Dans la première partie de cette thèse nous étudions la propagation des mesures de Wigner associées aux solutions de l'équation de Schrödinger à potentiels présentant des singularités coniques, et nous montrons qu'elles sont transportées par deux différents flots Hamiltoniens, l'un sur le fibré cotangent à la variété des singularités et l'autre ailleurs dans l'espace des phases, à moins d'un phénomène d'échange entre ces deux régimes qui peut se produire quand des trajectoires du flot extérieur atteignent le fibré cotangent. Nous décrivons en détail et le flot et la concentration de masse autour et sur la variété singulière, et illustrons avec des exemples quelques questions issues de la faute d'unicité des trajectoires classiques sur les singularités en dépit de l'unicité des solutions quantiques, ce qui refute tout principe de sélection classique, mais qui n'empêche dans certains cas de résoudre complètement le problème.Dans la deuxième partie nous présentons un travail mené en collaboration avec Dr. Clotilde Fermanian et Dr. Fabricio Macià où nous analysons une équation de type Schrödinger pertinente à l'étude semiclassique de la dynamique d'un électron dans un cristal avec impuretés et montrons que, dans la limite où la période caractérisique du réseau cristallin est sufisamment petite par rapport à la variation du potentiel extérieur représentant les impuretés, cette équation peut être approximée par une équation de masse effective, ou, plus généralement, que sa solution se décompose en modes de Bloch et que chacun d'eux satisfait une équation de masse effective spécifique à son énergie de Bloch / In the first part of this thesis we study the propagation of Wigner measures linked to solutions of the Schrödinger equation with potentials presenting conical singularities and show that they are transported by two different Hamiltonian flows, one over the bundle cotangent to the singular set and the other elsewhere in the phase space, up to a transference phenomenon between these two regimes that may arise whenever trajectories in the outsider flow lead in or out the bundle. We describe in detail either the flow and the mass concentration around and on the singular set and illustrate with examples some issues raised by the lack of uniqueness for the classical trajectories on the singularities despite the uniqueness of quantum solutions, dismissing any classical selection principle, but in some cases being able to fully solve the problem.In the second part we present a work in collaboration with Dr. Clotilde Fermanian and Dr. Fabricio Macià where we analyse a Schrödinger-like equation pertinent to the semiclassical study of the dynamics of an electron in a crystal with impurities, showing that in the limit where the characteristic lenght of the crystal's lattice can be considered sufficiently small with respect to the variation of the exterior potential modelling the impurities, then this equation is approximated by an effective mass equation, or, more generally, that its solution decomposes in terms of Bloch modes, each of them satisfying an effective mass equation specificly assigned to their Bloch energies

Este trabalho aborda o problema da inversão do vetor momento magnético, com ampla aplicação na Ressonância Nuclear Magnética (RNM). Em vez de uma sequência de impulsos e de abordarmos somente o problema de conduzir o vetor de -e3 para +e3, escolhemos uma lei de controle limitada e analisamos o processo de várias iterações (voltas completas). Através do método da medianização, obtemos uma solução explícita aproximada para o sistema e, através dela e de alguns teoremas auxiliares sobre rotações, discutimos a propagação do erro em módulo e fase cometido após a realização dessas iterações. / This dissertation considers the problem of inversion of the magnetic moment vector, with wide application in Nuclear Magnetic Resonance (NMR). Instead of a pulse sequence and only approach the problem of driving the vector from -e3 to +e3, we choose limited controls and we analyze several iterations of the process (laps). By the averaging method, we obtain an approximate explicit solution for the system and through this method, together with some auxiliary theorems on rotations, we discuss the propagation of error in magnitude and phase committed after performing these iterations.

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.

Es werden spektrale Eigenschaften des zweidimensionalen Schrödinger-Operators mit einem zweifach periodischen Potential und starkem magnetischem Feld untersucht mit Hilfe semiklassischer Methoden. Man beschreibt die spektrale Asymptotik durch Benutzung der Reeb-Graph-Technik. Im Falle des rationalen Flusses konstruiert man semiklassische Magneto-Bloch-Funktionen und beschreibt die Asymptotik des Spektrums auf dem physikalischen Beweisniveau. / Spectral properties of the two-dimensional Schroedinger operator with a two-periodic potential and a strong uniform magnetic field is studied with the help of semiclassical methods. The spectral asymptotics is described using the Reeb graph technique. In the case of the rational flux one constructs semiclassical magneto-Bloch functions and describes the asymptotics of the band spectrum on the physical level of proof.

A metodologia GMAX, (Gradient-Modulated Adiabatic Excitation), caracteriza-se pelo uso de pulsos adiabáticos para localização de volumes em espectroscopia e seleção de fatias em MRI. A sua utilidade surge do interessante perfil de inversão da magnetização transversal induzido ao longo da amostra. Entretanto, a interpretação desse comportamento tem sido dada apenas de forma qualitativa, através da utilização da condição de adiabaticidade como ponto de partida. Neste trabalho é apresentada uma descrição analítica partindo da solução em termos da função hipergeométrica para os pulsos sech e tanh. A partir desse procedimento encontramos um conjunto de resultados com os quais é possível inferir analiticamente o comportamento característico da magnetização, tendo como objetivo obter um maior controle da magnetização a partir dos parâmetros da metodologia que proporcionam interpretação física. / The Gradient-Modulated Adiabatic Excitation (GMAX) methodology is characterized by the use of adiabatic pulses for volume localization in spectroscopy and slice selection in MRI. Its use derives from the interesting nodal point transverse magnetization profile induced throughout the sample. Nevertheless, the interpretation of such behavior for the magnetization has been of qualitative purpose only, using the adiabatic condition as a starting point. Here, we present an analytical description, starting from the solution in terms of the hypergeometric functions for sech and tanh pulses. From this procedure we found a set of results with which is possible to infer analytically the characteristic behavior of the magnetization. This is on the purpose of obtaining greater control of the magnetization from parameters of the methodology that carry physical interpretation.

A K-teoria algébrica é um ramo da álgebra que associa para cada anel com unidade R, uma sequência de grupos abelianos chamados os n-ésimos K-grupos de R. Em 1970, Daniel Quillen dá uma definição geral dos K-grupos de um anel qualquer R a partir da +-construção do espaço classificante BGL(R). Por outro lado, considerando R um anel comutativo, obtém-se também a definição dos K-grupos de Milnor KMn (R). Usando o produto dos K-grupos de Quillen e Milnor e suas estruturas anti-comutativas, definimos o seguinte homomorfismo tn : KMn (R) &rarr; Kn(R): Mostraremos nesta dissertação que se R é um anel local com ideal maximal m tal que R / m é um corpo infinito, então esse homomorfismo é um isomorfismo para 0 &le; n &le; 2. Em geral tn nem sempre é injetor ou sobrejetor. Por exemplo quando n = 3, sabe-se que t3 não é sobrejetor e definimos a parte indecomponível de K3(R) como sendo o grupo Kind3 (R) := coker (KM3 (R) &rarr; t3 K3(R)). Usando alguns resultados de homologia dos grupos lineares, nesta dissertação mostraremos a existência da sequência exata de Bloch-Wigner para corpos infinitos. Esta sequência dá uma descrição explícita da parte indecomponível do terceiro K-grupo de um corpo infinito. TEOREMA (Sequência exata de Bloch-Wigner). Seja F um corpo infinito e seja p(F) o grupo de pre-Bloch de F, isto é, o grupo quociente do grupo abeliano livre gerado pelos símbolos [a], a &isin; F×, pelo subgrupo gerado por elementos da forma [a] - [b] + [b/a] - [1-a-1 /1-b-1] + [1-a /1-b] com a, b &isin; F× - {1}, a /= b. Então temos a sequência exata TorZ1 (&mu; (F), &mu; (F)) ~ &rarr; Kind3 (F) &rarr; p(F) &rarr; (F× &#8855; ZFx)&sigma; F×)&sigma; &rarr; K2(F) &rarr; 0 onde (F× &#8855; ZF×)&sigma; := (F×; &#8855; ZF×)/<a &#8855; b + b &#8855; a | a, b &isin; F×> e TorZ1 (&mu; (F); &mu; (F)) ~ é a única extensão não trivial de Z=2Z por TorZ1 (&mu; (F); &mu; (F)) se char(F) &ne; 2 e &mu; 2 &infin; (F) é finito e é TorZ1 (&mu; (F); &mu; (F)) caso contrário. O homomorfismo p(F) &rarr; (F× &#8855; ZF×) &sigma; é definido por [a] &rarr; a &#8855; (1-a). O estudo da sequência exata de Bloch-Wigner é justificada pela relação entre o segundo e terceiro K-grupo de um corpo F. / The algebraic K-theory is a branch of algebra that associates to any ring with unit R a sequence of abelian groups called n-th K-groups of R. In 1970, Daniel Quillen gave a general definition of K-groups of any ring R using the +-construction of the classifying space BGL(R). On the other hand, if we consider a commutative ring R, we can define the Milnors K-groups, KMn (R), of R. Using the product of the Quillen and Milnors K-groups and their anti-commutative structure, we define a natural homomorphism tn : KMn (R) &rarr; Kn(R): In this dissertation, we show that if R is a local ring with maximal ideal m such that R=m is infinite, then this map is an isomorphism for 0<= n<= 2. But in general tn is not injective nor is surjective. For example when n = 3, we know that t3 is not surjective and define the indecomposable part of K3(R) as the group Kind3 (R) := coker (KM3 (R) &rarr; t3 K3(R)). Using some results about the homology of linear groups, in this dissertation we will prove the Bloch-Wigner exact sequence over infinite fields. This exact sequence gives us a precise description of the indecomposable part of the third K-group of an infinite field. THEOREM (Bloch-Wigner exact sequence). Let F be an infinite field and let p(F) be the pre-Bloch group of F, that is, the quotient group of the free abelian group generated by symbols [a], a &isin; F× - [1}, by the subgroup generated by the elements of the form [a][b]+ b/a][ 1-a-1/1-b-1]+ [1-a/1-b] with a; b &isin; F×, a =/ b. Then we have the exact sequence TorZ1 (&mu; (F), &mu; (F)) ~ &rarr; Kind3 (F) &rarr; p(F) &rarr; (F× &#8855; ZF×)$sigma; &rarr; K2(F) &rarr; 0 where (F× &#8855; ZF×)&sigma; := (F× &#8855; ZF×) / a &38855; b +b &#8855; a | a; b &isin; F× and TorZ1(&mu;(F);&mu;(F)) is the unique non trivial extension of Z=2Z by TorZ1 (&mu; (F); &mu; (F)) if char(F) =/ 2 and &mu;2 &infin; is finite and is TorZ1 (&mu; (F);&mu; (F)) otherwise. The homomorphism p(F) &rarr; (F×ZF×)%sigma; is defined by [a] &rarr; a &#8855; (1-a). As it is shown, the study of the Bloch-Wigner exact sequence is also justified by the relation between the second and third K-group of a field F.

Au cours des dernières décennies, l'angiographie par résonance magnétique a été utilisée comme routine clinique pour l'exploration précise et non invasive des vaisseaux sanguins, ainsi que pour le diagnostic des affections neurovasculaires les plus courantes. Plusieurs méthodes spécifiques ont été développées pour simuler numériquement le procédé de formation des angiographies. Cependant, à ce jour, la plupart des logiciels de simulation IRM avancés sont exclusivement spécialisés dans l'imagerie des tissus statiques. Le présent travail a donc été réalisé pour étendre les possibilités d'un logiciel existant afin de proposer un outil complet pour la simulation IRM des écoulements fluides. L'efficacité de cette approche est démontrée en reproduisant les principales séquences angiographiques et les artéfacts de flux les plus courants. Pour finir, des applications sur des simulations de flux sanguins dans des géométries de vaisseaux réalistes sont présentées. / During the last decades, magnetic resonance angiography has been used as a clinical routine for precise and non-invasive exploration of vessels, as well as for diagnosis of the most common neurovascular diseases. Several dedicated methods were developed to simulate specifically the process of angiographic acquisitions. Though, currently, most of advanced MRI simulators are exclusively specialized in static tissues imaging. This work was carried out to expand the possibilities of one of those simulators in order to propose a complete tool for MRI simulation of flow motion.The efficiency of this approach is proven by replicating the main angiographic pulse sequences and the most common flow artifacts. Finally, applications are provided on simulations of blood flow in realistic vessels geometries.

Cette thèse concerne la mise en place de méthodes numériques pour déterminer les configurations magnétiques d'équilibre, et leur utilisation pour des systèmes possédant des hétérogénéités structurales et des singularités magnétiques.<br /><br />Nous décrivons tout d'abord un algorithme fondé sur une méthode multipolaire rapide et qui permet de calculer efficacement le champ dipolaire dans une assemblée de spins dans le cadre du modèle de Heisenberg classique. <br /><br />En utilisant ce modèle, nous étudions l'interaction de parois magnétiques avec des défauts structuraux dans des couches minces de FePt. Nous traitons le cas des parois d'antiphase et les micromacles. Nous analysons les valeurs des champs de décrochage des parois magnétiques, notamment en les comparant avec des données expérimentales.<br /><br />Nous détaillons ensuite une méthode multiéchelle que nous développée. Cette méthode permet, dans un formalisme unifié, de décrire un système avec le modèle de Heisenberg et le modèle micromagnétique.<br /><br />La dernière partie de cette thèse est consacrée à l'étude de systèmes de grande taille possédant des variations spatiales rapides d'aimantation, en utilisant la méthode multiéchelle : vortex dans un élément magnétique, configurations avec un point de Bloch dans un cube, bulle magnétique dans une couche mince de FePd. Dans ce dernier cas, les résultats sont comparés à des observations récentes par microscopie de Lorentz.

Made available in DSpace on 2014-06-11T19:26:35Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-02-26Bitstream added on 2014-06-13T20:54:47Z : No. of bitstreams: 1 lima_ago_me_assis.pdf: 670041 bytes, checksum: 728eab97f13eb45b86f5ab4fe15597ad (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Esta dissertação aborda dois cursos de Lucien Febvre, proferidos no Collège de France entre os anos de 1944 e 1947. A partir de seus dois livros publicados postumamente: L’Europe: genèse d’une civilisation (1999) e Honneur et Patrie (1996), procuramos mapear primeiramente o conceito de Europa, e posteriormente, o surgimento de uma pátria cristã, passando pela européia, e por fim, o fortalecimento de um ideal da pátria francesa, após o século XVII. Toda essa problemática levantada por Lucien Febvre foi dada como um reflexo do contexto em que a Europa, principalmente a França se encontrava, no período da ocupação alemã em seu solo. Observamos que ao realizar tais estudos, Lucien Febvre utilizou-se da psicologia histórica, aplicando em todos os seus cursos, o conceito de outillage mental para mapear essa Europa, mas, principalmente, para tentar compreender sua trágica França / This paper deals with two Lucien Febvre’s courses, held in Collège de France between 1944 and 1947. From his two after death published books, L’Europe: genèse d’une civilisation (1999) and Honneur et Patrie (1996), the intention is, first of all, to map the Europe’s concept and then the arising of a Christian country, going through the European country, to finally the strengthening of an ideal French’s country, after the XVII Century. All this subject matter proposed by Lucien Febvre was taken as a reflex of the context in which Europe, specially France was living at that time, with the German invasion in French’s soil. Carrying out his studies, Lucien Febvre has used historical psychology, providing in all his courses the “mental outillage” concept to map that Europe, but specially his “tragical” France

A K-teoria algébrica é um ramo da álgebra que associa para cada anel com unidade R, uma sequência de grupos abelianos chamados os n-ésimos K-grupos de R. Em 1970, Daniel Quillen dá uma definição geral dos K-grupos de um anel qualquer R a partir da +-construção do espaço classificante BGL(R). Por outro lado, considerando R um anel comutativo, obtém-se também a definição dos K-grupos de Milnor KMn (R). Usando o produto dos K-grupos de Quillen e Milnor e suas estruturas anti-comutativas, definimos o seguinte homomorfismo tn : KMn (R) &rarr; Kn(R): Mostraremos nesta dissertação que se R é um anel local com ideal maximal m tal que R / m é um corpo infinito, então esse homomorfismo é um isomorfismo para 0 &le; n &le; 2. Em geral tn nem sempre é injetor ou sobrejetor. Por exemplo quando n = 3, sabe-se que t3 não é sobrejetor e definimos a parte indecomponível de K3(R) como sendo o grupo Kind3 (R) := coker (KM3 (R) &rarr; t3 K3(R)). Usando alguns resultados de homologia dos grupos lineares, nesta dissertação mostraremos a existência da sequência exata de Bloch-Wigner para corpos infinitos. Esta sequência dá uma descrição explícita da parte indecomponível do terceiro K-grupo de um corpo infinito. TEOREMA (Sequência exata de Bloch-Wigner). Seja F um corpo infinito e seja p(F) o grupo de pre-Bloch de F, isto é, o grupo quociente do grupo abeliano livre gerado pelos símbolos [a], a &isin; F×, pelo subgrupo gerado por elementos da forma [a] - [b] + [b/a] - [1-a-1 /1-b-1] + [1-a /1-b] com a, b &isin; F× - {1}, a /= b. Então temos a sequência exata TorZ1 (&mu; (F), &mu; (F)) ~ &rarr; Kind3 (F) &rarr; p(F) &rarr; (F× &#8855; ZFx)&sigma; F×)&sigma; &rarr; K2(F) &rarr; 0 onde (F× &#8855; ZF×)&sigma; := (F×; &#8855; ZF×)/<a &#8855; b + b &#8855; a | a, b &isin; F×> e TorZ1 (&mu; (F); &mu; (F)) ~ é a única extensão não trivial de Z=2Z por TorZ1 (&mu; (F); &mu; (F)) se char(F) &ne; 2 e &mu; 2 &infin; (F) é finito e é TorZ1 (&mu; (F); &mu; (F)) caso contrário. O homomorfismo p(F) &rarr; (F× &#8855; ZF×) &sigma; é definido por [a] &rarr; a &#8855; (1-a). O estudo da sequência exata de Bloch-Wigner é justificada pela relação entre o segundo e terceiro K-grupo de um corpo F. / The algebraic K-theory is a branch of algebra that associates to any ring with unit R a sequence of abelian groups called n-th K-groups of R. In 1970, Daniel Quillen gave a general definition of K-groups of any ring R using the +-construction of the classifying space BGL(R). On the other hand, if we consider a commutative ring R, we can define the Milnors K-groups, KMn (R), of R. Using the product of the Quillen and Milnors K-groups and their anti-commutative structure, we define a natural homomorphism tn : KMn (R) &rarr; Kn(R): In this dissertation, we show that if R is a local ring with maximal ideal m such that R=m is infinite, then this map is an isomorphism for 0<= n<= 2. But in general tn is not injective nor is surjective. For example when n = 3, we know that t3 is not surjective and define the indecomposable part of K3(R) as the group Kind3 (R) := coker (KM3 (R) &rarr; t3 K3(R)). Using some results about the homology of linear groups, in this dissertation we will prove the Bloch-Wigner exact sequence over infinite fields. This exact sequence gives us a precise description of the indecomposable part of the third K-group of an infinite field. THEOREM (Bloch-Wigner exact sequence). Let F be an infinite field and let p(F) be the pre-Bloch group of F, that is, the quotient group of the free abelian group generated by symbols [a], a &isin; F× - [1}, by the subgroup generated by the elements of the form [a][b]+ b/a][ 1-a-1/1-b-1]+ [1-a/1-b] with a; b &isin; F×, a =/ b. Then we have the exact sequence TorZ1 (&mu; (F), &mu; (F)) ~ &rarr; Kind3 (F) &rarr; p(F) &rarr; (F× &#8855; ZF×)$sigma; &rarr; K2(F) &rarr; 0 where (F× &#8855; ZF×)&sigma; := (F× &#8855; ZF×) / a &38855; b +b &#8855; a | a; b &isin; F× and TorZ1(&mu;(F);&mu;(F)) is the unique non trivial extension of Z=2Z by TorZ1 (&mu; (F); &mu; (F)) if char(F) =/ 2 and &mu;2 &infin; is finite and is TorZ1 (&mu; (F);&mu; (F)) otherwise. The homomorphism p(F) &rarr; (F×ZF×)%sigma; is defined by [a] &rarr; a &#8855; (1-a). As it is shown, the study of the Bloch-Wigner exact sequence is also justified by the relation between the second and third K-group of a field F.