Bien que l'on ne sache pas (encore) quel phénomène unitarise la diffusion WLWL à l'échelle du TeV, les données indirecte actuelles favorise le boson de Higgs. Etant donné que cette particule scalaire pourrait être aussi lourde que la masse de Planck, comment peut-on expliquer sa légèreté ? La supersymmétrie (SUSY), brisée à l'échelle du TeV, peut effectuer cette stabilisation, et permettre du même coup l'existence de Théories de Grande Unifications (GUTs). Ces SUSY GUTs réalisées dans une dimension supplémentaire compactifiée, peuvent être particulièrement simples. De plus, elles peuvent être prises comme limite basse énergie d'une théorie de cordes. Cette thèse est consacrée à l'étude de tels modèles de SUSY GUTs. Nous avons étudié, développé et étendu certains aspects de la classe de modèle d'Unification Jauge-Higgs, et de la classe de modèle d'Unification Holographique. Différents aspects de la physique basse-énergie ont été étudiés, incluant spectre de masses, physique des saveur, matière noire, et phénoménologie au LHC. / Although one does not know (yet) which phenomenon unitarizes WLWL scattering at the TeV scale, indirect data currently favors the Higgs boson. Since such a scalar particle is susceptible to become as heavy as the Planck mass, how can one explain its lightness ? Supersymmetry (SUSY), broken at the TeV scale, can do this stabilization, providing in the same time models of Grand Uni fied Theories (GUTs). These SUSY GUTs, combined with extra spatial dimensions compacti fied on an interval, can be particularly simple. Moreover they can be seen as the low energy limit of some string theory. This thesis is devoted to the study of such models of SUSY GUTs on flat and warped orbifolds, trying to cover the range from models to experimental constraints. We studied, developed and extended certain aspects of two interesting frameworks of this type: a framework with gauge-Higgs uni fication, and the framework of holographic grand uni fication. We investigated several aspects of the low-energy implications, including mass spectra, flavour constraints, dark matter and LHC phenomenology

L’objet de cette thèse est l’étude de problèmes d’évaluation d’options dans les modèles à volatilité stochastique. La première partie est centrée sur les options américaines dans le modèle de Heston. Nous donnons d’abord une caractérisation analytique de la fonction de valeur d’une option américaine comme l’unique solution du problème d’obstacle parabolique dégénéré associé. Notre approche est basée sur des inéquations variationelles dans des espaces de Sobolev avec poids étendant les résultats récents de Daskalopoulos et Feehan (2011, 2016) et Feehan et Pop (2015). On étudie aussi les propriétés de la fonction de valeur d’une option américaine. En particulier, nous prouvons que, sous des hypothèses convenables sur le payoff, la fonction de valeur est décroissante par rapport à la volatilité. Ensuite nous nous concentrons sur le put américaine et nous étendons quelques résultats qui sont bien connus dans le monde Black-Scholes. En particulier nous prouvons la convexité stricte de la fonction de valeur dans la région de continuation, quelques propriétés de la frontière libre, la formule de Prime d’Exercice Anticipée et une forme faible de la propriété du smooth fit. Les techniques utilisées sont de type probabiliste. Dans la deuxième partie nous abordons le problème du calcul numérique du prix des options européennes et américaines dans des modèles à volatilité stochastiques et avec sauts. Nous étudions d’abord le modèle de Bates-Hull-White, c’est-à-dire le modèle de Bates avec un taux d’intérêt stochastique. On considère un algorithme hybride rétrograde qui utilise une approximation par chaîne de Markov (notamment un arbre “avec sauts multiples”) dans la direction de la volatilité et du taux d’intérêt et une approche (déterministe) par différence finie pour traiter le processus de prix d’actif. De plus, nous fournissons une procédure de simulation pour des évaluations Monte Carlo. Les résultats numériques montrent la fiabilité et l’efficacité de ces méthodes. Finalement, nous analysons le taux de convergence de l’algorithme hybride appliqué à des modèles généraux de diffusion avec sauts. Nous étudions d’abord la convergence faible au premier ordre de chaînes de Markov vers la diffusion sous des hypothèses assez générales. Ensuite nous prouvons la convergence de l’algorithme: nous étudions la stabilité et la consistance de la méthode hybride par une technique qui exploite les caractéristiques probabilistes de l’approximation par chaîne de Markov / We study option pricing problems in stochastic volatility models. In the first part of this thesis we focus on American options in the Heston model. We first give an analytical characterization of the value function of an American option as the unique solution of the associated (degenerate) parabolic obstacle problem. Our approach is based on variational inequalities in suitable weighted Sobolev spaces and extends recent results of Daskalopoulos and Feehan (2011, 2016) and Feehan and Pop (2015). We also investigate the properties of the American value function. In particular, we prove that, under suitable assumptions on the payoff, the value function is nondecreasing with respect to the volatility variable. Then, we focus on an American put option and we extend some results which are well known in the Black and Scholes world. In particular, we prove the strict convexity of the value function in the continuation region, some properties of the free boundary function, the Early Exercise Price formula and a weak form of the smooth fit principle. This is done mostly by using probabilistic techniques.In the second part we deal with the numerical computation of European and American option prices in jump-diffusion stochastic volatility models. We first focus on the Bates-Hull-White model, i.e. the Bates model with a stochastic interest rate. We consider a backward hybrid algorithm which uses a Markov chain approximation (in particular, a “multiple jumps” tree) in the direction of the volatility and the interest rate and a (deterministic) finite-difference approach in order to handle the underlying asset price process. Moreover, we provide a simulation scheme to be used for Monte Carlo evaluations. Numerical results show the reliability and the efficiency of the proposed methods.Finally, we analyze the rate of convergence of the hybrid algorithm applied to general jump-diffusion models. We study first order weak convergence of Markov chains to diffusions under quite general assumptions. Then, we prove the convergence of the algorithm, by studying the stability and the consistency of the hybrid scheme, in a sense that allows us to exploit the probabilistic features of the Markov chain approximation

Estudamos o efeito de Conjugação de Fase por Mistura Degenenerada de Quatro Ondas em Al2O3:Cr+3 (Rubi) e GdAlO3:Cr+3 com um laser de Ar (λ=5145 Å). Obtivemos eficiência aproximadamente quatro vezes maior no GdAlO3:Cr+3 (onde este trabalho é original) que no Rubi o que nos motivou a investigar as propriedades físicas que são relevantes para o fenômeno nestes sistemas (isto não foi bem compreendido no trabalho anterior em Rubi). Desenvolvemos um método interferométrico muito sensível para medida dos coeficientes não lineares do índice de refração n2 destes materiais (que não eram conhecidos) Com estes valores de n2 calculamos a eficiência de Conjugação de Fase em bom acordo com experiência. / We have studied the effect of Phase Conjugation by Degenerate Four Wave Mixing in Al2O3:Cr+3 (Rubi) and GdAlO3:Cr+3 with an Ar (λ=5145 Å). We obtained efficiency ?approximately? 4 times greater in GdAlO3:Cr+3 (where this work is original) than in Rubi and this have motivated us to investigate the physical properties that are important to explain this phenomenon in these materials (what wasn\'t well understood in the previous paper on Rubi(10)). We developed an interferometric method very sensitive to measure the nonlinear coeficient of refractive index n2 of these materials (what wasn\'t known). With these values of n2 we calculated the efficiency of the Phase Conjugation in good agreement with the experience.

We prove strong convergence of order 1/4 - E for arbitrarily small E > 0 of the Euler-Maruyama method for multidimensional stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is based on estimating the difference between the Euler-Maruyama scheme and another numerical method, which is constructed by applying the Euler-Maruyama scheme to a transformation of the SDE we aim to solve.

No description available.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste presente trabalho, faremos o estudo de importantes propriedades geomÃtricas e analÃticas de soluÃÃes de equaÃÃes diferenciais parciais elÃpticas totalmente nÃo-lineares do tipo: singulares e degeneradas. O estudo de processos de combustÃo que se degeneram ao longo do conjunto de anulamento da densidade de um gÃs, um caso particular de problemas do tipo "quenching", apresentam em sua modelagem equaÃÃes singulares que estÃo descritas neste trabalho. Nesta primeira parte iremos obter propriedades de uma soluÃÃo minimal, que vÃo desde o controle completo Ãtimo, atà a obtenÃÃo de estimativas de Hausdorff da fronteira livre singular. Por fim, iremos obter a regularidade Ãtima de soluÃÃes de equaÃÃes em que suas propriedades de difusÃo(elipticidade) se deterioram na ordem de uma potÃncia do seu gradiente ao longo do conjunto em que tal taxa de variaÃÃo se anula. / In this work we study important geometric and analytic properties to solutions of fully nonlinear elliptic partial differential equations, both singular and degenerate types. The study of combustion processes that degenerate along the null-set of the density of a gas, a particular case of quenching problems, present in their modeling, equations described in this work. In this first part we obtain properties of a minimal solution, since the complete optimal control until the Hausdorff estimates of the singular free boundary. Ultimately, we obtain the optimal regularity to equation solutions where their diffusion property (elipticity) deterorate in a power of their gradient along the set where such rate of variation nullifies.

Cette thèse porte sur l'étude des données $mathbb{Z}$-modulaires et leur catégorification, et particulièrement sur des données $mathbb{Z}$-modulaires reliées aux groupes de réflexions complexes, ainsi que sur la notion de caractère cellulaire pour ces derniers. Dans sa classification des caractères des groupes finis de type de Lie, Lusztig décrit une transformée de Fourier non abélienne et définit des données $mathbb{N}$-modulaires pour chaque famille de caractères unipotents. Dans des tentatives de généralisation aux Spetses, Broué, Malle et Michel introduisent des données $mathbb{Z}$-modulaires. On commence par donner une explication catégorique de certaines de ces données via la catégorie des représentations du double de Drinfeld d'un groupe fini, que l'on munit d'une structure pivotale non sphérique. Une étude approfondie de la notion de catégorie de fusion pivotale et légèrement dégénérée montre que l'on peut ainsi produire des données $mathbb{Z}$-modulaires. Afin de construire des exemples de telles catégories, on considère des extensions des catégories de fusion associées à $qgrroot{mathfrak{g}}$, où $mathfrak{g}$ est une algèbre de Lie simple, et $xi$ une racine de l'unité. Ces dernières sont construites comme des semi-simplifications de la catégorie des modules basculants de l'algèbre $qdblroot{mathfrak{g}}$, qui est une extension centrale de $qgrroot{mathfrak{g}}$. Dans le cas où $mathfrak{g}=mathfrak{sl}_{n+1}$, on relie cette catégorie à une des données $mathbb{Z}$-modulaires associée au groupe de réflexions complexes $Gleft(d,1,frac{n(n+1)}{2}right)$. Les groupes de réflexions exceptionnels sont également étudiés, et les catégorifications des données $mathbb{Z}$-modulaires associées font apparaître diverses catégories : des catégories de représentations de doubles de Drinfeld tordus ainsi que des sous-catégories des catégories de fusion des modules basculants en $qdblroot{mathfrak{g}}$ en type $A$ et $B$. / This work is a contribution to the categorification of $mathbb{Z}$-modular data and deals mainly with $mathbb{Z}$-modular data arising from complex reflection groups, as well as cellular characters for these groups. In his classification of representations of finite groups of Lie type, Lusztig defines a nonabelian Fourier transform, and associate a $mathbb{N}$-modular datum to each family of unipotent characters. In a generalization of Lusztig's theory to Spetses, Broué, Malle and Michel construct $mathbb{Z}$-modular data associated to some complex reflection groups. We first give a categorical explanation of some of these $mathbb{Z}$-modular data in terms of representation of the Drinfeld double of a finite group. We had to endow the category of representations with a non-spherical structure. The study of slightly degenerate categories shows that they naturally give rise to $mathbb{Z}$-modular data. In order to construct some examples, we consider an extension of the fusion categories associated to $qgrroot{mathfrak{g}}$, where $mathfrak{g}$ is a simple Lie algebra and $xi$ a root of unity. These categories are constructed as semisimplification of the category of tilting modules of $qdblroot{mathfrak{g}}$, which is a central extension of $qgrroot{mathfrak{g}}$. If $mathfrak{s}=mathfrak{sl}_{n+1}$, we show that this category is related to some $mathbb{Z}$-modular data associated to the complex reflection group $Gleft(d,1,frac{n(n+1)}{2}right)$. Exceptional complex reflection groups are also considered and many different categories appear in the categorification of the associated $mathbb{Z}$-modular data : modules categories over twisted Drinfeld doubles as well as some subcategories of fusion categories of tilting modules over $qdblroot{mathfrak{g}}$ in type $A$ and $B$.

We establish elements of a new approach to ellipticity and parametrices within operator algebras on manifolds with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaes. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The "full" calculus involves two separate theories, one near the tip of the corner and another one at the conical exit to infinity. However, concerning the conical exit to infinity, we establish here a new concrete calculus of edge-degenerate operators which can be iterated to higher singularities. / Wir führen einen neuen Zugang ein zu Elliptizität und Parametrices in Operatorenalgebren auf Mannigfaltigkeiten mit höheren Singularitäten, nur basierend auf allgemeinen axiomatischen Voraussetzungen über parameter-abhängige Operatoren in geeigneten Skalen von Räumen. Die Idee besteht darin, ein iteratives Verfahren zu modellieren mit neuen Generationen von parameter-abhängigen Operatortheorien, zusammen mit neuen Skalen von Räumen, die analoge Voraussetzungen erfüllen wie die ursprünglichen Objekte, jetzt auf dem entsprechenden höheren Niveau. Der „volle“ Kalkül besteht aus zwei separaten Theorien, eine nahe der Spitze der Ecke und eine andere am konischen Ausgang nach Unendlich. Allerdings, bezüglich des konischen Ausgangs nach Unendlich, bauen wir hier einen neuen konkreten Kalkül von kanten-entarteten Operatoren auf, der für höhere Singularitäten iteriert werden kann.

Chirality is a nuclear symmetry which is suggested to occur in nuclei when the total angular momentum of the system has an aplanar orientation [Fra97, Fra01]. It can occur for nuclei with triaxial shape, which have valence protons and neutrons with predominantly particle and hole nature. It is expected that the angular momenta of an odd particle and an odd hole (both occupying high-j orbitals) are aligned predominantly along the short and the long axes of the nucleus respectively, whereas the collective rotation occurs predominantly around the intermediate axis of a triaxially deformed nucleus in order to minimize the total energy of the system. Such symmetry is expected to be exhibited by a pair of degenerate DI = 1 rotational bands, i.e. all properties of the partner bands should be identical. The results suggested that spin independence of the energy staggering parameter S(I ) within two-quasiparticle chiral bands (previously suggested a fingerprint of chirality) is found only if the Coriolis interaction can be completely neglected. However, if the configuration is nonrestricted, the Coriolis interaction is often strong enough to create considerable energy staggering. It was also found that staggering in the intra- and inter-band B(M1) reduced transition probabilities (proposed as another fingerprint of chirality) may be a result of effects other than strongly broken chirality. Therefore, the use of the B(M1) staggering as a fingerprint of strongly broken chiral symmetry seems rather risky, in particular if the phase of the staggering is not checked.

The study of white dwarfs (WDs) provides insight into understanding WD formation rates, evolution, and space density. Individually, nearby WDs are excellent candidates for astrometric planetary searches because the astrometric signature is greater than for an identical, more distant WD system. As a population, a complete volume-limited sample is necessary to provide unbiased statistics; however, their intrinsic faintness has allowed some to escape detection. The aim of this dissertation is to identify nearby WDs, accurately characterize them, and target a subset of potentially interesting WDs for follow-up analyses. The most unambiguous method of identifying new WDs is by their proper motions. After evaluating all previous southern hemisphere proper motion catalogs and selecting viable candidates, we embarked on our own southern hemisphere proper motion survey, the SuperCOSMOS-RECONS (SCR) survey. A number of interesting objects were discovered during the survey, including the 24th nearest star system -- an M dwarf with a brown dwarf companion. After a series of spectroscopic observations, a total of 56 new WD systems was identified (18 from the SCR survey and 38 from other proper motion surveys). CCD photometry was obtained for most of the 56 new systems in an effort to model the physical parameters and obtain distance estimates via spectral energy distribution fitting. An independent distance estimate was also obtained by deriving a color-MV relation for several colors based on WDs with known distances. Any object whose distance estimate was within 25 pc was targeted for a trigonometric parallax via our parallax program, CTIOPI. Currently, there are 62 WD systems on CTIOPI. A subset of 53 systems has enough data for at least a preliminary parallax (24 are definitive). Of those 53 systems, nine are previously known WDs within 10 pc that we are monitoring for perturbations from unseen companions, and an additional 29 have distances within 25 pc. Previously, there were 109 known WDs with parallaxes placing them within 25 pc; therefore, our effort has already increased the nearby sample by 27%. In addition, at least two objects show hints of perturbations from unseen companions and need follow-up analyses.