Thesis (Ph. D.)--University of Texas at Austin, 2009. / Title from PDF title page (University of Texas Digital Repository, viewed on Sept. 9, 2009). Vita. Includes bibliographical references.

Thesis. 1977. M.S.--Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Includes bibliographical references. / M.S.

Los problemas de dispersión de partículas, como son los de dos y tres cuerpos, tienen una relevancia crucial en física atómica, pues permiten describir diversos procesos de colisiones. Hoy en día, los casos de dos cuerpos pueden ser resueltos con el grado de precisión numérica que se desee. Los problemas de dispersión de tres partículas cargadas son notoriamente más difíciles pero aún así algo similar, aunque en menor medida, puede establecerse. El objetivo de este trabajo es contribuir a la comprensión de procesos Coulombianos de dispersión de tres cuerpos desde un punto de vista analítico. Esto no solo es de fundamental interés, sino que también es útil para dominar mejor los enfoques numéricos que se actualmente se desarrollan dentro de la comunidad de colisiones atómicas. Para lograr este objetivo, proponemos aproximar la solución del problema con desarrollos en series de funciones adecuadas y expresables analíticamente. Al hacer esto, desarrollamos una serie de herramientas matemáticas relacionadas con funciones Coulombianas, ecuaciones diferenciales de segundo orden homogéneas y no homogéneas, y funciones hipergeométricas en una y dos variables. En primer lugar, trabajamos con las funciones de onda Coulombianas radiales y revisamos sus principales propiedades. Así, extendemos los resultados conocidos para dar expresiones analíticas de los coeficientes asociados al desarrollo, en serie de funciones de tipo Laguerre, de las funciones Coulombianas irregulares. También establecemos una nueva conexión entre los coeficientes asociados al desarrollo de la función Coulombiana regular y los polinomios de Meixner-Pollaczek. Esta relación nos permite deducir propiedades de ortogonalidad y clausura para estos coeficientes al considerar la carga como variable. Luego, estudiamos las funciones hipergeométricas de dos variables. Para algunas de ellas, como las funciones de Appell o las confluentes de Horn, presentamos expresiones analíticas de sus derivadas respecto de sus parámetros. También estudiamos un conjunto particular de funciones Sturmianas Generalizadas de dos cuerpos construidas considerando como potencial generador el potencial de Hulthén. Contrariamente al caso habitual, en el que las funciones Sturmianas se construyen numéricamente, las funciones Sturmianas de Hulthén poseen forma analítica. Sus propiedades matem´aticas pueden ser analíticamente estudiadas proporcionando una herramienta única para comprender y analizar los problemas de dispersión y sus soluciones. Además, proponemos un nuevo conjunto de funciones a las que llamamos funciones Quasi-Sturmianas. Estas funciones se presentan como una alternativa para expandir la solución buscada en procesos de dispersi´on de dos y tres cuerpos. Se definen como soluciones de una ecuación diferencial de tipo-Schrödinger, no homogénea. Por construcción, incluyen un comportamiento asintótico adecuado para resolver problemas de dispersión. Presentamos diferentes expresiones analíticas y exploramos sus propiedades matemáticas, vinculando y justificando los desarrollos realizados previamente. Para finalizar, utilizamos las funciones estudiadas (Sturmianas de Hulthén y Quasi-Sturmianas) en la resolución de problemas particulares de dos y tres cuerpos. La eficacia de estas funciones se ilustra comparando los resultados obtenidos con datos provenientes de la aplicación de otras metodologías. / Two and three-body scattering problems are of crucial relevance in atomic physics as they allow to describe different atomic collision processes. Nowadays, the two-body cases can be solved with any degree of numerical accuracy. Scattering problem involving three charged particles are notoriously difficult but something similar –though to a lesser extentcan be stated. The aim of this work is to contribute to the understanding of three-body Coulomb scattering problems from an analytical point of view. This is not only of fundamental interest, it is also useful to better master numerical approaches that are being developed within the collision community. To achieve this aim we propose to approximate scattering solutions with expansions on sets of appropriate functions having closed form. In so doing, we develop a number of related mathematical tools involving Coulomb functions, homogeneous and non-homogeneous second order differential equations, and hypergeometric functions in one and two variables. First we deal with the two-body radial Coulomb wave functions, and review their main properties. We extend known results to give in closed form the Laguerre expansions coefficients of the irregular solutions, and establish a new connection between the coefficients corresponding to the regular solution and Meixner-Pollaczek polynomials. This relation allows us to obtain an orthogonality and closure relation for these coefficients considering the charge as a variable. Then we explore two-variable hypergeometric functions. For some of them, such as Appell and confluent Horn functions, we find closed form for the derivatives with respect to their parameters. We also study a particular set of two-body Generalized Sturmian functions constructed with a Hulth´en generating potential. Contrary to the usual case in which Sturmian functions are numerically constructed, the Hulth´en Sturmian functions can be given in closed form. Their mathematical properties can thus be analytically studied providing a unique tool to investigate scattering problems. Next, we introduce a novel set of functions that we name Quasi-Sturmian functions. They constitute an alternative set of functions, given in closed form, to expand the sought after solution of two- and three-body scattering processes. Quasi-Sturmian functions are solutions of a non-homogeneous second order Schr¨odinger-like differential equation and have, by construction, the appropriate asymptotic behavior. We present different analytic expressions and explore their mathematical properties, linking and justifying the developed mathematical tools described above. Finally we use the studied Hulth´en Sturmian and Quasi-Sturmian functions to solve some particular two- and three-body scattering problems. The efficiency of these sets of functions is illustrated by comparing our results with those obtained by other methods

In this thesis uniform asymptotic approximations of integrals are discussed. In order to derive these approximations, two well-known methods are used i.e., the saddle point method and the Bleistein method. To start with this, examples are given to demonstrate these two methods and a general idea of how to approach these techniques. The asymptotics of the hypergeometric functions with large parameters are discussed i.e., 2F1 (a + e1λ, b + e2λ c + e3λ ; z)where ej = 0,±1, j = 1, 2, 3 as |λ|→ ∞, which are valid in large regions of the complex z-plane, where a, b and c are fixed. The saddle point method is applied where the saddle point gives a dominant contributions to the integral representations of the hypergeometric functions and Bleistein’s method is adopted to obtain the uniform asymptotic approximations of some cases where the coalescence takes place between the critical points of the integrals. As a special case, the uniform asymptotic approximation of the hypergeometric function where the third parameter is large, is obtained. A new method to estimate the remainder term in the Bleistein method is proposed which is created to deal with new type of integrals in which the usual methods for the remainder estimates fail. Finally, using the asymptotic property of the hypergeometric function when the third parameter is large, the uniform asymptotic approximation of the monic Meixner Sobolev polynomials Sn(x) as n → ∞ , is obtained in terms of Airy functions. The asymptotic approximations for the location of the zeros of these polynomials are also discussed. As a limit case, a new asymptotic approximation for the large zeros of the classical Meixner polynomials is provided.

No description available.

Student Number : 9606114D PhD Thesis School of Mathematics Faculty of Science / In this thesis, our primary interest lies in the investigation of the location of the zeros and the asymptotic zero distribution of hypergeometric polynomials. The location of the zeros and the asymptotic zero distribution of general hy- pergeometric polynomials are linked with those of the classical orthogonal polynomials in some cases, notably 2F1 and 1F1 hypergeometric polynomials which have been extensively studied. In the case of 3F2 polynomials, less is known about their properties, including the location of their zeros, because there is, in general, no direct link with orthogonal polynomials. Our intro- duction in Chapter 1 outlines known results in this area and we also review recent papers dealing with the location of the zeros of 2F1 and 1F1 hyperge- ometric polynomials. In Chapter 2, we consider two classes of 3F2 hypergeometric polynomials, each of which has a representation in terms of 2F1 polynomials. Our first result proves that the class of polynomials 3F2(−n, a, b; a−1, d; x), a, b, d 2 R, n 2 N is quasi-orthogonal of order 1 on an interval that varies with the values of the real parameters b and d. We deduce the location of (n−1) of its zeros and dis- cuss the apparent role played by the parameter a with regard to the location of the one remaining zero of this class of polynomials. We also prove re- sults on the location of the zeros of the classes 3F2(−n, b, b−n 2 ; b−n, b−n−1 2 ; x), b 2 R, n 2 N and 3F2 (−n, b, b−n 2 + 1; b − n, b−n+1 2 ; x), n 2 N, b 2 R by using the orthogonality and quasi-orthogonality of factors involved in its representation. We use Mathematica to plot the zeros of these 3F2 hypergeometric polynomials for different values of n as well as for different ranges of the pa- rameters. The numerical data is consistent with the results we have proved. The Euler integral representation of the 2F1 Gauss hypergeometric function is well known and plays a prominent role in the derivation of transformation identities and in the evaluation of 2F1(a, b; c; 1), among other applications (cf. [1], p.65). The general p+kFq+k hypergeometric function has an integral repre- sentation (cf. [37], Theorem 38) where the integrand involves pFq. In Chapter 3, we give a simple and direct proof of an Euler integral representation for a special class of q+1Fq functions for q >= 2. The values of certain 3F2 and 4F3 functions at x = 1, some of which can be derived using other methods, are deduced from our integral formula. In Chapter 4, we prove that the zeros of 2F1 (−n, n+1 2 ; n+3 2 ; z) asymptotically approach the section of the lemniscate {z : |z(1 − z)2| = 4 27 ;Re(z) > 1 3} as n ! 1. In recent papers (cf. [31], [32], [34], [35]), Mart´ınez-Finkelshtein and Kuijlaars and their co-authors have used Riemann-Hilbert methods to derive the asymptotic distribution of Jacobi polynomials P(an,bn) n when the limits A = lim n!1 an n and B = lim n!1 Bn n exist and lie in the interior of certain specified regions in the AB-plane. Our result corresponds to one of the transitional or boundary cases for Jacobi polynomials in the Kuijlaars Mart´ınez-Finkelshtein classification.

Nous donnons une condition nécessaire et suffisante pour que les coefficients de Taylor à l'origine de séries en plusieurs variables $q_i({mathbf z})=z_iexp(G_i({mathbf z})/F({mathbf z}))$ soient entiers, avec ${mathbf z}=(z_1,dots,z_d)$ et où $F({mathbf z})$ et $G_i({mathbf z})+log(z_i)F({mathbf z})$, $i=1,dots,d$, sont des solutions particulières de certains $A$-systèmes d'équations différentielles linéaires. Ce critère est basé sur les propriétés analytiques de l'application de Landau (classiquement associée aux suites de quotients de factorielles de formes linéaires). Pour démontrer ce critère, nous généralisons entre autres une version en plusieurs variables d'un théorème de Dwork concernant les congruences formelles entre séries formelles, démontrée par Krattenthaler et Rivoal dans og Multivariate $p$-adic formal congruences and integrality of Taylor coefficients of mirror maps fg [arXiv:0804.3049v3, math.NT]. Ce critère en plusieurs variables implique l'intégralité des coefficients de Taylor de nouvelles applications miroir d'une seule variable dans og Tables of Calabi--Yau equations fg [arXiv:math/0507430v2, math.AG] de Almkvist, van Enckevort, van Straten et Zudilin. Dans le cas particulier d'une variable, nous affinons notre critère et démontrons l'intégralité des coefficients de Taylor de racines d'applications miroir. Cela nous permet de démontrer une conjecture de Zhou énoncée dans og Integrality properties of variations of Mahler measures fg [arXiv:1006.2428v1 math.AG]. / We give a necessary and sufficient condition for the integrality of the Taylor coefficients at the origin of formal power series $q_i({mathbf z})=z_iexp(G_i({mathbf z})/F({mathbf z}))$, with ${mathbf z}=(z_1,dots,z_d)$ and where $F({mathbf z})$ and $G_i({mathbf z})+log(z_i)F({mathbf z})$, $i=1,dots,d$ are particular solutions of some $A$-systems of differential equations. This criterion is based on the analytical properties of Landau's function (which is classically associated to the sequences of factorial ratios). One of the techniques used to prove this criterion is a generalization of a version of a theorem of Dwork on the formal congruences between formal series, proved by Krattenthaler and Rivoal in og Multivariate $p$-adic formal congruences and integrality of Taylor coefficients of mirror maps fg [arXiv:0804.3049v3, math.NT]. This criterion involves the integrality of the Taylor coefficients of new univariate mirror maps listed in og Tables of Calabi--Yau equations fg [arXiv:math/0507430v2, math.AG] by Almkvist, van Enckevort, van Straten and Zudilin. In the particular case of one variable, we refine our criterion and demonstrate the integrality of the Taylor coefficients of roots of mirror maps. This allows us to prove a conjecture stated by Zhou in og Integrality properties of variations of Mahler measures fg [arXiv:1006.2428v1 math.AG]. STAR Date de soutenance : 6 septembre 2011 Thèse sur travaux: non

This thesis is a contribution to the analysis of the relationship between the economic growth and the usage of non-renewable energy resources. More precisely, it is studied the conditions under which energy-saving technologies can sustain long-run growth, even if energy is mainly produced by means of non-renewable energy resources, such as fossil fuels. A general equilibrium framework is considered, giving special attention to the dynamical properties of the economy. In accordance with the well-known debate of complementarity vs. substitutability between physical capital and energy as production inputs, this thesis is divided into two parts. The first part of this thesis assumes complementarity between physical capital and energy as production inputs, which captures the idea of the existence of a minimum energy requirement to use a machine. Even if in contrast with the standard literature on non-renewable energy resources, which assumes substitutability, the assumption of complementarity is indeed supported by various empirical studies. This relationship of complementarity allows one to introduce the assumption of different generations of machines coexisting in each period by adding a new variable to the firm's problem: physical capital replacement. In this first part of the thesis, it is provided a theoretical study of physical capital replacement, i.e., vintage effect, which is an important environmental policy when new machines are assumed to be more energy-saving. Following the standard literature on non-renewable energy resources, this second part of the thesis assumes substitutability between capital and energy. This branch of the literature gives central position to physical capital accumulation to offset the constraint on production possibilities due to use of non-renewable energy resources. This literature assumes the same technology for both physical capital accumulation and consumption, which implies (among other things) that the energy intensity of both sectors is the same. However, data do not support this implication and suggest that physical capital accumulation is relatively more energy-intensive than consumption. Following that, this second part of the thesis studies the implications of this hypothesis.

We develop the mathematical theory of the open/closed correspondence, proposed by Mayr in physics as a class of dualities between open strings on Calabi-Yau 3-folds and closed strings on Calabi-Yau 4-folds. Given an open geometry on a toric Calabi-Yau 3-orbifold relative to a framed Aganagic-Vafa outer brane, we construct a closed geometry on a toric Calabi-Yau 4-orbifold and establish the correspondence between the two geometries on the following levels across both the A- and B-sides of mirror symmetry: numerical Gromov-Witten invariants; generating functions of Gromov-Witten invariants; B-model hypergeometric functions and Givental-style mirror theorems; Picard-Fuchs systems and solutions; integral cycles on Hori-Vafa mirrors and periods; mixed Hodge structures.

The aim of this work is to study the interaction between two classical objects of mathematics: the modular group, and the absolute Galois group. The latter acts on the category of finite index subgroups of the modular group. However, it is a task out of reach do understand this action in this generality. We propose a lattice which parametrizes a certain system of &rdquo / geometric&rdquo / elements in this category. This system is setwise invariant under the Galois action, and there is a hope that one can explicitly understand the pointwise action on the elements of this system. These elements admit moreover a combinatorial description as quadrangulations of the sphere, satisfying a natural nonnegative curvature condition. Furthermore, their connections with hypergeometric functions allow us to realize these quadrangulations as points in the moduli space of rational curves with 8 punctures. These points are conjecturally defined over a number field and our ultimate wish is to compare the Galois action on the lattice elements in the category and the corresponding points in the moduli space.