Ce travail de thèse est consacré dans un premier temps à l'étude des propriétés spectrales de localisation dynamique pour des opérateurs de Schrödinger ainsi qu'à leur classification.Nous allons introduire trois classes de propriétés équivalentes en cherchant à établir le lien entre elles d'une façon optimale que nous illustrerons par des contre-exemples.Certaines de ces propriétés s'avèrent jouer un rôle crucial dans l'étude mathématique de plusieurs phénomènes issus de la physique, notamment la quantificationde la conductance de Hall et l'apparition des plateaux dûs aux états localisés.Nous nous intéressons ainsi dans la seconde partie, aux conductances de Hall et de bord pour des modèles désordonnés continus et en présence d'un mur électrique aussi bien que magnétique. Nous expliquons comment les murs entrent en jeu pour pouvoir définir la conductance de bord, en tenant compte de la contribution des états localisés et la régularisation que les systèmes désordonnés requièrent. Nous établissons l'égalité de ces deux conductances en dérivant l'une de l'autre, et non par quantification séparée. / The first part of this thesis is devoted to the study of spectral properties of dynamical localization for Schr\"odinger operators and their classification.We introduce three classes of equivalent properties and investigate the relationships between them in an optimal way.Moreover, some of these properties have been shown to play a crucial role in the mathematical proof of several phenomenon of physical interestsuch as the quantization of the Hall conductance and the existence of the plateaux due to localized states.Then, we are interested in the bulk and edge Hall conductances for continuous models in the presence of magnetic and electric walls. We explain how the walls come into play in order to define the edge conductance, taking into account the contribution of localized states and the regularization that a disordered media requires. We prove the equality of these conductances by deriving one from the other, and not by separate quantization.

The double hypernuclear systems are studied within the context of the hyperspherical approach. Possible bound states of these systems are sought as zeros of the corresponding three-body Jost function in the complex energy plane. Hypercentral potentials for the system are constructed from known potentials in order to determine bound states of the system. Calculated binding energies for double- hypernuclei having A = 4 − 20, are presented. / Physics / M.Sc. (Physics)

A dinâmica da equação diferencial com retardo x 2 pontos + 2ax ponto + bx = f(x ), para a função não linear f(x) = tanh(x), foi analisada como função dos parâmetros a, b, e do retardo , onde x = x(t ). Esse modelo descreve um oscilador harmônico amortecido sujeito a retroalimentação com retardo . Nesse estudo, examinamos os casos de retroalimentação negativa ( < 0) e positiva ( > 0). Usamos o método de passos para mostrar a propriedade de suavização da solução, da equação diferencial não linear com retardo, com o crescimento de t. Fizemos a análise da estabilidade local, construímos as cartas de estabilidade no espaço de parâmetros, e mostramos que o espectro de autovalores é discreto e, no máximo, enumerável. Foram construídos diagramas de bifurcação que exibiram a ocorrência da bifurcação de Hopf supercrítica, da bifurcação de forquilha supercrítica, e da bifurcação de Hopf dupla. Para alguns pontos de bifurcação de Hopf dupla, ressonantes e não ressonantes, foi calculada numericamente a série temporal, construído o espaço de fase e gerado o mapa de primeiro retorno para uma dada seção de Poincaré. Por fim, realizamos a discretização da equação do oscilador e fizemos uma breve análise da dinâmica da equação não linear de diferenças resultante. / The dynamics of the delay differential equation x 2 pontos + 2ax ponto + bx = f(x ), for the nonlinear function f(x) = tanh(x), has been analyzed as a function of the parameters a, b, and the delay , where x = x(t ). This model describes a damped harmonic oscillator subject to feedback with delay . Here, we have examined the cases of negative feedback (< 0) and positive feedback ( > 0). The method of steps have been used to show the property of solutions smoothing, for the nonlinear delay differential equation, for the increasing t. We have analyzed the local stability, made the stability charts, and showed that the spectrum of eigenvalues is discrete and at most enumerable. We have constructed the bifurcation diagrams that showed the occurrence of supercritical Hopf bifurcation, the supercritical pitchfork bifurcation and double Hopf bifurcation. For some points of resonant and non-resonant double Hopf bifurcation we have numerically calculated the time series, produced the phase space, and generated the first return map for a given Poincaré section. Finally, we have performed a discretization of the equation and made a brief analysis of the dynamics of the resulting nonlinear difference equation.

Nesta tese estudamos o modelo CP^(N-1) em (2+1) dimensões do espaco-tempo, onde o campo básico está na representacão fundamental. Diferentemente do caso em que o campo básico está na representacão adjunta, já estudado na literatura, o modelo por nós estudado se reduz ao modelo supersimétrico usual no limite comutativo. Analisamos a estrutura de fase e calculamos as correcões dominantes e subdominantes na expansão 1/N. Provamos que a teoria é livre de singularidades infravermelhas não integráveis e é renormalizável na ordem dominante. A funcão de vértice de dois pontos do campo básico é calculada e renormalizada de uma forma explicitamente supersimétrica na ordem subdominante. / In this thesis we sutudy the noncommutative supersymmetric CP^(N-1) model in (2+1) space-time dimensions, where the basic field is in the fundamental representation which, differently to the adjoint representation already studied in the literature, goes to the usual supersymmetric model in the commutative limit. We analyse the phase structure of the model and calculate the leading and subleading corrections in the 1/N expansion. We prove that the theory is free of non-integrable IR/UV infrared singularities and is renormalizable in the leading order. The two point vertex function of the basic field is also calculated and renormalized in an expliciitly supersymmetic way up to subleading order.

Estudamos as restrições impostas pelo princípio da covariância sobre o procedimento de quantização em espaços planos e curvos. Mostramos que o conjunto de todas as quantizações covariantes em espaços planos em coordenadas retangulares é composto de ordenamentos de operadores de posição e momento e exibimos uma parametrização funcional deste conjunto. Deduzimos regras para a quantização covariante em espaços planos em coordenadas gerais. Generalizamos estas quantizações para espaços curvos e mostramos que nestes espaços, além da ambiguidade de ordenamento, surge uma nova ambiguidade relacionada à curvatura. Este novo tipo de ambiguidade explica o surgimento de uma classe grande de potenciais quânticos no problema da quantização de uma partícula não relativística em um espaço curvo. / We study the restrictions imposed by the covariance principle on the quantization procedure in flat and curved spaces. We show that the set of all covariant quantizations in flat spaces in rectangular coordinates is composed of position and momentum operator orderings and exhibit a functional parametrization of this set. We deduce rules for the covariant quantization in flat spaces in general coordinates. We generalize these quantizations for curved spaces and show that in such spaces, besides the ordering ambiguity, it appears a new ambiguity related to the curvature. This new kind of ambiguity explains the appearence of a wide class of quantum potentials in the problem of quantization of a non-relativistic particle in curved space.

La simulation multiphysique de l'interaction rayonnement-matière, des effets thermiques et mécaniques induits dans un matériau nanostructuré à un intérêt notamment lorsqu'il s'agit d'élaborer des capteurs voire de les optimiser. En effet, les effets thermiques peuvent être utilisés pour des applications chimiques ou biologiques et les dilatations mécaniques peuvent influer sur la durabilité du capteur et sur son efficacité. A l’échelle nanométrique, les longueurs caractéristiques des effets thermo-électro-magnétique-mécaniques ne sont pas du même ordre de grandeur et la simulation éléments finis doit être adaptée à chaque problème avec un contrôle adapté à l'erreur de la solution physique. Une procédure utilisant un remailleur adaptatif 3D Optiform et Comsol Multiphysics permet une relaxation du maillage ou un raffinement adapté afin d'accélérer la résolution (RAM et CPU) et améliorer la solution physique. Des simulations numériques des nano-objets de formes simples et des nanoantennes pour lesquelles l'exaltation du champ électromagnétique est localisée dans des zones de quelques nanomètres, alors que le gradient de température est beaucoup plus homogène dans le domaine de calcul et les dilatations sont nanométriques sont effectuées / Multiphysics simulation of light-matter interaction, induced temperature and dilation in nanostructures is of interest especially when it comes to develop or optimize sensors. Indeed, thermal effects can be used for chemical or biological applications, and mechanical dilation can affect the durability of the sensor and its effectiveness.However, the characteristic lengths of electromagnetic fields, temperature and dilation are not of the same magnitude and the mesh used in a finite element multiphysics model must be adapted to each problem. An efficient numerical model for controlling the error in the computational domain is necessary while allowing the relaxation or the refinement of the mesh, in order to decrease the computational time and memory.The purpose of this thesis is to show that the adaptation loop of the mesh for solving a multiphysics 3D problem using Comsol Multiphysics in OPTIFORM mesher, based on the error estimation of the physical solution, is more efficient than a conventional remeshing process.The proposed procedure is applied to simulate nano-objects with simple shapes and to nanoantennas for which the confinement of the electromagnetic field is localized on a few nanometers, while the gradient of the temperature is much smoother in the computational domain but leading to nanoscale dilation

Scientific phenomena are often studied through collections of related scalar fields such as data generated by simulation experiments that are parameter or time dependent . Exploration of such data requires robust measures to compare them in a feature aware and intuitive manner. Topological data analysis is a growing area that has had success in analyzing and visualizing scalar fields in a feature aware manner based on the topological features. Various data structures such as contour and merge trees, Morse-Smale complexes and extremum graphs have been developed to study scalar fields. The extremum graph is a topological data structure based on either the maxima or the minima of a scalar field. It preserves local geometrical structure by maintaining relative locations of extrema and their neighborhoods. It provides a suitable abstraction to study a collection of datasets where features are expressed by descending or ascending manifolds and their proximity is of importance. In this thesis, we design a measure to understand the similarity between scalar fields based on the extremum graph abstraction. We propose a topological structure called the complete extremum graph and define a distance measure on it that compares scalar fields in a feature aware manner. We design an algorithm for computing the distance and show its applications in analyzing time varying data such as understanding periodicity, feature correspondence and tracking, and identifying key frames.

Sólitons são soluções clássicas de equações de campos não lineares, que possuem energia finita e densidade de energia localizada. Eles constituem pacotes de energia que se movem de maneira uniforme e não dispersiva, assemelhando-se a partículas estendidas. Quando se estuda um sistema à temperatura finita é possível tecer um paralelo entre a teoria quântica de campos e a mecânica estatística. Neste trabalho calculamos, na aproximação de um laço, a correção quântica à massa do kink do modelo 4 acoplado a um campo fermiônico. As contribuições bosônica e fermiônica são calculadas à temperatura zero e o comportamento das flutuações a temperatura finita também é analisado. / Solitons are classical solutions of non-linear field equations, that have finite energy and localised energy density. They constitute non-dispersive localised packages of energy moving uniformly, resembling extended particles. When studying a system at finite temperature one can make an analogy between quantum field theory and statistical mechanics. In this work we calculate, in one loop approximation, the quantum correction to the mass of the kink of the model 4 coupled to a fermionic field. The bosonic and fermionic contributions are calculated at zero temperature and the behavior of the finite temperature fluctuations are also analysed.

Le but à long terme de mon travail de recherche est de trouver une alternative viable à la méthode conforme, qui nous permettrait de mieux comprendre la structure géométrique de l'espace des solutions des équations de contrainte. L'avantage du modèle de Maxwell (the drift model) par rapport aux modèles plus classiques est la présence des paramètres supplémentaires. Le prix à payer, par contre, sera que la complexité analytique du système correspondant. Ma thèse a été structuré en deux parties : a. Existence sous la condition de petitesse des données initiales. Nous avons montré que le système de Maxwell est raisonnable dans le sens où nous pouvons le résoudre, malgré sa forte nonliniarité, sous des conditions de petitesse sur ses coefficients, en dimension 3, 4 et 5. Par conséquent, l'ensemble des solutions est non-vide. b. Stabilité Nous montrons la stabilité des solutions du système: ce résultat est obtenu en dimension 3,4 et 5, dans le cas où la métrique est conformément plate, et le drift et petit / The long-term goal of my work is to find a viable alternative to the conformal method, which would allow us to better understand the geometry of the space of solutions of the constraint equations. The advantage of Maxwell's model (the drift model) is the presence of additional parameters. Its downside, however, is that it proves to be much more difficult from an analytic standpoint. My thesis is structued in two parts: a. Existence under suitable smallness conditions. We show that Maxwell's system is sufficiently reasonable: it can be solved even given the presence of focusing non linearities. We prove this under smallness conditions of its coefficients, and in dimensions 3,4 and 5. An immediate consequence is that the set of solutions is non-empty. b. Stability. We verify that the solutions of the system are stable: this result holds in dimensions 3,4 and 5, when the metric is conformally flat and the drift is small

Prompted by the success of heterotic line bundle model building on Complete Intersection Calabi Yau (CICY) manifolds and the new developments regarding a generalization thereof, I analyze the possibility of model building on generalized CICY (gCICY) manifolds.  Ultimately this is realized on two examples of gCICYs, one of which topologically equivalent to a CICY and one inequivalent to any previously studied examples.  The first chapter is dedicated to reporting background information on CICYs and gCICYs.  The mathematical machinery of CICYs and their generalizations are introduced alongside explicit constructions of two examples.  The second chapter introduces the reader to heterotic line bundle model building on CICYs and gCICYs.  In the setting of gCICYs, similar to regular CICYs, model building is accomplished in two steps: first the larger $E_{8}$ gauge group is broken to an $SU( 5 )$ grand unified theory  through a line bundle model.  Then the GUT is broken using Wilson line symmetry breaking, for which the presence of a freely acting discrete symmetry must be established.  To that end, I proceed to show that the two previous examples benefit from a $\mathbb{Z}_{2}$ freely acting discrete symmetry.  Utilizing this symmetry I construct 20 and 11 explicit models for the two gCICY examples respectively, by scanning over a finite range of line bundle charges. / Ett av de största problemen i modern teoretisk fysik är att hitta en teori för kvantgravitation.För en konsekvent kvantteori gravitation skulle vara en väsentlig del i fysikens pussel, och koppla samman gravitationsfysiken för planeter och galaxer, som beskrivs av allmänna relativitetsteorin, till fysiken för partiklar, beskrivet av kvantfältteori.Bland de mest lovande teorierna finns strängteorin som föreslår att ersätta partiklar med strängar som materiens grundläggande beståndsdel.Förutom att lösa kvantgravitationproblemet hoppas teoretiska fysiker genom strängteorin att förenkla beskrivningen av partikelfysik.Detta skulle ske genom att ersätta hela partikelzoo med ett enda objekt: strängen.Olika vibrationer i strängen skulle motsvara olika partiklar och interaktioner mellan strängar skulle motsvara interaktioner mellan partiklar.För att vara motsägelsefri kräver dock strängteori att det finns minst sex fler dimensioner än de vi kan uppleva.En av strategierna som för närvarande studeras för att förlika extra dimensioner med och moderna experiment kallas ``kompaktifiering'' eller ``compactification'' på engelska.Strategin föreslår att dessa extra dimensioner ska vara kompakta och så små att de är osynliga för observationer.Interesant nog påverkar geometrin i det sexdimensionella kompakta rummet i stor utsträckning fysiken som strängteorin producerar: olika rum skulle producera olika partiklar och olika grundläggande naturkrafter.I den här uppsatsen studerar jag två exempel på sådana sexdimensionella rum som kommer från en uppsättning av rum som kallas `` generaliserade CICYs'' som nyligen har upptäckts.Med hjälp av de tekniker som liknar de som har utvecklats för andra liknade rum, visar jag att vissa aspekter av en strängteori kompaktifierad på generaliserade CICY återspeglar de som mäts genom moderna partikelfysikexperiment.