L'objet de cette thèse est l'étude d'un système différentielle non linéaire issu d'un modèle stellaire. Après réduction et changements d'inconnues et variables, on se ramène à un second membre analytique en chacune des variables du problème ainsi qu'en des fonctions bien choisies. Nous montrons ensuite que les solutions peuvent s'écrire dans un espace de séries absolument convergentes. Ce théorème d'existence servira alors de brique élémentaire à une méthode de réduction de type Fuchsienne. L'objectif étant d'obtenir un développement sous forme de série faisant apparaître de manière explicite les différentes constantes arbitraires inhérentes à ce type d'équations. / The object of this thesis is the study of a non linear differential equation stemming from a stellar model. After reduction and unknowns changes and variables, we achieve to an analytic second member in each of the problem variables and well chosen functions. Then we show that the solutions can be described in a space of absolute convergent series. This theorem of existence will be used as an elementary brick to a nearby method of Fuchsian reduction. The objective was to obtain a development which elicits arbitrary various constants inherent to this type of equations.

Laplacian differential operator -- Vibrations of plucked strings and Hollow cylinders.

This project deals with techniques to solve Markov Chains numerically.

In this thesis the author presents a history of non-convergent series which, in the past, played an important role in mathematics. Euler's formula, Stirling's series and Poincare's theory are examined to show the development of asymptotic series, a subdivision of divergent series.

Orientador: Adolfo Maia Junior / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-09-24T19:34:37Z (GMT). No. of bitstreams: 1 Gama_MarceloCristino_D.pdf: 779899 bytes, checksum: 7636e903a3f18cb64db479eab807a404 (MD5) Previous issue date: 2008 / Resumo: Neste trabalho buscamos soluções exatas não-triviais da Equação de LaneEmden-Fowler. Esta equação tem aplicações importantes em Teoria de Campos não-linear, bem como em Astrofísica Estelar. Inicialmente, a partir do formalismo da Integral Primeira, obtemos soluções para um modelo >.~n+1 com n = 2,3,5, utilizando Integrais Elípticas de Jacobi. Segue-se então o cálculo de flutuações no modelo >.~n+1 para um campo clássico ~ sujeito a um potencial da forma V(f) = -1/2 m2f2 + ?/n+1 fn+1, em torno de uma solução estática. Uma outra aplicação é no estudo das configurações de Equilíbrio Hidrostático de estrelas esféricas politrópicas. Mostramos que o método da Integral Primeira fornece uma série de soluções singulares na origem. Também é obtida a bem conhecida solução de Chandrasekhar, que é regular na origem / Abstract: In this work we search for non-trivial exact solutions to the Lane-Emden-Fowler¿s Equation. That equation has important applications in Non-linear Field Theory, as well in Stellar Astrophysics. Initially, from the First Integral formalism, we obtain solutions for a ?fn+1 model with n = 2, 3, 5, using Jacobian Elliptic Integrals. Follows the calculation of the fluctuations in the ?fn+1 model for a classical field f in a potential of the form..Another application is the study of the Hydrostatic Equilibrium configuration of politropic spherical stars. We show the First Integral Method gives solutions that are singular at the origin. In addition we obtain the well known Chandrasekhar¿s solution, which is regular at the origin / Doutorado / Fisica-Matematica / Doutor em Matemática Aplicada

Jede Zelle des Säugetierorganismus benötigt Eisen als Spurenelement für zahlreiche oxidativ-reduktive Elektronentransfer-Reaktionen und für Transport und Speicherung von Sauerstoff. Der Organismus unterhält daher ein komplexes Regulationsnetzwerk für die Aufnahme, Verteilung und Ausscheidung von Eisen. Die intrazelluläre Regulation in den verschiedenen Zelltypen des Körpers ist mit einer globalen hormonellen Signalstruktur verzahnt. Sowohl Eisenmangel wie Eisenüberschuss sind häufige und ernste menschliche Krankheitsbilder. Sie betreffen jede Zelle, aber auch den Organismus als Ganzes. In dieser Dissertation wird ein mathematisches Modell des Eisenstoffwechsels der erwachsenen Maus vorgestellt. In ihm wird die Flussbilanz des Eisens in den wichtigsten Zelltypen in Form von transmembranalen und intrazellulären kinetischen Gleichungen dargestellt, und es werden diese Zellmodelle mit dem zentralen Eisenaustausch-Kompartiment (Blutplasma) des Körpers integriert. Der Eisenstatus wird charakterisiert als Gehalt an labil gebundenen Eisen und an ferritin-gebundenen Eisen für jede Zelle. Der Stoffwechsel wird als Netzwerk von Flussdynamik formuliert. Der experimentelle Input in dieses Modells stammt von verschiedenen Quellen. Radioaktive Tracerdaten, gemessen am intakten Tier (Mausstamm C57BL6 – das am intensivsten studierte Tiermodell) unter varrierten physiologischen Bedingungen lieferten den experimentellen Hintergrund, von dem aus Clearance-Parameter durch numerisches Fitting ermittelt wurden. Es wird gezeigt, dass das Modell mit entsprechend adaptierten Parametersätzen die wichtigsten metabolischen und regulatorischen Ereignisse in Übereinstimmung mit den Messungen darstellen kann. In Zukunft soll die quantitative Übereinstimmung mit Daten aus weiteren genetischen Rekonstruktionen (globale und zell-spezifische knock-outs und konstitutive Expression relevanter Gene des Modellorganismus Maus) hergestellt werden. / Every cell of the mammalian organism needs iron as trace element in numerous oxido-reductive processes as well as for transport and storage of oxygen. The mammalian organism maintains therefore a complex regulatory network of iron uptake, excretion and intra-body distribution. Here a mathematical model of iron metabolism of the adult mouse is presented. It formulates the iron flux balance of the most important cell types of the organism in the form of transmembraneous and intracellular kinetic equations and integrates these cell models with the central exchange compartment (blood plasma) of the body. The iron status is represented as content of labile iron and of ferritin-bound iron in every cell type, and the metabolism is formulated as a network of flux dynamics. The experimental input into the model stems from different sources. Radioactive tracer data measured in the intact animal (mouse strain C57BL6 - the most intensively studied animal model) under various physiological conditions provided the experimental background from which clearance parameters could be obtained by numerical parameter fitting. Future research should render more precise the quantitative representation of genetic reconstructions (global and cell-type-addressed knock-out and constitutive expression of relevant genes of the model mouse strain).

Il est connu qu’une équation différentielle linéaire, x^(k+1)Y' = A(x)Y, au voisinage d’un point singulier irrégulier non-résonant est uniquement déterminée (à isomorphisme analytique près) par : (1) sa forme normale formelle, (2) sa collection de matrices de Stokes. La définition des matrices de Stokes fait appel à un ordre sur les parties réelles des valeurs propres du système, ordre qui peut être perturbé par une rotation en x. Dans ce mémoire, nous avons établi le caractère intrinsèque de cette relation : nous avons donc établi comment la nouvelle collection de matrices de Stokes obtenue après une rotation en x qui change l’ordre des parties réelles des valeurs propres dépend de la collection initiale. Pour ce faire, nous donnons un chapitre de préliminaires généraux sur la forme normale des équations différentielles ordinaires puis un chapitre sur le phénomène de Stokes pour les équations différentielles linéaires. Le troisième chapitre contient nos résultats. / It is well known that a linear differential equation, x^(k+1)Y' = A(x)Y, near a non-resonant irregular singular point is uniquely determined (up to analytic isomorphism) by : (1) its formal normal form, (2) the collection of its Stokes matrices. By definition, the Stokes matrices depend on an order defined on the real parts of the eigenvalues of the system which can be perturbed by a rotation in the x coordinate. In this paper, we have established the intrinsic character of the dependency : we have described how the new Stokes collection is obtained from the first collection after a rotation in x which changes the order on the real parts of the eigenvalues. The first chapter contains preliminaries concerning the normal form of an ordinary differential equation and a chapter on the Stokes phenomenon for linear differential equations. The third chapter contains our results.