Estudando o processo q q BARRA W Brown, Mikaelian, Sahdev, Samuel descobriram um zero na distribuição angular do W quando seu momento magnético tem o valor característico de uma partícula de gauge. Goebel, Halzen e Leveille mostraram que este zero é uma consequência da fatorização da amplitude em um termo que contém a dependência da carga ou outros índices de simetria interna, e outro que contém a dependência dos spins ou índices de polarização. Esta fatorização existe em geral para amplitudes de processos envolvendo quatro partículas na aproximação árvore, quando uma ou mais destas partículas é um campo de gauge. Portanto a existência de um zero na seção de choque é uma prova direta da estrutura de gauge da teoria. A fatorização baseia-se em uma identidade, identidade de Jacobi espacial generalizada, cuja demonstração ou significado físico ainda não fora elucidado. O objetivo do presente trabalho é estudar esta identidade de Jacobi espacial generalizada. Para tanto calculamos, no capítulo I, a amplitude de um processo de espalhamento gluon-gluon envolvendo cinco partículas e reorganizando esta amplitude por analogia com um processo de interação fóton-pion, mostramos que não existe, no caso de cinco partículas, a identidade de Jacobi espacial generalizada, mas sim uma série de identidades espaciais parciais, que se compõe, no processo de quatro partículas, em uma única identidade. No capítulo II estudamos um processo envolvendo quatro partículas, das quais três campos escalares, porém agora aproximação de um loop, e mostramos que também não existe identidade de Jacobi espacial generalizada. / Brown, Mikaelian, Sahdev, and Samuel discovered that the angular distribution of the process q q BARRA W in lowest order has a zero, if the magnetic moment f the W has the characteristic value of a gauge field. Goebel, Halzen and Leveille showed that this zero is a consequence of a factorizability of the amplitude into one factor which contains the dependence on the charge or other internal, symmetry indices, and another which contains the dependence on the spin or polarization indices. This factorization is found to hold for any four particle tree-approximation amplitude, when one or more of the four particles is a gauge-field. Therefore, the study of the angular distribution of the process q q BARRA W, directly probes the gauge structure of the theory. The factorization hinges on a spatial generalized Jacobi identity obeyed by the polarization-dependent factors of the vertices, whose physical significance or general demonstration was not known. The purpose of the present work is to study this identity. With this in mind we work out, in chapter I, the amplitude of a scattering gluon-gloun with five particles. Reorganizing this amplitude by analogy with an interaction process photon-pion, we show that does not exist, in this case, the spatial generalized Jacobi identity, but instead many spatial partial identities that compose themselves, in the case of a four particle process, in one single identity. In chapter II, we study a process with four particles, three of them scalar fields, but in the one loop approximation, and show that, in this case too, does not exist the spatial generalized Jacobi identity.

Artificial vision is the problem of creating systems capable of processing visual information. A fundamental sub-problem of artificial vision is image segmentation, the problem of detecting a structure from a digital image. Examples of segmentation problems include the detection of a road from an aerial photograph or the determination of the boundaries of the brain's ventricles from medical imagery. The extraction of structures allows for subsequent higher-level cognitive tasks. One of them is shape comparison. For example, if the brain ventricles of a patient are segmented, can their shapes be used for diagnosis? That is to say, do the shapes of the extracted ventricles resemble more those of healthy patients or those of patients suffering from schizophrenia? This thesis deals with the problem of image segmentation and shape comparison in the mathematical framework of partial differential equations. The contribution of this thesis is threefold: 1. A technique for the segmentation of regions is proposed. A cost functional is defined for regions based on a non-parametric functional of the distribution of image intensities inside the region. This cost is constructed to favor regions that are homogeneous. Regions that are optimal with respect to that cost can be determined with limited user interaction. 2. The use of direction information is introduced for the segmentation of open curves and closed surfaces. A cost functional is defined for structures (curves or surfaces) by integrating a local, direction-dependent pattern detector along the structure. Optimal structures, corresponding to the best match with the pattern detector, can be determined using efficient algorithms. 3. A technique for shape comparison based on the Laplace equation is proposed. Given two surfaces, one-to-one correspondences are determined that allow for the characterization of local and global similarity measures. The local differences among shapes (resulting for example from a segmentation step) can be visualized for qualitative evaluation by a human expert. It can also be used for classifying shapes into, for example, normal and pathological classes.

Habilitationsschrift--Hamburg. / Bibliography: p. 332-338.

Orientadores: Dimitar Kolev Dimitrov, Roberto Andreani / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-23T19:23:17Z (GMT). No. of bitstreams: 1 Yen_ChiLun_D.pdf: 3950162 bytes, checksum: 1812f3dd736abbe2d4ff070c7877fdff (MD5) Previous issue date: 2013 / Resumo: O objetivo deste trabalho é apresentar uma nova formulação do Teorema de comparação de Sturm e suas aplicações na teoria dos zeros de polinômios ortogonais, que são: monotonicidade dos zeros dos polinômios ortogonais X1-Jacobi, desigualdades de Gautschi sobre os zeros dos polinômios ortogonais de Jacobi e o comportamento assintótico dos zeros dos polinômios ultrasféricos / Abstract: In this thesis we state a new formulation of the Sturm comparison Theorem and its applications to the zeros of orthogonal polynomials. Specifically, these applications deal with the monotonicity of zeros of X1-Jacobi orthogonal polynomials, Gautschi's conjectures about inequalities of zeros of Jacobi polynomials and the asymptotic of zeros of ultrasphricals polynomials / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Doctor of Philosophy / Department of Mathematics / Christopher G. Pinner

We consider the numerical solution of discretised Hamilton-Jacobi-Bellman (HJB) equations with applications in finance. For the discrete linear complementarity problem arising in American option pricing, we study a policy iteration method. We show, analytically and numerically, that, in standard situations, the computational cost of this approach is comparable to that of European option pricing. We also characterise the shortcomings of policy iteration, providing a lower bound for the number of steps required when having inaccurate initial data. For discretised HJB equations with a finite control set, we propose a penalty approach. The accuracy of the penalty approximation is of first order in the penalty parameter, and we present a Newton-type iterative solver terminating after finitely many steps with a solution to the penalised equation. For discretised HJB equations and discretised HJB obstacle problems with compact control sets, we also introduce penalty approximations. In both cases, the approximation accuracy is of first order in the penalty parameter. We again design Newton-type methods for the solution of the penalised equations. For the penalised HJB equation, the iterative solver has monotone global convergence. For the penalised HJB obstacle problem, the iterative solver has local quadratic convergence. We carefully benchmark all our numerical schemes against current state-of-the-art techniques, demonstrating competitiveness.

Omniprésente mais jamais réellement thématisée, la notion de Schwärmerei apparaît comme un Leitfaden, un fil directeur dans l’oeuvre kantienne. Confusion, insupportable prétention mais aussi « raison négative » (antiparadigme) la Schwärmerei représente pour le philosophe le danger mortel d’une alternative possible à sa propre rationalité et il va se mettre en devoir de la dénoncer. Malgré cela Kant est fasciné, il est irrésistiblement attiré par le personnage du visionnaire (incarné par Swedenborg) et cette attirance est niée et refoulée : un combat interne à la rationalité kantienne apparaît alors et, partant de la Kritik der schwärmerischen Vernunft, Kant va faire naître le criticisme par la nécessité d’imposer à l’exaltation trop tentante des restrictions et des limites. Il oppose ainsi sa « métaphysique de la modestie » à l’orgueil des « favoris du ciel », limitation nécessaire si l’on ne veut pas voir la dialectique inévitable de la raison dériver vers la Schwärmerei. Puis ce combat contre l’illuminisme va devenir public, à l’occasion de la querelle du panthéisme et apparaître comme un affrontement entre Kant et Jacobi. Cette confrontation de deux « idiosyncrasies philosophiques » permet, in fine, d’examiner deux conceptions très différentes de la foi, du savoir et de la raison faisant ainsi du dialogue entre la Schwärmerei et la raison un thème fondamental qui appartient concomitamment au système de la raisn kantienne mais aussi (avec le Pantheismusstreit et les débuts de l’idéalisme allemand) à l’histoire de la raison elle-même / Omnipresent but never really conceptualized, the idea of Schwärmerei appears like a Leitfaden, a guiding principle, in Kant’s works. Confusion, unbearable conceit but also “negative reason”(antiparadigm), the Schwärmerei is, in Kant’s eyes, the serious danger of a possible alternative to his peculiar rationality and he sets about denouncing it. Nevertheless, Kant is fascinated and irresistibly attracted by the Schwärmer’s figure (represented by Swedenborg) and this attraction is denied and repressed : an internal struggle in Kant’s rationality then becomes apparent and, starting from the Kritik of schwärmerischen Vernunft, Kant gives birth to criticism, by need to limit the elation, too tempting. He then divides his “metaphysics of modesty” and the pride of the “heaven’s favourites”, necessary restriction to avoid the drift of the inevitable dialectic of the reason toward the Schwärmerei. Then, this struggle against illuminism becomes public on the occasion of the “Pantheismusstreit” and appears to be a confrontation between Kant and Jacobi. This clash of two “philosophical idiosyncrasies” gives, after all, rise to examine two very different conceptions of faith, knowledge and reason. It then transforms the dialogue between Schwärmerei and reason into a fundamental topic that belongs to the system of kantian reason and to the story of reason too, with Pantheismusstreit and the beginning of the“German idealism”

Orientador: Bruto Max Pimentel Escobar / Co-orientador: / Banca:Andrey Yuryevich Mikhaylov / Banca: Edmundo Capelas de Oliveira / Resumo: A teoria de Hamilton-Jacobi geralmente é apresentada como uma extensão da teoria de Hamilton através das transformações canônicas. No entanto, o matemático Constantin Carathéodory mostrou que essa teoria, sua existência e validade, independem do formalismo hamiltoniano. Neste trabalho, apresentaremos a abordagem de Carathéodory para a teoria de Hamilton-Jacobi. Partindo desse procedimento, construiremos uma teoria de vínculos para que se possa resolver problemas com vínculos involutivos e não-involutivos. Para isso, analisaremos a integrabilidade das equações e introduziremos a operação dos parênteses generalizados que, no lugar do parênteses de Poisson, passará a descrever a dinâmica de sistemas vinculados. Mostraremos uma aplicação dessa teoria de vínculos no modelo BF da teoria de campos. Para finalizar, trataremos da Termodinâmica Axiomática de Carathéodory e também da teoria de Hamilton-Jacobi na Termodinâmica, o que é válido para ilustrar a grande abrangência desse formalismo / Abstract: The Hamilton-Jacobi theory is usually presented as an extension of the Hamilton's theory through the canonical transformations. However, the mathematician Constantin Carathéodory showed this theory, its existence and validity, is independent of the Hamiltonian formalism. In this work, we present the Caratheodory's approach to the Hamilton-Jacobi theory. From this procedure, we build a theory of constraints which can solve problems with involutive and non-involutive constraints. For this, we analyze the integrability of the equations and introduce the operation of the generalized brackets that, instead of Poisson brackets, will describe the dynamics of constrained systems. We show an application of this theory in BF model of the field theory. Finally, we will discuss the Carathéodory's Axiomatic Thermodynamics and also show the Hamilton-Jacobi theory in Thermodynamics, which is valid to illustrate the wide coverage of this formalism / Mestre

Orientador: Geraldo Nunes Silva / Banca: Iguer Luis Domini dos Santos / Banca: Marko Antonio Rojas Medar / Resumo: Neste trabalho caracterizaremos sistemas dinâmicos na forma dos chamados domínios estratificados. Bressan e Hong[9] foram os primeiros a definir e trabalhar em domínios estratificados. Grosso modo, estes são uma coleção de domínios disjuntos, cada um tendo sua própria dinâmica; mas não se exige que seus domínios sejam proximamente suaves e nem wedged. Estes termos foram introduzidos por P. Wolenski e R. Barnard em[10]. Primeiramente, estabeleceremos condições Hamiltonianos para caracterizar invariância fraca e forte para sistemas não Lipschitz em domínios estratificados. Depois, estudamos condições Hamiltonianas para sistemas fracamente e fortemente decrescentes e apresentamos condições que garantem a estabilidade assintótica global para uma dinâmica estratificada e finalmente apresentamos o problema tipo Mayer em domínios estratificados em que mostramos que a função valor e a única solução semicontínua inferior de uma generalização adequada da clássica equação Hamilton-Jacobi-Bellman, para a dinâmica estratificada / Abstract: In this work will characterize dynamical systems in the form of the so-called strati ed domain. Bressan and Hong [9] were the rst to de ne and work in strati ed domains. Roughly speaking, these are a collection of disjoint domains, each having its own dynamics; but not requiring that their domains are proximally smooth and not wedged. These terms were introduced by P. Wolenski and R. Barnard in [10]. At rst, we will establish Hamiltonian conditions to characterize weak and strong invariance for systems non-Lipschitz in strati ed domains. Secondly, we study the Hamiltonian conditions for systems weakly and strongly de- creasing and present conditions that guarantee global asymptotic stability for a strati ed dynamics and nally we present the problem Mayer type in strati ed domains where we show that the value function is the unique lower semicontinuous solution of an appropriate generalization of the classical Hamilton-Jacobi-Bellman equation for strati ed dynamics / Mestre

Cette thèse comprend deux parties indépendantes. Dans la première partie, nous développons une approche fondée sur la théorie des formes de Jacobi dont l'indice est un réseau pour classifier les formes modulaires réflexives sur des réseaux de niveau arbitraire. Les formes modulaires réflexives ont des applications en géométrie algébrique, en algèbre de Lie et en arithmétique. La classification des formes modulaires réflexives est un problème ouvert et a été étudiée par Borcherds, Gritsenko, Nikulin, Scheithauer et Ma depuis 1998. Dans cette partie, nous établissons de nouvelles conditions nécessaires à l'existence d'une forme modulaire réflexive. Nous prouvons la non-existence de formes modulaires réflexives et de formes modulaires 2-réflexives sur des réseaux de grand rang. Nous donnons également une classification complète des formes modulaires 2-réflexives sur des réseaux contenant deux plans hyperboliques.La deuxième partie est consacrée à l’étude des formes de Jacobi de $W(E_8)$-invariantes. Ce type de formes de Jacobi a une signification dans les variétés de Frobenius, la théorie de Gromov-Witten et la théorie des cordes. En 1992, Wirthm\"{u}ller a prouvé que l’espace des formes de Jacobi pour tout système de racines irréductible excepté $E_8$ est une algèbre polynomiale. Très peu de choses sont connues dans le cas de $E_8$. Dans cette partie, nous montrons que l'anneau bigradué des formes de Jacobi $W(E_8)$-invariantes n'est pas une algèbre polynomiale et prouvons que chacune de ces formes de Jacobi peut être exprimée uniquement sous la forme d'un polynôme en neuf formes de Jacobi algébriquement indépendantes introduites par Sakai avec des coefficients méromorphes $\SL_2(\ZZ)$-modulaires. Ce dernier résultat implique que, à indice fixé, l’espace des formes de Jacobi $W(E_8)$-invariantes est un module libre sur l’anneau des formes $\SL_2(\ZZ)$-modulaires et que le nombre de générateurs peut être calculé via une série génératrice. Nous déterminons et construisons tous les générateurs pour des indices petits. Ces résultats étendent un théorème de type de Chevalley au cas du réseau $E_8$. / This thesis consists of two independent parts. In the first part we develop an approach based on the theory of Jacobi forms of lattice index to classify reflective modular forms on lattices of arbitrary level. Reflective modular forms have applications in algebraic geometry, Lie algebra and arithmetic. The classification of reflective modular forms is an open problem and has been investigated by Borcherds, Gritsenko, Nikulin, Scheithauer and Ma since 1998. In this part, we establish new necessary conditions for the existence of a reflective modular form. We prove non-existence of reflective modular forms and 2-reflective modular forms on lattices of large rank. We also give a complete classification of 2-reflective modular forms on lattices containing two hyperbolic planes. The second part is devoted to the study of Weyl invariant $E_8$ Jacobi forms. This type of Jacobi forms has significance in Frobenius manifolds, Gromov--Witten theory and string theory. In 1992, Wirthm\"{u}ller proved that the space of Jacobi forms for any irreducible root system not of type $E_8$ is a polynomial algebra. But very little has been known about the case of $E_8$. In this paper we show that the bigraded ring of Weyl invariant $E_8$ Jacobi forms is not a polynomial algebra and prove that every such Jacobi form can be expressed uniquely as a polynomial in nine algebraically independent Jacobi forms introduced by Sakai with coefficients which are meromorphic $\SL_2(\ZZ)$ modular forms. The latter result implies that the space of Weyl invariant $E_8$ Jacobi forms of fixed index is a free module over the ring of $\SL_2(\ZZ)$ modular forms and that the number of generators can be calculated by a generating series. We determine and construct all generators of small index. These results give a proper extension of the Chevalley type theorem to the case of $E_8$.