o loop de Wilson se apresenta como uma variável muito apropriada para servir de base para uma teoria das forças nucleares. Esta afirmação é justificada se lembrarmos que a apresentação usual da cromodinâmica quântica perturbativa parte de variáveis coloridas, um fato altamente criticável, pois elas não representam qualquer variável física, enquanto que na eletrodinâmica as variáveis que descrevem os elétrons e fótons são físicas. Esta possibilidade de estudar as interações entendimento da fortes implica na necessidade de um claro estrutura analítica dos loops de Wilson. O objetivo deste trabalho e usar a técnica da regularização dimensional para discutir a estrutura dos diagramas da expansão perturbativa do loop de Wilson em quarta ordem. A introdução do número de dimensões como variável analítica, isola as divergências sob a forma de polos no plano complexo da dimensão, facilita a discussão da renormalização, além de proporcionar um método invariante de gauge. Com esta técnica renormalizamos a constante de acoplamento das interações fortes até a quarta ordem e mostramos que em quatro dimensões 0 resíduo não depende nem da curva nem do área no qual ele e calculado. Também mostramos que é possível calcular a parte finita do loop de Wilson para 0 caso especial de duas semi-retas de mesma origem. / The Wilson loop shows itself as a very appropriate variable to be used as a base in a theory of nuclear forces. This statment is justified if we remember that the usual presentation of perturbative quantum chromodynamics starts from colored variables, a fact itself subject to critism since they do not represent any physical variable, whereas in the usual quantum eletrodynamics the variables that describes eletrons and photons are physical. This possibility of studying strong interactions implies the need of clear a understanding of the analytical structure of the Wilson loops. The purpose of this dissertation is to use the technique of dimensional regularization to discuss the structure of perturbative expansion diagrams of the Wilson loops up to fourth order. The introduction of the number of dimensions as an analytical variable isolates the divergences in the form of poles in the complex plane of the dimensions, make the renormalization procedure easier and also makes the method gauge invariant. With these techniques we have renormalized the coupling constant of strong interactions up to fourth order and have also show that in four dimension the residue is dependent neither on the curve nor on the arc on which it is calculated. We have also shown that it is possible to calculate the finit part of the Wilson loops for the special. Case of two half lines with the same origin.

A teoria de perturba ca o quiral (ChPT) e aceita, atualmente, como a teoria efetiva da cromodinamica quantica (QCD) para baixas energias. Ela foi colocada na sua versa o moderna com os artigos de Gasser e Leutwyler, na primeira metade da d ecada de 80 e, durante os 25 anos que se seguiram, ocorreu um aumento considera vel da variedade de fenomenos por ela descritos, sempre acompanhando a precisa o cres- cente dos resultados experimentais. Os trabalhos que apresentamos nesta tese de doutorado se inserem neste contexto e envolvem duas partes, ambas relacionadas a` determina c ao de alguns dos parametros que compo em a lagrangiana da ChPT. Por ser uma teoria efetiva, tais constantes s o podem ser fixadas por experimentos, modelos ou por c alculos da QCD na rede. Em um dos trabalhos, discutimos a cons- tante de decaimento do p on, F, e os efeitos decorrentes do acr escimo de intera co es eletromagn eticas a` ChPT. N os argumentamos que as incertezas estimadas para o valor mais aceito de F podem estar subestimadas. Mostramos, tamb em, que na o se pode determinar esta constante na presen ca das intera co es eletromagn eticas, pois a grandeza de onde ela e extra da adquire uma dependencia no calibre utilizado no ca lculo e tem suas propriedades alteradas drasticamente. No outro trabalho, abor- damos os fatores de forma escalares dos m esons pseudoescalares em tres sabores. A partir dos resultados obtidos com a ChPT e do uso de um modelo que trata dos fatores de forma no espa co das posi co es, conseguimos escreve-los em termos ape- nas das constantes presentes na lagrangiana em ordem dominante da ChPT, F e as massas dos m esons pseudoescalares. No s determinamos, tamb em, os respectivos raios quadra ticos m edios e, comparando-os com os calculados com a ChPT, obtive- mos as LECs L4(mu) = -0,26 · 10^-3 e L5(mu) = 0,85 · 10^-3, para mu = 770 MeV. Esses valores sa o compat veis com as principais estimativas vindas da ChPT. / At present, chiral perturbation theory (ChPT) is considered the effective theory of quantum chromodynamics (QCD) at low energies. It was established in its modern version by the papers of Gasser and Leutwyler written in the first half of the 80s. For the last 25 years, there has been considerable increase in the number of phe- nomena described by ChPT, always following the growing precision of experiments. The two works we present in this Ph.D. thesis are related to ChPT and discuss the determination of some of the parameters that appear in the ChPT lagrangian. As ChPT is an effective theory, such constants can only be fixed by experiments, models or calculations in the lattice. In the first presented work, we discuss the pion decay constant, F, and how it is changed by the inclusion of electromagne- tic interactions. We argue that the uncertainty of the most accepted value of F might be underestimated. We also show that we cannot determine this constant in the presence of electromagnetic interactions because the function from which it is extracted acquires a gauge dependence and the functions properties drastically change. In the other work, we deal with pseudoscalar meson scalar form factor in three flavors. We manage to write the form factors only in terms of constants present in ChPT lagrangian at leading order, F and masses of pseudoscalar mesons, using ChPT results and the model that deals with form factors in coordinate space. We also determine the respective square radii and, comparing these to those calculated using ChPT, we have obtained L4(mu) = -0.26 · 10^-3 and L5(mu) = 0.85 · 10^-3, for mu = 770 MeV. These values are compatible with the main estimates evaluated with ChPT.

o loop de Wilson se apresenta como uma variável muito apropriada para servir de base para uma teoria das forças nucleares. Esta afirmação é justificada se lembrarmos que a apresentação usual da cromodinâmica quântica perturbativa parte de variáveis coloridas, um fato altamente criticável, pois elas não representam qualquer variável física, enquanto que na eletrodinâmica as variáveis que descrevem os elétrons e fótons são físicas. Esta possibilidade de estudar as interações entendimento da fortes implica na necessidade de um claro estrutura analítica dos loops de Wilson. O objetivo deste trabalho e usar a técnica da regularização dimensional para discutir a estrutura dos diagramas da expansão perturbativa do loop de Wilson em quarta ordem. A introdução do número de dimensões como variável analítica, isola as divergências sob a forma de polos no plano complexo da dimensão, facilita a discussão da renormalização, além de proporcionar um método invariante de gauge. Com esta técnica renormalizamos a constante de acoplamento das interações fortes até a quarta ordem e mostramos que em quatro dimensões 0 resíduo não depende nem da curva nem do área no qual ele e calculado. Também mostramos que é possível calcular a parte finita do loop de Wilson para 0 caso especial de duas semi-retas de mesma origem. / The Wilson loop shows itself as a very appropriate variable to be used as a base in a theory of nuclear forces. This statment is justified if we remember that the usual presentation of perturbative quantum chromodynamics starts from colored variables, a fact itself subject to critism since they do not represent any physical variable, whereas in the usual quantum eletrodynamics the variables that describes eletrons and photons are physical. This possibility of studying strong interactions implies the need of clear a understanding of the analytical structure of the Wilson loops. The purpose of this dissertation is to use the technique of dimensional regularization to discuss the structure of perturbative expansion diagrams of the Wilson loops up to fourth order. The introduction of the number of dimensions as an analytical variable isolates the divergences in the form of poles in the complex plane of the dimensions, make the renormalization procedure easier and also makes the method gauge invariant. With these techniques we have renormalized the coupling constant of strong interactions up to fourth order and have also show that in four dimension the residue is dependent neither on the curve nor on the arc on which it is calculated. We have also shown that it is possible to calculate the finit part of the Wilson loops for the special. Case of two half lines with the same origin.

The study of cardiac action potentials has many medical applications. Dr. Dennis Noble first used mathematical models to study cardiac action potentials in the 1960s. We begin our study of cardiac action potentials with one form of the Fitzhugh-Nagumo partial differential equation. We use the non-classical method to produce a closed form solution for the decoupled Fitzhugh Nagumo equation. Using voltage recording data of action potentials in a cardiac myocyte and in purkinje fibers, we estimate parameter values for the closed form solution with standard linear and non-linear regression methods. Results are limited, thus leading us to perturb the solution to obtain a better fit. We turn to singular perturbation theory to justify our pole-based approach. Finally, we test our model on independent action potential data sets to evaluate our model and to draw conclusions on how our model can be applied.

La réponse linéaire de la théorie de la fonctionnelle de la densité dépendante du temps est étudiée dans le cadre du formalisme d'ondelettes du code BigDFT, qui permet d'exprimer les fonctions d'onde électroniques sur une grille de simulation dans l'espace réel. L'objectif est de déterminer un spectre d'excitations de référence pour un système et un potentiel d'échange-corrélation donnés.Il apparaît que seule une partie du spectre, concernant les transitions entre orbitales liées, peut être facilement amenée à convergence par rapport aux paramètres d'entrée de BigDFT, que sont l'extension de la grille de simulation et le nombre d'orbitales du continuum qui sont considérées pour le calcul des spectres. L'énergie de la dernière orbitale inoccupée utilisée dans les calculs se révèle d'ailleurs être un paramètre plus important que ce nombre d'orbitales inoccupées. La justification vient de l'étude de la complétude des bases formées par les orbitales de l'état fondamental du système. Tout ceci permet de porter un regard neuf sur les résultats obtenus avec le formalisme à base gaussienne, tel qu'implémenté dans le code NWChem.En ce qui concerne la convergence du spectre de plus haute énergie, concernant des transitions entre orbitales occupées et orbitales inoccupées du continuum, l'espoir d'une convergence se heurte au problème du tassement du continuum. Il faut alors songer à une manière différente de capter l'information contenue dans ce continuum.Le formalisme des états résonants, dont les fondements ont été posés lors de la première moitié du XXème siècle, est une piste très encourageante pour cela. Une étude préliminaire dans le cas du puits de potentiel carré à une dimension est donc présentée. La première étape a consisté en la détermination de ces états résonants, dont les énergies et fonctions d'onde sont complexes. Une normalisation a notamment pu leur être attribuée. Il est ensuite montré, sous certaines conditions, que la base formée par les états propres de ce potentiel, dont une partie est constituée par les états du continuum, peut être efficacement remplacée par une base discrète et complète faite d'états résonants. Des applications numériques montreront qu'ils peuvent être avantageusement utilisés pour définir la fonction de Green ou encore calculer la propagation temporelle d'un paquet d'onde. / The linear response on the time-dependent density functional theory is studied in the wavelets formalism used in the BigDFT code, that allows the representations of electronic wave-functions on a simulation grid in real space. The goal of this study is to determine a reference excitation spectrum for a given system and exchange-correlation potential.It appears that only one part of the spectrum can be easily brought to convergence with respect to the input parameters of BigDFT, which are the simulation grid extension and the number of unoccupied continuum orbitals considered in the spectrum calculation. The energy of the last unoccupied orbital used actually proves to be more important as a parameter than this number of unoccupied orbitals. This is justified by the study of the completeness of the basis sets made of the ground state orbitals of the system. This gives another point of view regarding spectrum obtained by using the Gausian basis sets formalism, as the one implemented in the code NWChem.As to the convergence of the spectrum at higher energy, concerning transitions between occupied orbitals and unoccupied orbitals of the continuum, the hope for a convergence faces the problem of the continuum collapse. One therefore has to think of another way of retrieving the data contained in this continuum.The resonant states formalism, whose foundations were laid in the first half of the 20th century, is very encouraging in this regard. A preliminary study in the case of the one-dimension square well potential is therefore presented. The first step consisted in the determination of these resonant states, whose energies and wavefunctions are complex valued in general. Their normalization was also clearly defined. It is then shown, under certain conditions, that the basis set formed by the eigenstates of this potential, including the continuum states, can be efficiently replaced by a discrete and complete basis set made of resonant states. Numerical applications also show that these states can also be advantageously used to define the Green's function or even compute the time propagation of a wavepacket.

The purpose of this paper is to extend, as much as possible, the modern theory of condition numbers for conic convex optimization: z* := minz ctx s.t. Ax - b Cy C Cx , to the more general non-conic format: z* := minx ctx (GPd) s.t. Ax-b E Cy X P, where P is any closed convex set, not necessarily a cone, which we call the groundset. Although any convex problem can be transformed to conic form, such transformations are neither unique nor natural given the natural description of many problems, thereby diminishing the relevance of data-based condition number theory. Herein we extend the modern theory of condition numbers to the problem format (GPd). As a byproduct, we are able to state and prove natural extensions of many theorems from the conic-based theory of condition numbers to this broader problem format.

In 2008 a large statistics sample of approximately 1·107 η-decays has been collected with the WASA detector at COSY using the pd→3Heη reaction at the beam kinetic energy of 1 GeV. These data are being used to study the not so rare η decays involving charged pions, like η→π+π-π0. This decay proceeds mainly via a strong isospin violating contribution, where the decay width is proportional to the light quark mass difference squared, (md-mu)2. In addition this decay can be used to search for C-violating effects. The analysis is presented and the Dalitz plot parameters with statistical and systematical uncertainties are determined from a sample of 1.33·105 η→π+π-π0 events in the Dalitz plot. The asymmetry parameters with statistical uncertainties are presented which show no evidence of C-violation. / WASA-at-COSY

The study of noncovalent interactions between aromatic rings and various functional groups is a very popular topic in current computational chemistry. The research presented in this thesis takes steps to bridge the gap between theoretical prototypes and real-world systems. The non-additive contributions to the interaction energy in stacked aromatic systems are measured by expanding the prototype benzene dimer into trimeric and tetrameric systems. We show that the three- and four-body interaction terms generally do not contribute significantly to the overall interaction energy, and that the two-body terms are essentially the same as in the isolated dimer. The sulfur-pi interaction is then studied by using the hydrogen sufide-benzene dimer as a prototype system for theoretical predictions. We obtain higly-accurate potential energy curves, as well as an interaction energy extrapolated to the complete basis set limit. Energy decomposition analysis using symmetry-adapted perturbation theory shows that the sulfur-pi interaction is primarily electrostatic in nature. These theoretical results are then compared to an analysis of real sulfur-pi contacts found by searching protein structures in the Brookhaven Protein DataBank. We find that the most frequently seen configuration does not correspond to the theoretically predicted equilibrium for sydrogen sulfide-benzene, but instead to a configuration that suggests an alkyl-pi interaction involving the carbon adjacent to the sulfur atom. We believe our findings indicate that environmental effects within proteins are altering the energetics of the sulfur-pi interaction so that other functional groups are preferred for interacting with the aromatic ring.

Large-scale structures we see in the universe, such as galaxies, galaxy clusters and structures beyond the scale of clusters, result from gravitational instability of almost isotropic and homogeneous density distribution in the early universe. The degree of the initial anisotropy of the universe and the subsequent growth of gravitational instability, coupled with the expansion rate of the universe, determine the scale and abundance of the structures formed in the universe at later times. A galaxy survey directly observes a distribution of structures in the sky using galaxies as a tracer of the underlying density distribution, and yields constraints on cosmological models when compared to a physical theory of structure formation based on a given cosmological model. Among many cosmological and astronomical phenomena to be understood from a galaxy survey, the nature of the observed accelerated expansion of the universe is the most profound problem in the modern physics. Motivated by various planned and on-going galaxy surveys, including our own Hobby-Ebery Telescope Dark Energy eXperiment (HETDEX), we show the way to fully exploit the data from a galaxy survey. We improve a model of structure formation to include the effect of baryonic pressure and the free-streaming of massive neutrinos at a mildly non-linear regime. Future galaxy surveys are to reach the level of accuracy, where the effect of massive neutrinos on the observed power spectrum is no longer negligible. Proper understanding of these effects gives a way to measure the absolute masses of neutrinos: one of the most fundamental particles, which, by itself, will be a major development in the field of particle physics. Yet, most of the space (~80%) observed by galaxy surveys is occupied by voids. An ellipticity probability distribution function of voids offers yet another way of probing cosmology. Especially, a distribution of ellipticities in the redshift space provides a unique way to measure a growth rate of the structure in the universe apart from other cosmological parameters when combined with the galaxy power spectrum. / text

The main goal of this thesis is to present cosmological perturbation theory (based on the standard Friedmann cosmological model) in volume-preserving coordinates, which then provides a suitable basis for studies in cosmological averaging. We review perturbation theory to second order, allowing for averaging to second order in future research. To solve the averaging problem we need a method of covariantly and gauge invariantly averaging tensorial objects on a background manifold. This is a very difficult problem. However, the definition of an average takes on a particularly simple form when written in a system of volume-preserving coordinates. Therefore, we develop a three dimensional and a four dimensional volume-preserving coordinate gauge in this thesis that can be used for averaging in cosmological perturbation theory.