Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Nesta Tese, apresentamos a mecÃnica quÃntica nÃo-aditiva (MQNA), uma teoria desenvolvida a partir de primeiros princÃpios com o intuito de entender quais sÃo os efeitos da mÃtrica do espaÃo na teoria quÃntica. Em espaÃos nÃo-euclideanos, uma translaÃÃo de comprimento ∆x nÃo leva necessariamente uma partÃcula de uma posiÃÃo x para outra x + ∆x. O resultado dessa translaÃÃo depende da mÃtrica. Esse à o ponto de partida para o desenvolvimento da MQNA. AtravÃs de uma redefiniÃÃo do operador translaÃÃo, obtivemos novas relaÃÃes de comutaÃÃo entre os operadores posiÃÃo e momentum e uma equaÃÃo tipo equaÃÃo de SchrÃdinger que descreve a evoluÃÃo temporal do estado da partÃcula. Mostramos que essa equaÃÃo, juntamente com certas condiÃÃes de contorno, pode ser vista como um problema de Sturm-Liouville, garantindo que as energias da partÃcula sÃo reais e que os autoestados da hamiltoniana sÃo ortonormais e formam uma base no espaÃo dos estados. Apesar dessas modificaÃÃes, mostramos que continuam vÃlidos o determinismo na evoluÃÃo temporal, o princÃpio da superposiÃÃo e a conservaÃÃo local e global da probabilidade. Em contrapartida, generalizamos o teorema de Ehrenfest, mostrando que, para os valores mÃdios das grandezas fÃsicas, a MQNA cai na mecÃnica clÃssica em um referencial nÃo inercial, e demonstramos a existÃncia de uma incerteza mÃnima diferente de zero no momentum. AlÃm disso, investigamos, tanto classicamente como quanticamente, os efeitos dinÃmicos da mÃtrica na evoluÃÃo temporal de uma partÃcula livre. Para realizar a simulaÃÃo quÃntica tivemos que adaptar a tÃcnica split operator para resolver numericamente a nova equaÃÃo de SchrÃdinger. Por fim, exploramos a possibilidade de mapearmos diversos problemas fÃsicos de naturezas distintas atravÃs do surgimento de um potencial efetivo, consequÃncia de uma simples mudanÃa de coordenadas. / In this thesis, we study the nonadditive quantum mechanics (NAQM), which is a theory developed from first principles in order to understand the effects of the space metric in the quantum theory. In non-Euclidean spaces, the translation of length ∆x does not necessarily take a particle from the position x to x + ∆x. The result of this translation depends on the metric. This is the starting point for the development of the NAQM. Through a redefinition of the translation operator, we obtain new commutation relations between the position operator and the momentum operator, and a SchrÃdinger-like equation which describes the time evolution of the state of a particle. We show that this equation, with appropriate boundary conditions, can be seen as a Sturm-Liouville problem, ensuring that the energies of the particle are real and that the eigenstates of the hamiltonian are orthonormal and form a basis in the space of the states. In spite of these modifications, we show the determinism in the time evolution, the superposition principle and the local and global probability conservation remain valid. On the other hand, we generalize the Ehrenfest theorem, showing that, for the average values of the physical quantities, the NAQM is identical to the classical mechanics in a non-inertial reference frame, and we demonstrate the existence of a nonzero minimum uncertainty for the momentum. Besides, we investigate, classically as well as quantically, the dynamical effects of the metric in the time evolution of a free particle. In order to perform the quantum simulation, we adapt the split operator technique to solve numerically the new SchrÃdinger equation. Lastly, we explore the possibility of mapping of several physical problems of different nature through the arising of an effective potential which appears due to a simple change of coordinates.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / O propÃsito da dissertaÃÃo à apresentar um pouco da Teoria dos NÃmeros Transcendentes, em especial, explicitar exemplos de nÃmeros transcendentesusando alguns resultados desta teoria. Este trabalho tenta aparecer como um pequeno survey" em Teoria Transcendente, e nele figuram alguns dos principais resultados dessa teoria. / The purpose of the dissertation is to present a little of the theory of transcendent numbers,in particular, explicit examples of transcendental numbers some results using this theory. This paper attempts to appear as a little "survey" in the transcendental theory, and it included some of main results of this theory.

Estudamos a resolubilidade global de uma classe de sistemas involutivos com n campos vetoriais suaves definidos no toro de dimensão n + 1. Obtemos uma caracterização completa para o caso desacoplado desta classe em termos de formas de Liouville e da conexidade de todos os subníveis e superníveis, no espaço de recobrimento minimal, de uma primitiva global da 1-forma associada ao sistema. Além disso, apresentamos uma situação especial na qual o sistema não é globalmente resolúvel e usamos isso para obter alguns resultados em um caso com acoplamento mais forte / We study the global solvability of a class of involutive systems with n smooth vector fields on the torus of dimension n + 1. We obtain a complete characterization for the uncoupled case of this class in terms of Liouville forms and of the connectedness of all sublevel and superlevel sets of the primitive of a certain 1-form in the minimal covering space. Also, we exhibit a special situation where the system is not globally solvable and we use this to obtain some results in a more general case

En este trabajo de memoria se demostró un teorema de existencia para la ecuación de Liouville con condición de borde no lineal: El primer paso en esta demostración consiste en la aproximación del problema original usando un ansatz de la solución que explota en m puntos cuando el parámetro épsilon tiende a cero, más un término de corrección, sobre el cual se obtienen un conjunto de ecuaciones que van a caracterizar la solución del problema principal. En el capítulo 4 se analizó el operador lineal asociado a estas ecuaciones y se encontró un resultado de solubilidad al modificar la ecuación con términos aditivos de coeficientes cj, j = 1, . . . , m. A continuación se estableció la existencia de una solución al problema no lineal con la modificación aditiva y se estudió su comportamiento en función de los puntos singulares. Se demostró que la solución del problema principal, dada por el hecho de encontrar un conjunto de puntos tales que cj = 0, ∀ j, puede ser reducida al análisis de los puntos críticos de una función φm. En el capítulo final se mostró que existen al menos dos de estos puntos críticos y en consecuencia al menos dos soluciones del problema principal que explotan en m puntos.

Orientador: Yuri Dimitrov Bozhkov / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T01:04:06Z (GMT). No. of bitstreams: 1 SilvaJunior_ValterAparecido_M.pdf: 1356319 bytes, checksum: e64224c844e48a46487c6d387ec0f3e7 (MD5) Previous issue date: 2015 / Resumo: Nesta dissertação, obteremos as soluções exatas do Problema de Liouville-Gelfand (em uma e em duas dimensões) via grupos de Lie de simetrias / Abstract: In this dissertation, we shall obtain the exact solutions of the Liouville-Gelfand Problem (in one and in two dimensions) via Lie groups of symmetries / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

In this work, we make a brief review of the Conformal Field Theory in two dimensions,in order to understand some basic definitions in the study of the Liouville Field Theory, which has many application in theoretical physics like string theory, general relativity and supersymmetric gauge field theories. In particular, we focus on the analytic continuation of the Liouville Field Theory, context in which an interesting relation with the Chern-Simons Theory arises as an extension of its well-known relation with the Wess-Zumino-Witten model. Thus, calculating correlation functions by using the complex solutions of the Liouville Theory will be crucial aim in this work in order to test the consistency of this analytic continuation. We will consider as an application the time-like version of the Liouville Theory, which has several applications in holographic quantum cosmology and in studying tachyon condensates. Finally, we calculate the three-point function for the Wess-Zumino-Witten model for the standard Kac-Moody level k > 2 and the particular case 0 < k < 2, the latter has an interpretation in time-dependent scenarios for string theory. Here we will find an analogue relation we find by comparing the correlation function of the time-like and space-like Liouville Field Theory. / Neste trabalho, nós fazemos uma breve revisão da Teoria de Campo Conforme em duas dimensões, a fim de entender algumas denições básicas do estudo da Teoria de Campo de Liouville, que tem muitas aplicações em física teórica como a teoria das cordas, a relatividade geral e teorias de campo de calibre supersimétricas. Em particular, vamos nos concentrar sobre a continuação analítica da Teoria de Campo de Liouville, contexto no qual uma interessante relação com a Teoria de Chern-Simons surge como uma extensão de sua relação conhecida com o modelo de Wess-Zumino-Witten. Assim, o cálculo das funções de correlação usando as soluções complexas da Teoria Liouville será o objectivo fundamental neste trabalho, a fim de testar a consistência da continuação analítica. Vamos considerar como uma aplicação a versão time-like da Teoria de Liouville, que tem várias aplicações em cosmologia quântica holográfica e no estudo de condensados de tachyon. Finalmente, calculamos a função de três pontos para o modelo de Wess-Zumino-Witten no nível de Kac-Moody k > 2 e o caso particular 0 < k < 2, este último tem uma interpretação em cenários dependentes do tempo para a teoria das cordas. Aqui nós vamos encontrar uma relação análoga ao que temos para a função de correlação do space-like e time-like na Teoria de Campo de Liouville.

In this thesis, we study the inverse nodal problem and inverse spectral problem for various Sturm-Liouville operators, in particular, Hill's operators. We first show that the space of Schr"odinger operators under separated boundary conditions characterized by ${H=(q,al, e)in L^{1}(0,1) imes [0,pi)^{2} : int_{0}^{1}q=0}$ is homeomorphic to the partition set of the space of all admissible sequences $X={X_{k}^{(n)}}$ which form sequences that converge to $q, al$ and $ e$ individually. The definition of $Gamma$, the space of quasinodal sequences, relies on the $L^{1}$ convergence of the reconstruction formula for $q$ by the exactly nodal sequence. Then we study the inverse nodal problem for Hill's equation, and solve the uniqueness, reconstruction and stability problem. We do this by making a translation of Hill's equation and turning it into a Dirichlet Schr"odinger problem. Then the estimates of corresponding nodal length and eigenvalues can be deduced. Furthermore, the reconstruction formula of the potential function and the uniqueness can be shown. We also show the quotient space $Lambda/sim$ is homeomorphic to the space $Omega={qin L^{1}(0,1) : int_{0}^{1}q = 0, q(x)=q(x+1) mbox{on} mathbb{R}}$. Here the space $Lambda$ is a collection of all admissible sequences $X={X_{k}^{(n)}}$ which form sequences that converge to $q$. Finally we show that if the periodic potential function $q$ of Hill's equation is single-well on $[0,1]$, then $q$ is constant if and only if the first instability interval is absent. The same is also valid for convex potentials. Then we show that similar statements are valid for single-barrier and concave density functions for periodic string equation. Our result extends that of M. J. Huang and supplements the works of Borg and Hochstadt.

In this thesis, we study the Pokornyi's model of a Sturm-Liouville operator defined on graphs. The model, proposed by Pokornyi and Pryadiev in 2004, is derived from the consideration of minimal energy of a system of interlocking springs oscillating in a medium with resistance. Here the system of springs is defined as a graph $Gamma$ with edges $R(Gamma)={gamma_i:i=1,dots,n}$ and set of internal vertices $J(Gamma)$. Let $partialGamma$ denote the set of boundary vertices of $Gamma$. For each vertex ${f v}in J(Gamma)$, we let $Gamma({f v})={gamma_iin R(Gamma):~{f v}$ is an endpoint of $ gamma_i}$. The related eigenvalue problem of the model is as follows: egin{eqnarray*} -(p_iy_i')'+q_iy_i&=&lambda y_i,~~~~~qquad mbox{on}~gamma_i, y_i({f v})&=&y_j({f v}),~~~~~~~~forall {f v}in J(Gamma)~ mbox{and}~gamma_i,gamma_jin Gamma({f v}), sum_{gamma_iin Gamma({f v})}p_i({f v})frac{dy({f v})}{dgamma_i}+q({f v})y({f v})&=&lambda y({f v}),qquad ~~forall {f v}in J(Gamma), end{eqnarray*} equipped with Neumann or Dirichlet boundary conditions. This model is also a special case of some quantum graphs defined by Kuchment . par We shall derive the model and discuss the spectral properties. We shall also solve several Ambarzumyan problems on the model. In particular, we show that for a $n$-star shaped graph of uniform length $a$ with $p_iequiv1$, if ${frac{(m+frac{1}{2})^2)pi^2}{a^2}:min Ncup{0}}$ are Neumann eigenvalues, $0$ is the least Neumann eigenvalue, and $q_i({f v})=0$ for ${f v}in J(Gamma)$, then $q=0$ on $Gamma$.

In this thesis, we study a Bäcklund transformation for minimal surfaces - surfaces with vanishing mean curvature - transforming a given minimal surface into a possible infinity of new ones. The transformation, also carrying with it mappings between solutions to the elliptic Liouville equation, is first derived by using geometrical concepts, and then by using algebraic methods alone - the latter we have not been able to find elsewhere. We end by exploiting the transformation in an example, transforming the catenoid into a family of new minimal surfaces.

This dissertation focuses on a novel approach for characterizing the mechanical behavior of an elastic body. In particular, we develop a mathematical tool for the estimation of residual stress field in an elastic body that has mechanical properties similar to that of the arterial wall, by making use of intravascular ultrasound (IVUS) imaging techniques. This study is a preliminary step towards understanding the progression of a cardiovascular disease called atherosclerosis using ultrasound technology. It is known that residual stresses play a significant role in determining the overall stress distribution in soft tissues. The main part of this work deals with developing a nonlinear inverse spectral technique that allows one to accurately compute the residual stresses in soft tissues. Unlike most conventional experimental, both in vivo and in vitro, and theoretical techniques to characterize residual stresses in soft tissues, the proposed method makes fundamental use of the finite strain non- linear response of the material to a quasi-static harmonic loading. The arterial wall is modeled as a nonlinear, isotropic, slightly compressible elastic body. A boundary value problem is formulated for the residually stressed arterial wall, the boundary of which is subjected to a constant blood pressure, and then an idealized model for the IVUS interrogation is constructed by superimposing small amplitude time harmonic infinitesimal vibrations on large deformations via an asymptotic construction of its solution. We then use a semi-inverse approach to study the model for a specific class of deformations. The analysis leads us to a system of second order differential equations with homogeneous boundary conditions of Sturm-Liouville type. By making use of the classical theory of inverse Sturm-Liouville problems, and root finding and optimization techniques, we then develop several inverse spectral algorithms to approximate the residual stress distribution in the arterial wall, given the first few eigenfrequencies of several induced blood pressures.