Thesis (M.A.)--University of Wisconsin--Madison, 1967. / eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (M.S.)--Montana State University--Bozeman, 2006. / Typescript. Chairperson, Graduate Committee: William Wyckoff. Includes bibliographical references (leaves 235-248).

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

Thesis (doctoral)--University of Leipzig. / Cover title. Includes bibliographical references (p. 48-49).

Supramolecular chemistry (chemistry beyond the molecule) is the study and synthesis of complex molecular architectures from simple subunits using non-covalent interactions. The types of non-covalent interactions that are used for the self-assembly of these complex molecular architectures include electrostatic interactions (e.g. ionic, halogen, and hydrogen bonding), π-effects, van der Waals interactions, metal coordination, and hydrophobic effects. While these interactions are often used in concert, some of the most successful and ubiquitous approaches for the design and construction of new host–guest architectures are the incorporation of hydrogen bonding motifs. A popular class of molecules capable of making strong, highly directional hydrogen bonds is barbiturates. Barbiturates have a well-known reputation as potent hypnotics, anticonvulsants, and anxiolytics but recent years have seen a renewed interest in these molecules due to their unique, symmetric acceptor-donor-acceptor hydrogen bonding motif. In addition, receptors with complementary hydrogen bonding motifs capable of binding barbiturates have also been reported, namely those based on the work of Hamilton et al. Collectively, barbiturates and their receptors have seen widespread use in a variety of applications including sensing, optoelectronics, catalysis, and the design of soft materials. The work presented in this dissertation describes the development of novel Hamilton receptors for supramolecular catalysis and barbiturate sensing, as well as the design of new synthetic barbiturates. Together this body of research aims to extend the utility of these types of host–guest systems as well as continue to develop and refine the supramolecular design principles that govern the binding interactions between barbiturates and a variety of Hamilton-type receptors. This dissertation includes both previously published/unpublished and co-authored material.

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.

This thesis explores the algebraic and geometric structure of the Legendre transform and its application in various field of mathematics and physics. Specifically linear transformation as a mathematical process and motivating it in terms related to phenomena in mathematics and physics. The Legendre transform provides a change of variables to express equations of the motion or other physical relationships in terms of most convenient dynamical quantities for a given experiment or theoretical analysis. In classical mechanics the Legendre transform generates the Hamiltonian function of a system from the Lagrangian function or vice versa. In thermodynamics the Legendre transform allows thermodynamic relationships to be written in terms of alternative sets of independent variables. Here we review the properties of Legendre transform and why it is so important in mathematics and physics.

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.

El documento digital no refiere asesor / Estudia la existencia y unicidad de la ecuación de Hamilton Jacobi, donde Rn × [0,∞) −→ R, t ∈ R, H : Rn −→ R es una función llamada Hamiltoniano Du = (ux1 , . . . . . . . . . , uxn). Para alcanzar el objetivo planteado, se empleó el cálculo variacional, las ecuaciones de Hamilton, la transformada de Legendre y la fórmula de Hopf Lax. / Trabajo de suficiencia profesional