Thesis (Ph. D.)--University of Missouri-Columbia, 2005. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (May 23, 2006) Vita. Includes bibliographical references.

Techn. University, Diss., 2004--Kaiserslautern.

Thesis (Ph.D.)--Baylor University, 2009. / Subscript in abstract: n and n=0 in {Pn([alpha],[beta])(x)} [infinity] n=0, [mu] in (f,g)[mu], and R in [integral]Rfgd[mu]. Superscript in abstract: ([alpha],[beta]) and [infinity] in {Pn([alpha],[beta])(x)} [infinity] n=0. Includes bibliographical references (p. 115-119).

Cette thèse est consacrée à l’étude d’équation aux dérivée partielles dy type Hamilton- Jacobi ∂tu - Δu + |∇u|q = 0, in Ω × (0,T), (1) où Ω est un ouvert borné regulier dans ℝN contenant le point 0, ou Ω = ℝN; et q > 0. De plus, nous considérons l’équation parabolique avec un terme singulier {ut - Δu + χ{u>0}u-β = 0; in Ω × (0,T), u = 0, on ∂Ω × (0,T), u(0) = u0, (2) où Ω est un ouvert borné regulier dans ℝN, β ∈ (0,1), χw(x) = { 1, if x ∈ w, 0, if x ∉ w. , et u0 ∈ L1(Ω). Pour l’équation (1), nous étudions les solutions nonnégative avec une donnée initiale mesure de Radon bornée dans Ω, ou mesure de Borel dans Ω. En particulier, nous considérons l’existence de solution très singulière en (x,t) = (0,0) (voir [33]). Nous montrons qu’il existe une solution très singulière unique quand 1 < q < N+2/N+1. Par contre, on prouve la nonexistence d’une solution très singulière dans le cas q ≥ N+2/N+1. Ceci mène à un résultat de singularité éliminable pour solutions singulières qui satisfont la condition en bas u(x,0) = 0, in Ω\{0}, Les résultats ci-dessus nous permettent d’aller plus loin pour étudier le problème de trace initiale (voir chapitre 3). Nous montrons que chaque solution nonnégative faible admet une trace initiale u(0) = (S,µ) comme q > 1, où S est un compact dans Ω, et µ est une mesure nonnegative de Radon dans R = Ω\S. De plus, la condition initiale est compris ensuite lim t→0 ∫R u(x,t)v(x)dx = ∫R v(x)dµ(x), v ∈ Cc(R). lim t→0 ∫w u(x,t)dx = ∞, pour chaque x0 ∈ S, et pour chaque w une voisinage de x0 dans Ω. Par contre, chaque solution nonnegative faible reçoit une initial trace v ∈ M+(Ω) comme q ∈ (0,1]. Par ailleurs, on s’intéresse aussi la regularité de solution faible. On va démontrer que chaque solution faible est une solution classique comme q ≤ 2. De plus, on est consacré à étudier L∞-estimates pour solution faible. Ce résultat joue un role important de montrer la regularité et aussi prouver l’unicité de solution faible (chapitre 4). Enfin, nous considéron l’existence de solution nonnegative d’équation (2). On va démontrer qu’il existe une solution maximal d’équation (2) telle que cela disparaît après un certain temps T* qui dépend seulement de N et ||u0||L1(Ω). / This thesis is devote to study the viscous Hamilton-Jacobi equation ∂tu - Δu + |∇u|q = 0, in Ω × (0,T), (3) where Ω ⊂ ℝN is a bounded smooth domain containing 0 ∈ ℝN, or Ω = ℝN; and q > 0. Moreover, we also consider the singular problem {ut - Δu + χ{u>0}u-β = 0; in Ω × (0,T), u = 0, on ∂Ω × (0,T), u(0) = u0, (4) where Ω is a bounded domain in ℝN, β ∈ (0,1), χw(x) = { 1, if x ∈ w, 0, if x ∉ w. , and u0 ∈ L1(Ω). Concerning equation (3), we study nonnegative solutions with a given initial data which is the nonnegative Radon measure on Ω, even the regular Borel measure on Ω. Particularly, we study the existence of very singular solution at (x,t) = (0,0) (see in [33]). We prove that there exists a unique very singular solution when q ∈ (1, N+2/N+1). While q ≥ N+2/N+1 we show the nonexistence of very singular solution. This leads to a result of removable singularity for singular solutions which satisfy u(x,0) = 0, in Ω\{0}, Besides, we also consider the existence of initial trace of equation (3) (see chapter 3). We demonstrate that any nonnegative solution u admits an initial trace which is presented by a couple (S,µ) as q > 1; where S is a compact in Ω, and µ is a nonnegative Radon measure on R, the complement of S in Ω. Moreover, the initial condition u(0) is described in the following sense lim t→0 ∫R u(x,t)v(x)dx = ∫R v(x)dµ(x), v ∈ Cc(R). lim t→0 ∫w u(x,t)dx = ∞, for any x0 ∈ S, and for any w neighborhood of x0 in Ω. While q ∈ (0,1], we show that any nonnegative solution of equation (3) receives an initial trace which is the nonnegative Radon measure on Ω. In this case, we observe that the set of singular points S = Ø, and the set of regular points R = Ω.

Orientador: Bruto Max Pimentel Escobar / Resumo: Neste trabalho apresentamos um estudo detalhado do formalismo de Hamilton-Jacobi para sistemas singulares, fazendo sua generalização para sistemas com variáveis dinâmicas pertencentes à álgebra de Berezin. Analisamos, em especial, as condições de integrabilidade e sua relação com as condições de consistência no formalismo Hamiltoniano de Dirac. Por fim, estudamos o processo de quantização relacionado a esse formalismo, usualmente interpretado como uma quantificação relacionado a esse formalismo, usualmente interpretado como uma quantificação a "gauge livre", e os cuidados que devemos ter com esta interpretação. / Abstract: In this work we present a detailed study of the Hamilton-Jacobi formalism for singular systems, making its generalization for systems containing dynamical variables which belongs to the Berezin algebra. We analyse, with particular care, the integrability conditions and its relation with consistency conditions in Dirac's Hamiltonian formalism. Finally, we study the quantization process related to this formalism, usually interpreted as a "gauge free" quantization, and comment the necessity of caution with this interpretation. / Doutor

Dissertação (mestrado)—Universidade de Brasília, Instituto de Física, 2013. / Submitted by Alaíde Gonçalves dos Santos (alaide@unb.br) on 2014-02-20T12:27:50Z No. of bitstreams: 1 2013_GesielGomesSilva.pdf: 1349422 bytes, checksum: cb5bcd352224968d7157b23bf86e233e (MD5) / Approved for entry into archive by Guimaraes Jacqueline(jacqueline.guimaraes@bce.unb.br) on 2014-02-21T15:29:39Z (GMT) No. of bitstreams: 1 2013_GesielGomesSilva.pdf: 1349422 bytes, checksum: cb5bcd352224968d7157b23bf86e233e (MD5) / Made available in DSpace on 2014-02-21T15:29:39Z (GMT). No. of bitstreams: 1 2013_GesielGomesSilva.pdf: 1349422 bytes, checksum: cb5bcd352224968d7157b23bf86e233e (MD5) / Nosso trabalho consistiu em encontrar os níveis de energia do átomo de hidrogênio sob a ação de um campo magnético externo constante. Utilizamos o formalismo de Hamilton-Jacobi relativístico para introduzir o campo magnético e para obter uma equação para o átomo de hidrogênio sob a ação de um campo magnético uniforme. Propusemos também uma função, com base em uma expansão polinomial, como solução da equação obtida a partir do formalismo de Hamilton-Jacobi possibilitando assim a solução numérica do problema. A simetria do nosso sistema muda com a intensidade do campo magnético: a simetria é esférica quando a intensidade do campo aproxima de zero e é cilíndrica quando tende a infinito. Essa função permitiu obter resultados nestes extremos sem a necessidade de alterações na sua forma, bem como, permitiu obter resultados para campos intermediários. Utilizando o método variacional obtivemos um sistema de equações que nos permitiu obter os autovalores de energia. Os resultados obtidos concordam com os encontrados na literatura mostrando que o nosso método, ainda em evolução, é consistente. _______________________________________________________________________________________ ABSTRACT / In this work, we find the energy levels the energy levels of the hydrogen atom submitted to an external constant magnetic field. It was used the relativistic formalism of Hamilton-Jacobi to introduce the magnetic field and to obtain an equation for the hydrogen atom under the action of a uniform magnetic field. A function also was proposed, based on a polynomial expansion, as a solution of the equation obtained from the Hamilton-Jacobi formalism allowing the numerical solution of the problem. The symmetry of the system changes with the intensity of the magnetic field: the symmetry is spherical when the field strength approaches zero and is cylindrical when the field strength tends to infinity. This function allowed results in these extremes without the need of changes in form but has also enabled us to obtain results for other intermediary fields. Using the variational method it was possible to obtain a system of equations that has enabled us to obtain the eigenvalues of energy. The agreement of the results with other findings in the literature demonstrates that the method proposed here, still under development, is consistent with expected values.

Let G be a Lie group acting on a homogeneous space G/K. The center of the universal enveloping algebra of the Lie algebra of G maps homomorphically into the center of the algebra of differential operators on G/K invariant under the action of G. In the case that G is a Jacobi Lie group of rank 2, we prove that this homomorphism is surjective and hence that the center of the invariant differential operator algebra is the image of the center of the universal enveloping algebra. This is an extension of work of Bringmann, Conley, and Richter in the rank 1case.

The theory of viscosity solutions was developed for certain types of nonlinear first-order and second-order partial differential equations. It has been particularly useful in describing the solutions of partial differential equations associated with deterministic and stochastic optimal control problems [16], [53]. In its classical formulation, see [16], the theory deals with solutions which are continuous functions. The concept of continuous viscosity solutions was further generalized in various ways to include discontinuous solutions with the definition of Ishii given in [71] playing a pivotal role. In this thesis we propose a new approach for the treatment of discontinuous solutions of first-order Hamilton-Jacobi equations, namely, by involving Hausdorff continuous interval valued functions. The advantages of the proposed approach are justified by demonstrating that the main ideas within the classical theory of continuous viscosity solutions can be extended almost unchanged to the wider space of Hausdorff continuous functions and the existing theory of discontinuous viscosity solutions is a particular case of that developed in this thesis in terms of Hausdorff continuous interval valued functions. Two approaches to numerical solutions for Hamilton-Jacobi equations are presented. The first one is a monotone scheme for Hamilton-Jacobi equations while the second is based on preserving total variation diminishing property for conservation laws. In the first approach, we couple the finite element method with the nonstandard finite difference method which is based on the Mickens’ rule of nonlocal approximation [9]. The scheme obtained in this way is unconditionally monotone. In the second approach, computationally simple implicit schemes are derived by using nonlocal approximation of nonlinear terms. Renormalization of the denominator of the discrete derivative is used for deriving explicit schemes of first or higher order. Unlike the standard explicit methods, the solutions of these schemes have diminishing total variation for any time step size. / Thesis (PhD (Mathematical Science))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted

No description available.

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.