Submitted by LUCAS KENJY BAZAGLIA KURODA null (lucaskuroda@ibb.unesp.br) on 2016-04-15T18:44:12Z No. of bitstreams: 1 Dissertacao_Lucas.pdf: 6780478 bytes, checksum: 57d28e20a68fa0c65cd5e519b8657401 (MD5) / Approved for entry into archive by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br) on 2016-04-18T19:59:31Z (GMT) No. of bitstreams: 1 kuroda_lkb_me_bot.pdf: 6780478 bytes, checksum: 57d28e20a68fa0c65cd5e519b8657401 (MD5) / Made available in DSpace on 2016-04-18T19:59:31Z (GMT). No. of bitstreams: 1 kuroda_lkb_me_bot.pdf: 6780478 bytes, checksum: 57d28e20a68fa0c65cd5e519b8657401 (MD5) Previous issue date: 2016-02-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Um dos problemas de saúde mais conhecidos e temidos atualmente e o câncer. Hoje em dia, existem diversos estudos e trabalhos acerca do tratamento e combate desta doença. Nesse sentido, este trabalho utiliza o Cálculo Fracionário (generalização do cálculo usual, com integração e diferenciação de ordens arbitrárias) para descrever o comportamento do número de células tumorais sob a ação do sistema imunológico e do tratamento quimioterápico. Para isso, dividimos a apresentação deste trabalho em três etapas. Primeiramente, é apresentado um estudo do Cálculo Fracionário, suas principais definições, transformadas de Laplace e funções especiais relacionadas. No segundo momento, é apresentado o método "Multi-Step Generalized Differential Transform Method" (MSGDTM), utilizado para resolver sistemas de equações diferenciais fracionárias. Por fim, a versão fracionária de um modelo de dinâmica tumoral e apresentado e discutido. É observado que uma mudança na ordem da derivada fracionária gera uma mudança no comportamento da dinâmica tumoral, apresentada pelo modelo clássico. / One of the most known and feared health problems is cancer. Nowadays, there are several studies and works about the treatment and combat to this disease. In this sense, this work uses the fractional calculus (generalization of the usual calculus, with integration and differentiation of arbitrary orders) to describe the behavior of the number of tumor cells under the action of the immune system and chemotherapy. To do this, we divide the presentation of this work in three stages. First, a study of Fractional Calculus, its main de nitions, Laplace transforms and special functions, are presented. In the second phase, the "Multi-Step Generalized Differential Transform Method" (MSGDTM) is displayed and used to solve fractional differential equations. Finally, the fractional version of a tumor dynamics model is presented and discussed. It is observed that a change in the order of fractional derivative generates a change in behavior of the tumor dynamics, presented by the classical model.

Submitted by Arianne Vellasco Gomes (ariannevellasco@gmail.com) on 2018-06-15T18:52:22Z No. of bitstreams: 1 Arianne_Vellasco_Gomes_TESE_POSMAT_2018.pdf: 4211125 bytes, checksum: 16221f3149817fbc6e4db2f2026f2f14 (MD5) / Approved for entry into archive by Lucilene Cordeiro da Silva Messias null (lubiblio@bauru.unesp.br) on 2018-06-18T17:39:32Z (GMT) No. of bitstreams: 1 gomes_av_dr_bauru.pdf: 3510911 bytes, checksum: 2abe98b4f93107bb6dc267a184ebef70 (MD5) / Made available in DSpace on 2018-06-18T17:39:32Z (GMT). No. of bitstreams: 1 gomes_av_dr_bauru.pdf: 3510911 bytes, checksum: 2abe98b4f93107bb6dc267a184ebef70 (MD5) Previous issue date: 2018-04-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Tópicos fundamentais da estrutura eletrônica de materiais cristalinos, são investigados de forma generalizada mediante o Cálculo Fracionário. São calculadas as bandas de energia, as funções de Bloch e as funções de Wannier, para a equação de Schrödinger fracionária com derivada de Riesz. É apresentado um estudo detalhado do caráter não local desse tipo de derivada fracionária. Resolve-se a equação de Schrödinger fracionária para o modelo de Kronig-Penney e estuda-se os efeitos da ordem da derivada e da intensidade do potencial. Verificou-se que, ao passar da derivada de segunda ordem para derivadas fracionárias, o comportamento assintótico das funções de Wannier muda apreciavelmente. Elas perdem o decaimento exponencial, e exibem um decaimento acentuado em forma de potência. Fórmulas simples foram dadas para as caudas das funções de Wannier. A banda de energia mais baixa mostrou-se estar relacionada ao estado ligado de um único poço quântico. Sua função de onda também apresentou decaimento em lei de potência. As bandas de energia superiores mudam de comportamento em função da intensidade do potencial. No caso inteiro, a largura de cada uma dessas bandas diminui. No caso fracionário, diminui inicialmente e depois volta a aumentar, aproximando-se de um valor infinito à medida que a intensidade do potencial tende ao infinito. O grau de localização das funções de Wannier, expresso pelo desvio padrão da posição, mostra um comportamento similar ao da largura das bandas de energia. Além dos cristais perfeitos a Ciência de Materiais estuda cristais com defeito. Os defeitos são responsáveis por muitas propriedades de interesse tecnológico e podem induzir estados localizados. Neste trabalho, calculado o estado localizado de menor energia no modelo de Kronig-Penney fracionário com defeito, mediante método das transformadas de Fourier e das funções de Wannier. Verificou-se que este estado também decai em forma de lei de potência. / Basics topics on the electronic structure of crystalline materials are investigated in a generalized fashion through Fractional Calculus. The energy bands, the Bloch and Wannier functions for the fractional Schr odinger equation with Riesz derivative are calculated. The non-locality of the Riesz fractional derivative is analyzed. The fractional Schr odinger equation is solved for the Kronig-Penney model and the e ects of the derivative order and the potential intensity are studied. It was shown that moving from the integer to the fractional order strongly a ects the asymptotic behavior of the Wannier functions. They lose the exponential decay, gaining a strong power-law decay. Simple formulas have been given for the tails of the Wannier functions. A close relatim between the lowest energy band and the bound state of a single quantum well was found. The wavefunction of the latter decays as a power law. Higher energy bands change their behavior as the periodic potential gets stronger. In the integer case, the width of each one of those bands decreases. In the fractional case, it initially decreases and then increases. The width approaching a nite value as the strength tends to in nity. The degree of localization of the Wannier functions, as expressed by the position standard deviation, behaves similarly to the width of the energy bands. In addition to perfect crystals, Materials Science studies defective crystals. Defects are responsible for many properties of technological interest and can induce localized states. In this work, the localized state of lowest energy in the fractional Kronig-Penney model with defect is calculated through of the Fourier transform method and the Wannier functions. It was shown that is decays as a power law.

Orientador: Rubens de Figueiredo Camargo / Resumo: Um dos problemas de saúde mais conhecidos e temidos atualmente e o câncer. Hoje em dia, existem diversos estudos e trabalhos acerca do tratamento e combate desta doença. Nesse sentido, este trabalho utiliza o Cálculo Fracionário (generalização do cálculo usual, com integração e diferenciação de ordens arbitrárias) para descrever o comportamento do número de células tumorais sob a ação do sistema imunológico e do tratamento quimioterápico. Para isso, dividimos a apresentação deste trabalho em três etapas. Primeiramente, é apresentado um estudo do Cálculo Fracionário, suas principais definições, transformadas de Laplace e funções especiais relacionadas. No segundo momento, é apresentado o método "Multi-Step Generalized Differential Transform Method" (MSGDTM), utilizado para resolver sistemas de equações diferenciais fracionárias. Por fim, a versão fracionária de um modelo de dinâmica tumoral e apresentado e discutido. É observado que uma mudança na ordem da derivada fracionária gera uma mudança no comportamento da dinâmica tumoral, apresentada pelo modelo clássico. / Abstract: One of the most known and feared health problems is cancer. Nowadays, there are several studies and works about the treatment and combat to this disease. In this sense, this work uses the fractional calculus (generalization of the usual calculus, with integration and differentiation of arbitrary orders) to describe the behavior of the number of tumor cells under the action of the immune system and chemotherapy. To do this, we divide the presentation of this work in three stages. First, a study of Fractional Calculus, its main de nitions, Laplace transforms and special functions, are presented. In the second phase, the "Multi-Step Generalized Differential Transform Method" (MSGDTM) is displayed and used to solve fractional differential equations. Finally, the fractional version of a tumor dynamics model is presented and discussed. It is observed that a change in the order of fractional derivative generates a change in behavior of the tumor dynamics, presented by the classical model. / Mestre

Orientador: Edmundo Capelas de Oliveira / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-16T18:34:10Z (GMT). No. of bitstreams: 1 Oliveira_HeronSilva_M.pdf: 1078106 bytes, checksum: 9eb6e7bdc70150b5e616010bdfc9ab58 (MD5) Previous issue date: 2010 / Resumo: Efetuamos um levantamento histórico concernente ao cálculo integral e diferencial de ordem arbitrária, também conhecido como cálculo de ordem fracionária ou ainda cálculo fracionário, com o intuito de justificar sua importância, nos dias de hoje, a partir de uma audaciosa e profética frase proferida por Leibniz. A partir das várias definições para derivada de ordem arbitrária, em particular, as definições de Riemann, Liouville, Riemann-Liouville, Grünwald-Letnikov, Weyl e Caputo, elucidamos e justificamos a importância de cada uma delas, nas aplicações, quando associadas ao estudo de uma equação diferencial parcial de ordem arbitrária. Justificamos que, para problemas modelados pelas assim chamadas equações diferenciais de ordem arbitrária, o enfoque conforme proposto por Caputo parece ser o mais conveniente / Abstract: We propose a hystorical review associated with the integral and differential calculus of arbitrary order, known as calculus of fractional order or also fractional calculus with the objective to justify its importance nowadays as of an audacious and profetic phrasis said by Leibniz. By means of several definitions associated with the derivative of fractional order, specifically, the definitions of Riemann, Liouville, Riemann-Liouville, Grünwald-Letnikov,Weyl and Caputo, we discuss and justify the importance of each one, in the applications, when associated with the study to the so-called differential equations of arbitrary order. We also justify that the derivative as proposed by Caputo is the most convenient in problems modelled by a fractional differential equation / Mestrado / Mestre em Matemática

Orientador: Edmundo Capelas de Oliveira / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T00:53:38Z (GMT). No. of bitstreams: 1 Oliveira_DanieladosSantosde_M.pdf: 3702602 bytes, checksum: c0b05792ff3ac3c5bdd5fad1b7586dd5 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho apresentamos um estudo sobre as funções de Mittag-Leffler de um, dois e três parâmetros. Apresentamos a função de Mittag-Leffler como uma generalização da função exponencial bem como a relação que esta possui com outras funções especiais, tais como as funções beta, gama, gama incompleta e erro. Abordamos, também, a integração fracionária que se faz necessária para introduzir o conceito de derivação fracionária. Duas formulações para a derivada fracionária são estudadas, as formulações proposta por Riemann-Liouville e por Caputo. Investigamos quais regras clássicas de derivação são estendidas para estas formulações. Por fim, como uma aplicação, utilizamos a metodologia da transformada de Laplace para resolver a equação diferencial fracionária associada ao problema do oscilador harmônico fracionário / Abstract: This work presents a study about the one- two- and three-parameters Mittag-Leffler functions. We show that the Mittag-Leffler function is a generalization of the exponential function and present its relations to other special functions beta, gamma, incomplete gamma and error functions. We also approach fractional integration, which is necessary to introduce the concept of fractional derivatives. Two formulations for the fractional derivative are studied, the formulations proposed by Riemann-Liouville and by Caputo. We investigate which classical derivatives rules can be extended to these formulations. Finally, as an application, using the Laplace transform methodology, we discuss the fractional differential equation associated with the harmonic oscillator problem / Mestrado / Matematica Aplicada / Mestra em Matemática Aplicada

Cette thèse porte sur l’analyse statistique de quelques modèles de processus stochastiques gouvernés par des bruits de type fractionnaire, en temps discret ou continu.Dans le Chapitre 1, nous étudions le problème d’estimation par maximum de vraisemblance (EMV) des paramètres d’un processus autorégressif d’ordre p (AR(p)) dirigé par un bruit gaussien stationnaire, qui peut être à longue mémoire commele bruit gaussien fractionnaire. Nous donnons une formule explicite pour l’EMV et nous analysons ses propriétés asymptotiques. En fait, dans notre modèle la fonction de covariance du bruit est supposée connue, mais le comportement asymptotique de l’estimateur (vitesse de convergence, information de Fisher) n’en dépend pas.Le Chapitre 2 est consacré à la détermination de l’entrée optimale (d’un point de vue asymptotique) pour l’estimation du paramètre de dérive dans un processus d’Ornstein-Uhlenbeck fractionnaire partiellement observé mais contrôlé. Nous exposons un principe de séparation qui nous permet d’atteindre cet objectif. Les propriétés asymptotiques de l’EMV sont démontrées en utilisant le programme d’Ibragimov-Khasminskii et le calcul de transformées de Laplace d’une fonctionnellequadratique du processus.Dans le Chapitre 3, nous présentons une nouvelle approche pour étudier les propriétés du mouvement brownien fractionnaire mélangé et de modèles connexes, basée sur la théorie du filtrage des processus gaussiens. Les résultats mettent en lumière la structure de semimartingale et mènent à un certain nombre de propriétés d’absolue continuité utiles. Nous établissons l’équivalence des mesures induites par le mouvement brownien fractionnaire mélangé avec une dérive stochastique, et en déduisons l’expression correspondante de la dérivée de Radon-Nikodym. Pour un indice de Hurst H > 3=4, nous obtenons une représentation du mouvement brownien fractionnaire mélangé comme processus de type diffusion dans sa filtration naturelle et en déduisons une formule de la dérivée de Radon-Nikodym par rapport à la mesurede Wiener. Pour H < 1=4, nous montrons l’équivalence de la mesure avec celle la composante fractionnaire et obtenons une formule pour la densité correspondante. Un domaine d’application potentielle est l’analyse statistique des modèles gouvernés par des bruits fractionnaires mélangés. A titre d’exemple, nous considérons le modèle de régression linéaire de base et montrons comment définir l’EMV et étudié son comportement asymptotique. / This thesis focuses on the statistical analysis of some models of stochastic processes generated by fractional noise in discrete or continuous time.In Chapter 1, we study the problem of parameter estimation by maximum likelihood (MLE) for an autoregressive process of order p (AR (p)) generated by a stationary Gaussian noise, which can have long memory as the fractional Gaussiannoise. We exhibit an explicit formula for the MLE and we analyze its asymptotic properties. Actually in our model the covariance function of the noise is assumed to be known but the asymptotic behavior of the estimator ( rate of convergence, Fisher information) does not depend on it.Chapter 2 is devoted to the determination of the asymptotical optimal input for the estimation of the drift parameter in a partially observed but controlled fractional Ornstein-Uhlenbeck process. We expose a separation principle that allows us toreach this goal. Large sample asymptotical properties of the MLE are deduced using the Ibragimov-Khasminskii program and Laplace transform computations for quadratic functionals of the process.In Chapter 3, we present a new approach to study the properties of mixed fractional Brownian motion (fBm) and related models, based on the filtering theory of Gaussian processes. The results shed light on the semimartingale structure andproperties lead to a number of useful absolute continuity relations. We establish equivalence of the measures, induced by the mixed fBm with stochastic drifts, and derive the corresponding expression for the Radon-Nikodym derivative. For theHurst index H > 3=4 we obtain a representation of the mixed fBm as a diffusion type process in its own filtration and derive a formula for the Radon-Nikodym derivative with respect to the Wiener measure. For H < 1=4, we prove equivalenceto the fractional component and obtain a formula for the corresponding derivative. An area of potential applications is statistical analysis of models, driven by mixed fractional noises. As an example we consider only the basic linear regression setting and show how the MLE can be defined and studied in the large sample asymptotic regime.

There has been a rapid increase interest in computational number theory ever since the invention of public-key cryptography. Various attempts to solve the underlying hard problems behind public-key cryptosystems has led to interesting problems in computational number theory. One such problem, called the cubic sieve congruence problem, arises in the context of the cubic sieve method for solving the discrete logarithm problem in prime fields. The cubic sieve method requires a nontrivial solution to the Cubic Sieve Congruence (CSC)x3 y2z (mod p), where p is a given prime. A nontrivial solution must satisfy x3 y2z (mod p), x3 ≠ y2z, 1≤ x, y, z < pα , where α is a given real number ⅓ < α ≤ ½. The CSC problem is to find an efficient algorithm to obtain a nontrivial solution to CSC. This thesis is concerned with the CSC problem. Recently, the parametrization x y2z (mod p) and y υ3z (mod p) of CSC was introduced. We give a deterministic polynomial-time (O(ln3p) bit-operations) algorithm to determine, for a given υ, a nontrivial solution to CSC, if one exists. Previously it took Õ(pα) time to do this. We relate the CSC problem to the gap problem of fractional part sequences. We also show in the α = ½ case that for a certain class of primes the CSC problem can be solved deterministically Õ(p⅓) time compared to the previous best of Õ(p½). It is empirically observed that about one out of three primes are covered by this class, up to 109

La circulation artérielle coronaire est un système complexe dont les méthodes de mesures invasivespermettent une évaluation en pratique clinique.Matériels et méthodes Nous développons, dans deux revues, les principes et méthodes des différentes techniques invasivesde mesure du flux coronaire en pratique clinique. Puis nous étudions l’impact clinique de l’utilisationde la Fractional Flow Reserve (FFR) dans l’évaluation des sténoses coronaires intermédiaires, lesmoyens pharmacologiques pour mesurer la FFR et sa corrélation avec l’évaluation angiographique enfonction des facteurs de risque cardiovasculaires. Enfin, nous décrivons les principes et méthodesd’une technique de mesure du flux coronaire segmentaire permettant d’obtenir la FFR, le flux et lesrésistances absolues avec un microcathéter de perfusion qui, sur un principe de thermodilutionpermet d’évaluer distinctement la macro et la microcirculation coronaire.Résultats Nous recommandons une valeur seuil de FFR de 0,80 pour guider la revascularisation car le nombred’événements cardiovasculaires et la mortalité sont spontanément supérieurs chez les patients avecune FFR <0,80 comparativement à une FFR ≥0,80 (9,4 vs. 4,8%, P=0,06 et 7,5 vs. 3,2%, P=0,06;respectivement). Nous avons étudié différents agents hyperémiants permettant de mesurer la FFR:l’adénosine (100 μg à 200 μg) permettant d’obtenir une hyperémie maximale, et le produit decontraste permettant d’atteindre 65% de cette hyperémie maximale. La mesure de la FFR avec duproduit de contraste permet de meilleures performances diagnostiques que les indices de reposcomparé à la FFR sous adénosine. Nous avons établi que la corrélation entre la FFR et le degré desténose angiographique est faible et inversement proportionnel au nombre de facteurs de risquecardiovasculaires, particulièrement chez les patients diabétiques. Enfin, nous avons décrit dans troisétudes, le principe de thermodilution coronaire et la méthode de mesures du flux coronaire et desrésistances microvasculaires avec un microcathéter de perfusion intracoronaire spécifique. Nousavons montré que cette technique est précise (R=0,98), qu’elle induit une hyperémie maximale etlocale sans agent hyperémiant et quelle est reproductible chez l’homme (R=0,91).Conclusion La compréhension de la circulation coronaire et l’application chez l’homme des techniques demesure du flux coronaire segmentaire sont essentielles tant en pratique clinique courante qu’enrecherche. / Coronary circulation is complex and highly regulated while invasive coronary flow measurements techniques allow the assessment of coronary physiology in clinical practice. Material et methods We describe in two reviews the principles and methods of different invasive coronary flowmeasurements techniques in clinical practice. We study the clinical impact of fractional flow reserve(FFR) in intermediate coronary stenosis, the hyperemic agents and dosage to measure FFR and FFRcorrelation with angiographic indices according to risk factors accumulation. Finally, we describe the principle and method of coronary flow and microvascular resistances measurements with a dedicated infusion microcatheter for coronary thermodilution to obtain assessment of macro and microvascular components of coronary circulation. Results We recommend the FFR cut off value of 0.80 to guide revascularization based on our study showing higher myocardial infarction and death rate in patients treated with medical therapy and FFR<0.80compared to those with FFR>0.80, respectively 9.4 versus 4.8%, P=0.06 and 7,5 versus 3,2%, P=0.06. We studied different hyperemic agents and dosages and showed that intracoronary adenosine at 100μg to 200 μg induce maximal hyperemia while contrast medium induce 65% of maximal hyperemia. Therefore, FFR measurements with contrast medium is feasible and has better accuracy than restindices compared to FFR. We establish the weak correlation between FFR and angiographic indicesand weakens correlation as risk factors accumulates, especially in diabetic patients. Finally, we described in three studies the method of absolute coronary flow and microvascular resistancesmeasurements based on thermodilution principle with a dedicated infusion catheter. We showed anaccurate measurement with this technique (R=0.98), which induces maximal hyperemia without theneed of hyperemic agent with reproducible measurements in humans (R=0,91).Conclusion The use of invasive coronary flow measurements to study the coronary circulation is essential inclinical practice and in research.

Grafomotorick© obt­e (GD) vraznÄ ovlivuj­ kvalitu ivota koln­m vÄkem poÄ­naj­c, kde se vyv­jej­ grafomotorick© schopnosti, a do dchodov©ho vÄku. VÄasn diagnza tÄchto obt­­ a terapeutick zsah maj­ velk vznam k jejich zlepen­. Vzhledem k tomu, e GD souvis­ z v­cermi symptomy v oblasti kinematiky, zkladn­ kinematick© parametry jako rychlost, zrychlen­ a vih prokzaly efektivn­ kvantizaci tÄchto symptom. Objektivn­ vpoÄetn­ syst©m podpory rozhodovn­ pro identifikaci a vyeten­ GD vak nen­ dostupn. A proto je hlavn­m c­lem m© disertaÄn­ prce vzkum pokroÄil© metody parametrizace online p­sma pro analzu GD se speciln­m zamÄen­m na vyuit­ metod zlomkov©ho kalkulu. Tato prce je prvn­, kter experimentuje s vyuit­m derivac­ neceloÄ­seln©ho du (FD) pro analzu GD pomoc­ online p­sma z­skan©ho od pacient s Parkinsonovou nemoc­ a u dÄt­ koln­ho vÄku. Byla navrena a evaluovna nov metoda parametrizace online p­sma zaloena na FD vyuit­m Grnwald-Letnikova p­stupu. Bylo dokzno, e navren metoda vznamnÄ zlepuje diskriminaÄn­ s­lu a deskriptivn­ schopnosti v oblasti Parkinsonick© dysgrafie. StejnÄ tak metoda pozitivnÄ ovlivnila i nejmodernÄj­ techniky v oblasti analzy GD u dÄt­ koln­ho vÄku. Vyvinut parametrizace byla optimalizovna s ohledem na vpoÄetn­ nroÄnost (a o 80 %) a tak© na vyladÄn­ du FD. Ke konci prce byly porovnny v­cer© p­stupy vpoÄtu FD, jmenovitÄ Riemann-Liouvillv, Caputv spoleÄnÄ z Grnwald-Letnikovm p­stupem za Äelem identifikace tÄch nejvhodnÄj­ch pro jednotliv© oblasti analzy GD.

In this paper it is illustrated that currently there is a new element in electronics - fractional order filter consisting of fractal devices. The circuits selected are presented and analyzed in this diploma thesis. The calculation of the basic characteristics and parameters of the filter of fractal elements was made. An analysis of the functions and their graphs were presented. The simulation of fractional-order filter for its design was made in the program Matlab. The graphs of frequency characteristics (magnitude and phase response) were obtained in the program OrCAD. The examples of using this fractional order filter are described.