Nous nous intéressons, dans cette thèse, à l'étude des méthodes d'extrapolation polynômiales et à l'application de ces méthodes dans l'accélération de méthodes de points fixes pour des problèmes donnés. L'avantage de ces méthodes d'extrapolation est qu'elles utilisent uniquement une suite de vecteurs qui n'est pas forcément convergente, ou qui converge très lentement pour créer une nouvelle suite pouvant admettreune convergence quadratique. Le développement de méthodes cycliques permet, deplus, de limiter le coût de calculs et de stockage. Nous appliquons ces méthodes à la résolution des équations de Navier-Stokes stationnaires et incompressibles, à la résolution de la formulation Kohn-Sham de l'équation de Schrödinger et à la résolution d'équations elliptiques utilisant des méthodes multigrilles. Dans tous les cas, l'efficacité des méthodes d'extrapolation a été montrée.Nous montrons que lorsqu'elles sont appliquées à la résolution de systèmes linéaires, les méthodes d'extrapolation sont comparables aux méthodes de sous espaces de Krylov. En particulier, nous montrons l'équivalence entre la méthode MMPE et CMRH. Nous nous intéressons enfin, à la parallélisation de la méthode CMRH sur des processeurs à mémoire distribuée et à la recherche de préconditionneurs efficaces pour cette même méthode. / In this thesis, we study polynomial extrapolation methods. We discuss the design and implementation of these methods for computing solutions of fixed point methods. Extrapolation methods transform the original sequance into another sequence that converges to the same limit faster than the original one without having explicit knowledge of the sequence generator. Restarted methods permit to keep the storage requirement and the average of computational cost low. We apply these methods for computing steady state solutions of incompressible flow problems modelled by the Navier-Stokes equations, for solving the Schrödinger equation using the Kohn-Sham formulation and for solving elliptic equations using multigrid methods. In all cases, vector extrapolation methods have a useful role to play. We show that, when applied to linearly generated vector sequences, extrapolation methods are related to Krylov subspace methods. For example, we show that the MMPE approach is mathematically equivalent to CMRH method. We present an implementation of the CMRH iterative method suitable for parallel architectures with distributed memory. Finally, we present a preconditioned CMRH method.

Cette thèse est principalement consacrée à l’étude du comportement en temps long de solutions de certaines équations aux dérivées partielles hamiltoniennes, du type i∂_t u=X_H (u), en particulier l’existence globale, la croissance des normes de Sobolev, la diffusion et l’approximation par la dynamique résonante.Dans ce contexte, nous considérons d’abord une perturbation de l’équation de Szegő cubique par un potentiel linéaire, i∂_t u=∏ |u|² u+α∫ u,α∈R, (α-Szegő) où ∏▒ désigne le projecteur de Szegő sur les fréquences positives. Pour α=0, cette équation est l’équation de Szegő cubique, étudiée récemment par Gérard et Grellier comme modèle mathématique d’équation non linéaire et non dispersive. Pour l’équation (α–Szegő), nous établissons le caractère bien posé et la complète intégrabilité, et étudions la dynamique des valeurs singulières des opérateurs de Hankel associés. En outre, nous montrons les propriétés suivantes pour cette équation, sur une classe de sous–variétés invariantes de dimensions finies arbitrairement grandes : si α<0, toute trajectoire est relativement compacte, et toute norme de Sobolev est bornée le long de cette trajectoire. Siα>0, il existe des trajectoires le long desquelles toutes les normes de Sobolev de régularité plus grande que ½ tendent exponentiellement vers l’infini en temps.Dans une seconde partie, nous étudions un système mixte Schrödinger–ondes sur le cylinder (x,y)∈R×T , i∂_t U+∂_xx U-|D_y |U=|U|² U,(WS)En adaptant une idée de Hani–Pausader–Tzvetkov–Visciglia, nous établissons une théorie du scattering modifiée reliant les petites solutions de cette équation et les petites solutions de l’équation de Szegő cubique. En combinant cette théorie du scattering avec un résultat récent de Gérard–Grellier, nous en déduisons l’existence de solutions globales de (WS) qui sont non bornées dans l’espace L_x² H_y^s (R×T) pour tout s>½ . / The main purpose of this Ph.D. thesis is to study the long time behavior of solutionsto some Hamiltonian PDEs, i∂_t u=X_H (u), including global existence, growth of high Sobolev norms, scattering and long time approximation by resonant dynamics.In this context, at first we consider the Szegő equation on the circle S1 perturbed bya linear potential, i∂_t u=∏ |u|² u+α∫ u,α∈R, (α-Szegő) where ∏ is the projector onto the non-negative frequencies. For α=0, it turns out tobe the cubic Szegő equation, which was recently introduced by Gérard and Grellier as amathematical toy model of a non-linear totally non dispersive equation.We study the global well-posedness, the integrability and the dynamics of the singularvalues of the related Hankel operators of the α –Szegő equation. Moreover, we establishthe following properties for this equation on a class of invariant submanifolds, with anarbitrary large dimension. For α<0, any trajectory is relatively compact, and all theSobolev norms are bounded on it. For α>0, there exist trajectories on which everySobolev norm of regularity s>½ , exponentially tends to infinity in time.Second, we study the wave-guide Schrödinger equation posed on the spatial domain(x,y)∈R×T ，i∂_t U+∂_xx U-|D_y |U=|U|² U,(WS)Adapting an idea by Hani–Pausader–Tzvetkov–Visciglia, we establish a modified scattering theory between small solutions to this equation and small solutions to the cubic Szegő equation. Combining this scattering theory with a recent result by Gérard–Grellier, we infer existence of global solutions to (WS) which are unbounded in the space L_x^2 H_y^s (R×T) for every s>½ .

Mit dem Begriff Wellenleiterquantenelektrodynamik (WQED) wird gemeinhin die Physik des quantisierten und in eindimensionalen Wellenleitern geführten Lichtes in Wechselwirkung mit einzelnen Emittern bezeichnet. In dieser Arbeit untersuche ich Effekte der WQED für einzelne Dreiniveausysteme (3NS) bzw. Paare von Zweiniveausystemen (2NS), die in den Wellenleiter eingebettet sind. Hierzu bediene ich mich hauptsächlich numerischer Methoden und betrachte die Modellsysteme im Rahmen der Drehwellennäherung. Ich untersuche die Dynamik der Streuung einzelner Photonen an einzelnen, in den Wellenleiter eingebetteten 3NS. Dabei analysiere ich den Einfluss dunkler bzw. nahezu dunkler Zustände der 3NS auf die Streuung und zeige, wie sich mit Hilfe stationärer elektrischer Treibfelder gezielt auf die Streuung einwirken lässt. Ich quantifiziere Verschränkung zwischen dem Lichtfeld im Wellenleiter und den Emittern mit Hilfe der Schmidt-Zerlegung und untersuche den Einfluss der Form der Einhüllenden eines Einzelphotonpulses auf die Ausbeute der Verschränkungserzeugung bei der Streuung des Photons an einem einzelnen Lambda-System im Wellenleiter. Hier zeigt sich, dass die Breite der Einhüllenden im k-Raum und die Emissionszeiten der beiden Übergänge des 3NS die maßgeblichen Parameter darstellen. Abschließend ergründe ich die Emissionsdynamik zweier im Abstand L in den Wellenleiter eingebetteter 2NS. Diese Dynamik wird insbesondere durch kavitätsartige und polaritonische Zustände des Systems aus Wellenleiter und Emitter ausschlaggebend beeinflusst. Bei der kollektiven Emission der 2NS treten - abhängig vom Abstand L - Sub- bzw. Superradianz auf. Dabei nimmt die Intensität dieser Effekte mit längerem Abstand L zu. Diese Eigenart lässt sich auf die Eindimensionalität des Wellenleiters zurückführen. / The field of waveguide quantum electrodynamics (WQED) deals with the physics of quantised light in one-dimensional (1D) waveguides coupled to single emitters. In this thesis, I investigate WQED effects for single three-level systems (3LS) and pairs of two-level systems (2LS), respectively, which are embedded in the waveguide. To this end, I utilise numerical techniques and consider all model systems within the rotating wave approximation. I investigate the dynamics of single-photon scattering by single, embedded 3LS. In doing so, I analyse the influence of dark and almost-dark states of the 3LS on the scattering dynamics. I also show, how stationary electrical driving fields can control the outcome of the scattering. I quantify entanglement between the waveguide''s light field and single emitters by utilising the Schmidt decomposition. I apply this formalism to a lambda-system embedded in a 1D waveguide and study the generation of entanglement by scattering single-photon pulses with different envelopes on the emitter. I show that this entanglement generation is mainly determined by the photon''s width in k-space and the 3LS''s emission times. Finally, I explore the emission dynamics of a pair of 2LS embedded by a distance L into the waveguide. These dynamics are primarily governed by bound states in the continuum and by polaritonic atom-photon bound-states. For collective emission processes of the two 2LS, sub- and superradiance appear and depend strongly on the 2LS''s distance: the effects increase for larger L. This is an exclusive property of the 1D nature of the waveguide.

Dans cette thèse, nous étudions l’influence d’un plasma non-stationnaire produit par des impulsions laser en régime d’auto-focalisation. Cette auto-focalisation est couplée à des non-linéarités Brillouin pour des impulsions nanosecondes dans les verres de silice. Elle excite différents canaux d’ionisation dans les cristaux de KDP irradiées par des impulsions femtosecondes. Tout d’abord, nous dérivons les équations de propagation des ondes optiques laser et Stokes sujettes à la filamentation due à l’effet Kerr, la rétrodiffusion Brillouin et à la génération de plasma. Dans une deuxième partie, nous présentons des résultats numériques sur la propagation non-linéaire de faisceaux LIL. Ceux-ci révèlent l’importance de la distribution temporelle de l’impulsion pompe dans la compétition entre auto-compression Kerr et la rétrodiffusion Brillouin stimulée. Ces simulations préliminaires permettent de valider le système anti-Brillouin opté pour le LMJ sur la base de faisceaux millimétriques.Dans une troisième partie, nous présentons des résultats théoriques et numériques sur la filamentation d’impulsions nanosecondes opérant dans l’ultraviolet et l’infrarouge. L’influence d’un plasma inertiel sur la dynamique de couplage de deux ondes en contre-propagation est examinée. Dans une configuration à une onde, une analyse variationnelle reproduit les caractéristiques globales d’un équilibre quasi-stationnaire entre auto-compression Kerr et défocalisation plasma. Toutefois, cet équilibre cesse pour faire place à des instabilités modulationnelles induites par rétroaction du plasma sur l’onde de pompe. Nous montrons que des modulations de phase supprimant la rétrodiffusion Brillouin permettent d’inhiber ces instabilités plasma. La robustesse de ces modulations de phase est testée en présence d’un bruit aléatoire dans le profil de  l’impulsion laser.Enfin, nous étudions numériquement la dynamique non-linéaire d’impulsions femtosecondes se propageant dans la silice et le KDP. Premièrement, nous montrons que la présence de défauts impliquant moins de photons pour exciter un électron de la bande de valence à la bande de conduction promeut des intensités de filamentation plus élevées. Ensuite, nous comparons la dynamique de filamentation dans la silice avec celle dans un cristal KDP. Le modèle d’ionisation pour le KDP prend en compte la présence de défauts et la dynamique électrons-trous. Nous montrons que la dynamique de propagation dans la silice et le KDP présente des analogies remarquables pour des rapports de puissance incidente sur puissance critique équivalents.La conclusion nous permet de résumer les résultats originaux obtenus dans le cadre de cette thèse et d’en discuter des développements ultérieurs possibles. / In this thesis, we study the role of an inertial plasma reponse produced by laser pulses in self-focusing regime. Self-focusing is coupled with Brillouin nonlinearities for nanosecond pulses in silica glasses. For femtosecond pulses propagating in KDP crystals, self-focusing excites various ionization chanels. First of all, we derive the propagation equations for the pump and Stokes waves, subjected to filamentation due to optical Kerr effect, stimulated Brillouin scattering and plasma generation. In the second part, we present numerical results on the nonlinear propagation of LIL laser beams. These results show that temporal distribution of the pump pulse play a key role in the competition between self-focusing and stimulated Brillouin scattering. These preliminary results valide the anti-Brillouin system opted on the MegaJoule laser (LMJ) on the basis of milimetric-size laser beam.In a third part, we present numerical and theoretical results on the filamentation in fused silica of nanosecond light pulses operating in ultraviolet and infrared range. Emphasis is put on the action of a dynamical plasma reponse on two counterpropagating waves. For a single wave, we develop a variational analysis which reproduces global propagation features for a quasistationary balance between self-focusing and plasma defocusing. However, such a quasistionary balance ceases to clean up modulational instabilites induced by plasma retroaction on the pump wave. We show that phase modulations supress both simulated Brillouin scattering and plasma instabilities. The robustness of phase modulations is evaluated in presence of random fluctuations in the input pump pulse profile.Finally, we study numerically the nonlinear propagation of femtosecond pulses in fused silica and KDP. First, we show that the presence of defects involving less photons for exciting electrons from the valence band to the conduction band promotes higher filamentation intensity levels. Then, we compare the filamentation dynamic in silica and KDP crystal. The ionization model for KDP crystal takes into account the presence of defects and the electron-hole dynamics. We show that the propagation dynamics in silica and KDP are almost identical at equivalent ratios of input power over the critical power self-focusing.The summary of this thesis recalls the original results obtained and discusses the possibility of future developments.

Made available in DSpace on 2015-05-15T11:46:17Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2216718 bytes, checksum: 2b03ed1c154fa751c5c18afd31a144ad (MD5) Previous issue date: 2014-04-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work addresses a class of Trudinger-Moser type inequalities in weighted Sobolev spaces in R2. As an application of these inequalities and by using variational methods, we establish sufficient conditions for the existence, multiplicity and nonexistence of solutions for some classes of nonlinear Schrödinger elliptic equations (and systems of equations) with unbounded, singular or decaying radial potentials and involving nonlinearities with exponential critical growth of Trudinger-Moser type. / Este trabalho aborda uma classe de desigualdades do tipo Trudinger-Moser em espaços de Sobolev com peso em R2. Como aplicação destas desigualdades e usando métodos variacionais, estabeleceremos condições suficientes para a existência, multiplicidade e não-existência de soluções para algumas classes de equações (e sistemas de equações) de Schrödinger elípticas não-lineares com potenciais radiais ilimitados, singulares na origem ou decaindo a zero no infinito e envolvendo não-linearidades com crescimento crítico exponencial do tipo Trudinger-Moser.

Made available in DSpace on 2015-05-15T11:46:19Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2353317 bytes, checksum: d71ead9d4e0f785df35982fc9318c7da (MD5) Previous issue date: 2014-07-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this thesis, we study controllability results of some phenomena modeled by Partial Differential Equations (PDEs): Multi objective control problem, for parabolic equations, following the Stackelber-Nash strategy is considered: for each leader control which impose the null controllability for the state variable, we find a Nash equilibrium associated to some costs. The leader control is chosen to be the one of minimal cost. Null controllability for the linear Schrödinger equation: with a convenient space-time discretization, we numerically construct boundary controls which lead the solution of the Schrödinger equation to zero; using some arguments of Fursikov-Imanuvilov (see [Lecture Notes Series, Vol 34, 1996]) we construct controls with exponential decay at final time. Null controllability for a Schrödinger-KdV system: in this work, we combine global Carleman estimates with energy estimates to obtain an observability inequality. The controllability result holds by the Hilbert Uniqueness Method (HUM). Controllability results for a Euler type system, incompressible, inviscid, under the influence of a temperature are obtained: we mainly use the extension and return methods / Nesta tese, estudaremos resultados de controle para alguns problemas da teoria das equações diferenciais parciais (EDPs): Problema de controle multi objetivo para um problema parabólico, seguindo estratégias do tipo Stackelberg-Nash: para cada controle líder, que impõe a controlabilidade nula para o estado, encontramos seguidores, em equilíbrio de Nash, associados a funcionais custo. Em seguida, determinamos o líder de menor custo. Controlabilidade nula para a equação de Schrödinger linear: com uma discretização espaço-tempo adequada, construímos numericamente controles-fronteira que conduzem a solução de Schrödinger a zero; utilizando técnicas de Fursikov-Imanuvilov (veja [Lecture Notes Series, Vol 34, 1996]) contruímos controles que decaem exponencialmente no tempo final. Controlabilidade nula para um sistema acoplado Schrödinger-KdV: neste trabalho, combinando estimativas globais de Carleman com estimativas de energia, obtemos uma desigualdade de observabilidade. O resultado de controlabilidade segue pelo método de unicicade Hilbert (HUM). Controlabilidade para um sistema do tipo Euler, incompressível, invíscido, sob influência de uma temperatura: Utilizamos os métodos de extensão seguido do método do retorno para provar resultados de controlabilidade para este sistema

Ces travaux de thèse rapportent l’étude des instabilités non-linéaires et des évènements extrêmesse développant lors de la propagation guidée d’un champ électromagnétique au sein de fibresoptiques. Après un succinct rappel des divers processus linéaires et non-linéaires menant à lagénération de super continuum optique, nous montrons que le spectre de celui-ci peut présenterde larges fluctuations, incluant la formation d’événements extrêmes, dont les propriétés statistiqueset l’analogie avec les vagues scélérates hydrodynamiques sont abordées en détail. Nous présentonsune preuve de principe de l’application de ces fluctuations spectrales à la génération de nombres etde marches aléatoires et identifions le phénomène d’instabilité de modulation, ayant lieu lors de laphase initiale d’expansion spectrale du super continuum, comme principale contribution à la formationd’événements extrêmes. Ce mécanisme est étudié numériquement et analytiquement, en considérantune catégorie de solutions exactes de l’équation de Schrödinger non-linéaire présentant descaractéristiques de localisations singulières. Les résultats obtenus sont vérifiés expérimentalement,notamment grâce à un système de caractérisation spectrale en temps réel et à l’utilisation conjointede métriques statistiques innovantes (ex : cartographie de corrélations spectrales). L’excellent accordentre simulations et expériences a permis de valider les prédictions théoriques et d’accéder àune meilleure compréhension des dynamiques complexes inhérentes à la propagation non-linéaired’impulsions optiques. / This thesis reports the study of nonlinear instabilities and extreme events occurring during the guidedpropagation of an electromagnetic field into optical fibers. After a short overview of the various linearand nonlinear processes leading to optical supercontinuum generation, we show that its spectrumcan exhibit large fluctuations, including the formation of extreme events, whose statistical propertiesas well as hydrodynamic rogue waves analogy are studied in detail. We provide a proof of principle ofusing these spectral fluctuations for random number and random walk generation and identify modulationinstability, associated with the onset phase of supercontinuum spectral broadening, as themain phenomenon leading to extreme event formation. This mechanism is studied both numericallyand analytically, considering a class of exact solutions of nonlinear Schrödinger equation which exhibitsingular localization characteristics. The results are experimentally verified, especially througha real-time spectral characterization system along with the use of innovative statistical metrics (e.g.spectral correlation maps). The excellent agreement between simulations and experiments allowedus to validate the theoretical predictions and get further insight into the complex dynamics associatedto nonlinear optical pulse propagation.

Das Verstehen der elektronischen Prozesse in Nano-Hybridsystemen, bestehend aus Molekülen und Halbleiterstrukturen, eröffnet neue Möglichkeiten für optoelektronische Bauteile. Dafür benötigt es nanoskopische und gleichzeitig atomare Modelle und somit angepasste Rechenmethoden. Insbesondere "Standard"-Ansätze für die Dynamik offener Quantensysteme werden mit zunehmender Systemgröße jedoch sehr ineffizient. In dieser Arbeit wird eine neue Methode basierend auf einer stochastischen Schrödinger-Gleichung etablieren. Diese umgeht die numerischen Limits der Quanten-Mastergleichung und ermöglicht Simulationen von imposanter Größe. Ihr enormes Potenzial wird hier in Studien zu Anregungsenergietransfer und Ladungsseparation an zwei realistischen Nano-Hybridsystemen demonstriert: para-sexiphenyl Moleküle auf einer flachen ZnO Oberfläche (6P/ZnO), und ein tubuläres C8S3 Farbstoffaggregat gekoppelt an einen CdSe Nanokristall (TFA/NK). Im 6P/ZnO System findet nach optischer Anregung Energietransfer vom 6P Anteil zum ZnO statt. Direkt an der Grenzfläche können Frenkel-Exzitonen zusätzlich Ladungsseparation initiieren, wobei Elektronen ins ZnO transferiert werden und Löcher im 6P Anteil verbleiben. Beide Mechanismen werden mittels laserpulsinduzierter ultraschneller Wellenfunktionsdynamik simuliert. Danach wird die langsamere dissipative Lochkinetik im 6P Anteil studiert. Hierfür wird die eigene Simulationstechnik der stochastischen Schrödinger-Gleichung verwendet. Die Studie an der TFA/NK Grenzfläche basiert auf einer gigantischen equilibrierten Aggregatstruktur aus 4140 Molekülen. Ein generalisiertes Frenkel-Exzitonenmodell wird benutzt. Der Ansatz der stochastischen Schrödinger-Gleichung ermöglicht bemerkenswerte Einblicke in die Aggregat-interne Exzitonenrelaxation. Danach werden inkohärente Raten des Exzitonentransfers zum NK berechnet. Unterschiedliche räumliche Konfigurationen werden untersucht und es wird diskutiert, warum das Förster-Modell hier keine Gültigkeit besitzt. / Understanding the electronic processes in hybrid nano-systems based on molecular and semiconductor elements opens new possibilities for optoelectronic devices. Therefore, it requires for models which are both nanoscopic and atomistic, and so for adapted computational methods. In particular, "standard" methods for open quantum system dynamics however become very inefficient with increasing system size. In this regard, it is a key challenge of this thesis, to establish a new stochastic Schrödinger equation technique. It bypasses the computational limits of the quantum master equation and enables dissipative simulations of imposing dimensionality. Its enormous potential is demonstrated in studies on excitation energy transfer and charge separation processes in two realistic nanoscale hybrid systems: para-sexiphenyl molecules deposited on a flat ZnO surface (6P/ZnO), and a tubular dye aggregate of C8S3 cyanines coupled to a CdSe nanocrystal (TDA/NC). After optical excitation, the 6P/ZnO system exhibits exciton transfer from the 6P part to the ZnO. Close to the interface, Frenkel excitons may further initiate charge separation where electrons enter the ZnO and holes remain in the 6P part. Both mechanisms are simulated in terms of laser-pulse induced ultrafast wave packet dynamics. Afterwards, slower dissipative hole motion in the 6P part is studied. For this purpose, the own stochastic Schrödinger equation simulation technique is applied. The study on the TDA/NC interface is based on a gigantic equilibrated nuclear structure of the aggregate including 4140 dyes. A generalized Frenkel exciton model is employed. Thanks to the stochastic Schrödinger equation approach, energy relaxation in the exciton band of the TDA is simulated in outstanding quality and extend. Then, incoherent rates for exciton transfer to the NC are computed. Different spatial configurations are studied and it is discussed why the Förster model possesses no validity here.

In this thesis, we present some new results in distribution theory for both discrete and continuous random variables, together with their motivating applications. We start with some results about the Multivariate Gaussian Distribution and its characterization as a maximizer of the Strichartz Estimates. Then, we present some characterizations of discrete and continuous distributions through ideas coming from optimal transportation. After this, we pass to the Simpson's Paradox and see that it is ubiquitous and it appears in Quantum Mechanics as well. We conclude with a group of results about discrete and continuous distributions invariant under symmetries, in particular invariant under the groups $A_1$, an elliptical version of $O(n)$ and $\mathbb{T}^n$. As mentioned, all the results proved in this thesis are motivated by their applications in different research areas. The applications will be thoroughly discussed. We have tried to keep each chapter self-contained and recalled results from other chapters when needed. The following is a more precise summary of the results discussed in each chapter. In chapter \ref{chapter 2}, we discuss a variational characterization of the Multivariate Normal distribution (MVN) as a maximizer of the Strichartz Estimates. Strichartz Estimates appear as a fundamental tool in the proof of wellposedness results for dispersive PDEs. With respect to the characterization of the MVN distribution as a maximizer of the entropy functional, the characterization as a maximizer of the Strichartz Estimate does not require the constraint of fixed variance. In this chapter, we compute the precise optimal constant for the whole range of Strichartz admissible exponents, discuss the connection of this problem to Restriction Theorems in Fourier analysis and give some statistical properties of the family of Gaussian Distributions which maximize the Strichartz estimates, such as Fisher Information, Index of Dispersion and Stochastic Ordering. We conclude this chapter presenting an optimization algorithm to compute numerically the maximizers. Chapter \ref{chapter 3} is devoted to the characterization of distributions by means of techniques from Optimal Transportation and the Monge-Amp\`{e}re equation. We give emphasis to methods to do statistical inference for distributions that do not possess good regularity, decay or integrability properties. For example, distributions which do not admit a finite expected value, such as the Cauchy distribution. The main tool used here is a modified version of the characteristic function (a particular case of the Fourier Transform). An important motivation to develop these tools come from Big Data analysis and in particular the Consensus Monte Carlo Algorithm. In chapter \ref{chapter 4}, we study the \emph{Simpson's Paradox}. The \emph{Simpson's Paradox} is the phenomenon that appears in some datasets, where subgroups with a common trend (say, all negative trend) show the reverse trend when they are aggregated (say, positive trend). Even if this issue has an elementary mathematical explanation, the statistical implications are deep. Basic examples appear in arithmetic, geometry, linear algebra, statistics, game theory, sociology (e.g. gender bias in the graduate school admission process) and so on and so forth. In our new results, we prove the occurrence of the \emph{Simpson's Paradox} in Quantum Mechanics. In particular, we prove that the \emph{Simpson's Paradox} occurs for solutions of the \emph{Quantum Harmonic Oscillator} both in the stationary case and in the non-stationary case. We prove that the phenomenon is not isolated and that it appears (asymptotically) in the context of the \emph{Nonlinear Schr\"{o}dinger Equation} as well. The likelihood of the \emph{Simpson's Paradox} in Quantum Mechanics and the physical implications are also discussed. Chapter \ref{chapter 5} contains some new results about distributions with symmetries. We first discuss a result on symmetric order statistics. We prove that the symmetry of any of the order statistics is equivalent to the symmetry of the underlying distribution. Then, we characterize elliptical distributions through group invariance and give some properties. Finally, we study geometric probability distributions on the torus with applications to molecular biology. In particular, we introduce a new family of distributions generated through stereographic projection, give several properties of them and compare them with the Von-Mises distribution and its multivariate extensions. / Thesis / Doctor of Philosophy (PhD)

No description available.