented and example problems are solved for each type of PDE to showcase the methods in practice. Numerical techniques and considerations are discussed and results are presented. To demonstrate the practical applicability of this work, we study two real-world applications. First, we investigate a model for the migration of three fish species through floodplain waters. A development of the mathematical model is presented and a collage-based method is applied to this model to recover the diffusion parameters. Theoretical and numerical particulars are discussed and results are presented. Finally, we investigate a model for the “Gao beam”, a nonlinear beam model that incorporates the possibility of buckling. The mathematical model is developed and the weak formulation is discussed. An inverse problem that seeks the flexural rigidity of the beam is solved and results are presented. Finally, we discuss avenues of future research arising from this work. / Natural Sciences and Engineering Research Council of Canada, Department of Mathematics & Statistics

Made available in DSpace on 2016-03-15T19:40:19Z (GMT). No. of bitstreams: 1 Stephanny Maria Rampazo.pdf: 1203090 bytes, checksum: 35e6ca9b23021e367117e2c29b576f87 (MD5) Previous issue date: 2015-02-09 / The diagnosis of Autism Spectrum Disorder (ASD) is oriented to people with losses in areas such as communication and social interaction and by the presence of repetitive stereotyped behaviors. It is a clinical diagnosis which follows standard protocols of indirect evaluation and clinical observation (direct evaluation) in order to track signals and symptoms within the 3 areas mentioned previously and having stereotyped behaviors as a clinical indicator of this evaluation. Amongst these indicators, motors stereotypes are usually observed in people with ASD, however, there are a few Brazilian researches that consider broad clinical diversity evaluation as well as its functions, once its tools and common scales explore partially this symptomatology. The objective of this study is to show the registered results of motors stereotypes of people with ASD and discuss about possible ways of stereotypes evaluation regarding its functions and presentation. The sample was built with 22 children (age varying from 3 to 12 years old) diagnosed by Unidade de Referência em Autismo Prof° Marcos Tomanik Mercadante Irmandade Santa Casa de Misericórdia de São Paulo with ASD. The stereotype´s evaluations were based on direct and indirect behavior and tracking of ADS from data base tools: a) Brazilian version of The Behavior Problems Inventory-BPI-01; b) Brazilian version of Autism Behavior Checklist - ABC; c) Video register of Stereotypes Evaluation (RAEV). d) Experimental Functional Evaluation of Stereotype. A descriptive analysis was conducted on a case by case basis with non parametric tests and correlation with the used tools. The main conclusions of this study were: 1) the three methods of assessment, recorded the stereotypies in different relative proportions: 12 behaviors that indicate motor stereotypes at ABC were recorded in varying number of subjects in the sample (only one individual did not score), and the record of percentage relative to the total possible was 43.8%. All subjects scored at BPI-01 at least one of the 24 behaviors recorded, and the record of percentage out of the total possible was 67.4%. The direct evaluation was able to register 79 behaviors considered motor stereotypes, and therefore it was considered more effective than other methods; 2) the comparison between ABC and BPI-01 shows similar behaviors, but with slightly different questions, which led to different answers. These two questionnaires are easy to apply, the use of both is recommended because they are complementary to the registration of stereotypies. No difference was found when six similar ABC and BPI-01 records were compared; 3) You can identify the function of stereotyping for each individual from an experimental functional analysis with only a 30 minute session, divided into three situations with 10 minutes each: on demand situation (the children are requested to perform a task); attention situation (the child is praised when showing stereotypes); alone situation (free observation); 4) stereotyping may have multiple functions, i.e., stereotyped behavior can begin with a self-stimulation function, maintained by automatic reinforcement, but it can vary and serve other functions, such as obtaining social attention; 5) evaluating the function of stereotyping rather than the simple topographical record enables different, more detailed and meticulous interventions. / O Diagnóstico de Transtorno do Espectro Autista (TEA) é designado às pessoas com prejuízos nas áreas de comunicação, interação social e pela presença de comportamentos repetitivos e estereotipados. Seu diagnóstico é clinico e a avaliação é realizada a partir de protocolos padronizados de avaliação indireta e de observação clínica (avaliação direta), a fim de rastrear sinais e sintomas dentro das três áreas citadas, sendo um dos indicadores clínicos nessa avaliação, os comportamentos estereotipados. Dentre eles, nas pessoas com TEA, as estereotipias motoras comumente são manifestadas, porém existem poucas pesquisas brasileiras que contemplam avaliações que abranjam sua diversidade clínica, bem como sua função, já que os instrumentos e escalas usuais exploram de maneira parcial essa sintomatologia. O objetivo desse estudo é demonstrar os resultados do registro de estereotipias motoras em pessoas com TEA e discutir sobre as possíveis formas de avaliação de estereotipias, em relação à sua topografia e à sua função. A amostra foi constituída por 22 crianças com diagnóstico de TEA pela Unidade de Referência em Autismo Prof° Marcos Tomanik Mercadante Irmandade Santa Casa de Misericórdia de São Paulo, com idade entre 3 e 12 anos. As formas de avaliação das estereotipias se deram por avaliação comportamental direta e indireta e de rastreamento de TEA, a partir de instrumentos de coleta de dados: a) Versão Brasileira do Inventário de Problemas de Comportamentais (The Behavior Problems Inventory-BPI-01); b) Autism Behavior Checklist - ABC; c) Registro de Avaliação de Estereotipias por Vídeo (RAEV). d) Análise Funcional de estereotipia. Foi realizada análise descritiva da casuística, com testes não paramétricos e correlação entre os instrumentos utilizados. As principais conclusões deste estudo foram: 1) os três métodos de avaliação, registraram as estereotipias em proporções relativas diferentes: os 12 comportamentos que indicam estereotipias motoras no ABC foram registrados em número variados de sujeitos da amostra (apenas um indivíduo não pontuou), sendo que o percentual de registro em relação ao total possível foi de 43,8%. No BPI-01 todos os sujeitos pontuaram em pelo menos um dos 24 comportamentos registrados, sendo o percentual de registro em relação ao total possível de 67,4%. Já a avaliação direta foi capaz de registrar 79 comportamentos considerados estereotipias motoras, sendo, portanto mais eficaz que os demais métodos; 2) a comparação entre o ABC e o BPI-01 mostra que comportamentos similares, mas com perguntas ligeiramente diferentes levam a respostas diversas. Como são dois questionários de fácil aplicação, recomenda-se a utilização de ambos, pois são complementares para o registro das estereotipias. A comparação entre o registro no ABC e BPI-01 entre seis comportamentos cuja redação é muito semelhante não mostrou diferença; 3) é possível obter a função da estereotipia para cada indivíduo a partir de uma análise funcional com apenas uma sessão de 30 minutos, dividida em 3 situações de 10 minutos: demanda (solicita-se que a crianças execute uma tarefa); atenção (elogia-se quando a criança exibe a estereotipia); sozinho (observação livre); 4) uma estereotipia pode ter múltiplas funções, ou seja, um comportamento estereotipado pode iniciar por uma função de auto-estimulação, mantida por reforçamento automático, mas pode variar de função e adquirir outras funções simultâneas, como a obtenção de atenção social; 5) avaliar a função da estereotipia ao invés do simples registro topográfico possibilita intervenções diferentes, mais detalhadas e minuciosas.