Este trabalho propõe uma extensão do método de propagação de feixe (BPM - Beam Propagation Method) para a análise de guias de ondas ópticos e acopladores baseados em materiais não-lineares do tipo Kerr. Este método se destina à investigação de estruturas onde a utilização da equação escalar de Helmholtz (EEH) em seu limite paraxial não mais se aplica. Os métodos desenvolvidos para este fim são denominados na literatura como métodos de propagação de feixe de ângulo largo. O formalismo aqui desenvolvido é baseado na técnica das diferenças finitas e nos esquemas de Crank-Nicholson (CN) e Douglas generalizado (GD). Estes esquemas apresentam como característica o fato de apresentarem um erro de truncamento em relação ao passo de discretização transversal, &#916x, proporcional a O(&#916x2) para o primeiro e O(&#916x4). A convergência do método em ambos esquemas é otimizada pela utilização de um algoritmo interativo para a correção do campo no meio não-linear. O formalismo de ângulo largo é obtido pela expansão da EEH para os esquemas CN e GD em termos de polinômios aproximantes de Padé de ordem (1,0) e (1,1) para CN e GD, e (2,2) e (3,3) para CN. Os aproximantes de ordem superior a (1,1) apresentam sérios problemas de estabilidade. Este problema é eliminado pela rotação dos aproximantes no plano complexo. Duas condições de contorno nos extremos da janela computacional são também investigadas: 1) (TBC - Transparent Boundary Condition) e 2) condição de contorno absorvente (TAB - Transparent Absorbing Boundary). Estas condições de contorno possuem a facilidade de evitar que reflexões indesejáveis sejam transmitidas para dentro da janela computacional. Um estudo comparativo da influência destas condições de contorno na solução de guias de ondas ópticos não-lineares é também abordada neste trabalho. / This work introduces an extension of the beam propagation method (BPM) for the analysis of optical waveguides and couplers based on Kerr-type nonlinear materials. This method is intended for the investigation of structures where the paraxial scalar Helmholtz equation (EEH) no longer holds. The numerical methods developed for this situation are known in the literature as wide-angle beam propagation methods. The formulation developed in this work is based on finite differences and on the Crank-Nicholson (CN) and Generalized Douglas (GD) schemes. These schemes are characterized by a truncation error with respect to the transverse discretization step, &#916x, proporcional to O(&#916x2) for the CN and to O(&#916x4) for the GD scheme. The convergence of the method for both schemes is optimized by the application of an iterative algorithm for the correction of the field in the nonlinear medium. The wide-angle formalism is obtained by the expansion of the EEH for the CN and GD schemes in terms of Padé approximant polynomials. The expansions addressed in this work utilize Padé approximants of order (1,0) and (1,1) for the CN and GD scheme, and (2,2) and (3,3) for the CN scheme. Approximants orders higher than (1,1) show serious stability problems. This problem is circumvented by rotating the approximants in the complex plane. Two boundary conditions on the edge of the computational window are also investigated: 1) transparent boundary condition (TBC) and 2) transparent absorbing boundary (TAB). These boundary conditions are necessary in order to avoid unwanted reflections back to computational domain. A comparative study of the influence of these boundary conditions on the solution of nonlinear optical waveguides is also addressed in this work.

Abstract This dissertation consists of four articles on irrationality measures. In the first paper we derive explicit irrationality measures by using the simple continued fraction expansions in a completely new way. In the second and third articles we use Padé approximations to construct irrationality measures. In the second paper we obtain an explicit irrationality measure for the values of q-exponential series, for which the earlier corresponding results are not as explicit. Furthermore, we construct a restricted irrationality measure for the values of q-exponential series, which is an improvement on the earlier results in the restricted case. In the third article we derive the best possible asymptotic restricted irrationality exponent for the values of Jacobi's triple product. In the last paper we consider Cantor series. We generalize the earlier results by deriving Sondow's irrationality measure for some Cantor series. / Tiivistelmä Tämä väitöskirja koostuu neljästä artikkelista, jotka kaikki käsittelevät irrationaalisuusmittoja. Ensimmäisessä artikkelissa irrationaalisuusmittoja johdetaan uudella tavalla irrationaalilukujen yksinkertaisista ketjumurtolukuesityksistä. Toisessa ja kolmannessa artikkelissa irrationaalisuusmitat konstruoidaan Padé-approksimaatioiden avulla. Toisessa artikkelissa saadaan eksplisiittinen irrationaalisuusmitta q-eksponenttisarjan arvoille, joiden vastaavat aikaisemmat irrationaalisuusmitat eivät ole näin eksplisiittisiä. Lisäksi samassa artikkelissa konstruoidaan q-eksponenttisarjan arvoille rajoitettu eksplisiittinen irrationaalisuusmitta, mikä parantaa aikaisempia tuloksia rajoitetussa tapauksessa. Kolmannessa artikkelissa johdetaan paras mahdollinen asymptoottinen irrationaalisuuseksponentti Jacobin kolmitulon arvoille. Viimeisessä artikkelissa käsitellään Cantorin sarjoja. Siinä yleistetään aikaisempia tuloksia johtamalla Sondowin irrationaalisuusmitta tietylle joukolle Cantorin sarjoja.

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Les échos radar provenant des diverses réflexions du signal émis sur les éléments de l'environnement (le fouillis) ont longtemps été modélisés par des vecteurs Gaussiens. La procédure optimale de détection se résumait alors en la mise en oeuvre du filtre adapté classique.<br />Avec l'évolution technologique des systèmes radar, la nature réelle du fouillis s'est révélée ne plus être Gaussienne. Bien que l'optimalité du filtre adapté soit mise en défaut dans pareils cas, des techniques TFAC (Taux de Fausses Alarmes Constant) ont été proposées pour ce détecteur, dans le but d'adapter la valeur du seuil de détection aux multiples variations locales du fouillis. Malgré leur diversité, ces techniques se sont avérées n'être ni robustes ni optimales dans ces situations.<br />A partir de la modélisation du fouillis par des processus complexes non-Gaussiens, tels les SIRP (Spherically Invariant Random Process), des structures optimales de détection cohérente ont pu être déterminées. Ces modèles englobent de nombreuses lois non-Gaussiennes, comme la K-distribution ou la loi de Weibull, et sont reconnus dans la littérature pour modéliser de manière pertinente de nombreuses situations expérimentales. Dans le but d'identifier la loi de leur composante caractéristique qu'est la texture, sans a priori statistique sur le modèle, nous proposons, dans cette thèse, d'aborder le problème par une approche bayésienne. <br />Deux nouvelles méthodes d'estimation de la loi de la texture en découlent : la première est une méthode paramétrique, basée sur une approximation de Padé de la fonction génératrice de moments, et la seconde résulte d'une estimation Monte Carlo. Ces estimations sont réalisées sur des données de fouillis de référence et donnent lieu à deux nouvelles stratégies de détection optimales, respectivement nommées PEOD (Padé Estimated Optimum Detector) et BORD (Bayesian Optimum Radar Detector). L'expression asymptotique du BORD (convergence en loi), appelée le "BORD Asymptotique", est établie ainsi que sa loi. Ce dernier résultat permet d'accéder aux performances théoriques optimales du BORD Asymptotique qui s'appliquent également au BORD dans le cas où la matrice de corrélation des données est non singulière.<br />Les performances de détection du BORD et du BORD Asymptotique sont évaluées sur des données expérimentales de fouillis de sol. Les résultats obtenus valident aussi bien la pertinence du modèle SIRP pour le fouillis que l'optimalité et la capacité d'adaptation du BORD à tout type d'environnement.

The human brain consists of a myriad of chemical compounds critical to its functioning. A group of these compounds, collectively known as metabolites, have been a research interest for years because the pathogenesis of neurodegenerative diseases, a tumours classification, the effectiveness of a drug, etc., can be investigated via variations in brain metabolite concentration levels. Nuclear Magnetic Resonance Spectroscopy (NMRS) enables investigators to conduct non-invasive in vivo studies of metabolites in the human brain and the rest of the body. However a number of problems have hindered the usage of NMRS as a clinical diagnostic tool. One is the non-uniqueness of the most widely used analysis methods, i.e. as the parameters and/or prior knowledge data of an analysis method are changed, the results also change. A second problem is the lack of a method that can automatically classify the signal components estimated via signal decomposition based signal analysis methods. Additionally, some of the most widely used analysis methods, by virtue of their algorithms, intrinsically assume the nature of NMRS signals, e.g. stationary, linear, Lorentzian, etc. Hence, this thesis explores a new analysis approach, based on a theoretical and practical understanding of NMRS, that (a) avoids making assumptions about the nature of experimentally acquired NMRS signals, (b) relies on a unique decomposition analysis method, and (c) automatically classifies the estimated peaks of an analysis. Unique decomposition analysis was conducted via the rarely used unique and non-linear signal decomposition method − the Fast Pad´e Transform (FPT). The FPT is compared with the main decomposition based NMRS analysis methods via a detailed mathematical analysis, and a comparative analysis. Automatic classification was conducted via a novel classification method, which is introduced herein, and which is based on quantum mechanical predictions of metabolite NMRS behaviour.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-02-19Bitstream added on 2014-06-13T20:27:27Z : No. of bitstreams: 1 lima_maf_me_sjrp.pdf: 528569 bytes, checksum: 3cad2d8f7175d945b2ead7fb45a5c4e1 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O principal objetivo deste trabalho é estudar métodos para construir os numeradores e denominadores parciais da fração contínua que corresponde a duas expansões em série de potências de uma função analítica f(z); em z =0 e em z = 00. / The main purpose of this work is to two series expansions of an analytic function f(z); in z =0 and z =00 simultaneously. Furthermore we considered the case when there are zero coefficients in the series and also whwn there is symmetry in the coefficients of the two series. Some examples are given.

Orientador : Hugo Enrique Hernandez Figueroa / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-01T22:09:13Z (GMT). No. of bitstreams: 1 RubioMercedes_CosmeEustaquio_D.pdf: 5866050 bytes, checksum: 68669f333d337e8c63e289efcad5b166 (MD5) Previous issue date: 2002 / Resumo: Foram desenvolvidos códigos numéricos, baseados em elementos finitos 2D no domínio da freqüência, para a simulação de junçalimentadas por guias fotônicos, supostamente muito longos. Sobre as fronteiras da janela computacional, nas regiões de radiação não guiada, foram utilizadas PMLs (Perfectly Matched Layers). Sobre as portas ou regiões de acesso, cinco esquemas foram desenvolvidos, utilizando-se cinco condições de contorno diferentes: a primeira baseada na expansão modal dos campos; a segunda considerando-se uma variação de onda TEM (transversal eletromagnética); a terceira baseada na aproximação paraxial dos campos, e as duas ultimas baseadas em aproximações de Padé (Padé(l,l) e Padé(2,2)) ou de ângulo largo. Estas três últimas, até onde sabemos, estão sendo propostas pela primeira vez, e representam alternativas eficientes (relativamente à primeira) e confiáveis (relativamente à segunda), para a simulação de uma extensa gama de junções planares. O desempenho dos esquemas propostos, foi avaliado através da análise de um número considerável de ef) exemplos. Uma estratégia para analisar circuitos fotônicos, combinando os esquemas apresentados e lC) o BPM (Beam Propagation Method), é discutida e fartamente ilustrada. Estruturas de interesse atual, como ressoadores altamente compactos, utilizados como elementos add/drop em sistemas DWTIM) (Dense WavelengthDivision Multiplexing), foram analisados em detalhe / Abstract: Novel 2D finite element formulations in the frequency domain, for the simulation of arbitrary planar photonic junctions, were developed and thoroughly described. Such junctions are composed by multiple access ports, fed by supposedly very long photonics guides. On the boundaries of the computational window, placed over not guided radiation regions, PMLs (Perfectly Matched Layers) wereadopted. On the ports or access wave guides,five different boundary conditions were implemented: The first, basedon a modal expansionof the fields ; the second, considering apure modal wave variation the third, based on the paraxial approach of the fields,and the last two, based on the Padé (Padé(l,l) and Padé (2,2)) or wide angle approximations. These three last boundary conditions, to the best of our knowledge,were proposed here for the first time, and represent efficient alternatives (relatively to the first) and more reliable (relatively to the second) for the simulation of a wide range of planar junctions. The performanceof the proposed schemes,was evaluated through the analysis of a considerable number of examples. A strategy to analyze photonic circuits, combining the presented schemes with the BPM (Beam Propagation Method), is also discussed and widely illustrated. Structures of current interest , such as highly compact resonators, used as add/drop devices in DWDM (Dense Wavelength Division Multiplexing) systems were analyzed in detail / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

Orientador: Luiz Fernando Milanez / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica / Made available in DSpace on 2018-08-27T23:42:49Z (GMT). No. of bitstreams: 1 Dalmas_Sergio_D.pdf: 5052580 bytes, checksum: 18c083502953c8bf9d4aa3a089a26162 (MD5) Previous issue date: 2015 / Resumo: Excluídos problemas simples de condução de calor nos quais a temperatura depende apenas do tempo ou apenas de uma coordenada de posição, os demais levam a equações diferenciais parciais, as quais tem soluções em termos de séries obtidas de vários métodos como a separação de variáveis, a superposição, a função de Green, a técnica da transformada integral, a transformada de Laplace e o teorema de Duhamel. Estas soluções dependem de autovalores que são obtidos das raízes de equações transcendentais que na maioria dos casos não podem ser expressas em forma fechada, mas podem ser obtidas de tabelas, expressões aproximadas, e expressões iterativas. O objetivo desse estudo é encontrar novas expressões para estas raízes, que sejam mais simples ou que tenham mais exatidão do que as já existentes. As três equações transcendentais que são consideradas aqui são as mais frequentemente utilizadas entre as que não tem solução fechada, e surgem quando as condições de contorno são convectivas. Uma nova família de funções iterativas é obtida, a qual inclui várias funções clássicas e, em particular, toda a família de métodos de Householder. Um novo método obtido é o que tem convergência mais rápida para as presentes equações. Apesar das tabelas de raízes apresentarem valores com vários dígitos significativos, problemas reais dificilmente levam a um valor da variável independente que pode ser diretamente encontrado, tornando-se necessário o uso de interpolação. Então, a exatidão de raízes obtidas por estas tabelas é limitada pela exatidão da interpolação, a qual pode ser comparada com a das expressões aproximadas. As expressões existentes são analisadas utilizando propriedades das raízes. Uma expressão aproximada desenvolvida para a primeira raiz das três equações é baseada no método do ponto fixo, outra é obtida da aplicação do conceito de MiniMax para se reajustar expressões de outros autores, e uma final tem forma algébrica. O conceito de MiniMax não é obtido através de algum método que possa ser considerado elementar, e dois novos métodos são desenvolvidos para aplicá-lo. Modernos sistemas algébricos computacionais são utilizados para gerar novas expressões aproximadas para a primeira raiz, mas encontrou-se que elas podem ser melhoradas através de métodos analíticos. Expansão em frações contínuas e novamente a aproximação de Padé são utilizadas para se obter expressões de grande exatidão. Expressões que levam a bons resultados para a primeira raiz são generalizadas para que elas sirvam para as demais raízes. Finalmente, uma comparação é feita considerando todas expressões aproximadas, indicando quais são consideradas as melhores / Abstract: Apart from simple problems of heat conduction in which the temperature depends only on the time or just on a position coordinate, the others lead to partial differential equations, which have solutions in terms of series obtained from various methods such as separation variables, superposition, the Green's function, the technique of integral transform, the Laplace transform and Duhamel's theorem. These solutions depend on eigenvalues, which are obtained from the roots of transcendental equations that in most cases cannot be expressed in closed form, but they can be obtained from tables, approximate expressions and iterative expressions. The objective of this study is to find new expressions for these roots, which are simpler or have more accuracy than the existing ones. The three transcendental equations that are considered here are the most frequently used among those that have not closed solution, and appear when the boundary conditions are convective. A new family of iterative functions is proposed, which includes several classical functions and, in particular, the entire family of Householder methods. A new method is obtained which has faster convergence to the present equations. Although the tables of roots present values with various significant digits, real problems hardly lead to a value of the independent variable that can be directly found, making it necessary to use interpolation. Then, the accuracy of the roots obtained from these tables is limited by the accuracy of the interpolation, which can be compared with the approximate expressions. Existing expressions are analyzed using the root properties. An approximate expression developed for the first root of the three equations is based on the fixed point method, another is obtained from the application of the concept of MiniMax to readjust expressions of others authors, and the last one has an algebraic form. The MiniMax concept is not obtained through any method that can be considered elementary, and two new methods are developed to apply it. Modern computer algebra systems are used to generate new approximate expressions for the first root, but it is found that they can be improved by analytical methods. Expansion in continuous fractions is adopted and the Padé approximation to obtain expressions of greater accuracy. Expressions leading to good results for the first root are generalized so that they serve for the other roots. Finally, a comparison is made considering all approximate expressions, indicating what are considered the best / Doutorado / Termica e Fluidos / Doutor em Engenharia Mecânica

We consider orthogonal polynomials P_n satisfying orthogonality relations where the measure of orthogonality is, in general, a complex-valued Borel measure supported on subsets of the complex plane. In our consideration we will focus on measures of the form d\mu(z) = \rho(z) dz where the function \rho may depend on other auxiliary parameters. Much of the asymptotic analysis is done via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method, and relies heavily on notions from logarithmic potential theory.

No / The four vector extrapolation methods, minimal polynomial extrapolation, reduced rank extrapolation, modified minimal polynomial extrapolation and the topological epsilon algorithm, when applied to linearly generated vector sequences are Krylov subspace methods and it is known that they are equivalent to some well-known conjugate gradient type methods. However, the vector -algorithm is an extrapolation method, older than the four extrapolation methods above, and no similar results are known for it. In this paper, a determinantal formula for the vector -algorithm is given. Then it is shown that, when applied to a linearly generated vector sequence, the algorithm is also a Krylov subspace method and for a class of matrices the method is equivalent to a preconditioned Lanczos method. A new determinantal formula for the CGS is given, and an algebraic comparison between the vector -algorithm for linear systems and CGS is also given.