This study compared the results of his chi square text of independence and the corrected chi square statistic against Fisher's exact probability test (the hypergeometric distribution) in contection with sampling from a finite population. Data were collected by advancing the minimum call size from zero to a maximum which resulted in a tail area probability of 20 percent for sample sizes from 10 to 100 by varying increments. Analysis of the data supported the rejection of the null hypotheses regarding the general rule-of-thumb guidelines concerning sample size, minimum cell expected frequency and the continuity correction factor. it was discovered that the computation using Yates' correction factor resulted in values which were so overly conservative (i.e. tail area porobabilities that were 20 to 50 percent higher than Fisher's exact test) that conclusions drawn from this calculation might prove to be inaccurate. Accordingly, a new correction factor was proposed which eliminated much of this discrepancy. Its performance was equally consistent with that of the uncorrected chi square statistic and at times, even better.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016. / In this thesis, various partition functions with respect to Rogers-Ramanujan identities and George Andrews' partitions with initial repetitions are studied. Agarwal and Goyal gave a three-way partition theoretic interpretation of the Rogers- Ramanujan identities. We generalise their result and establish certain connections with some work of Connor. Further combinatorial consequences and related partition identities are presented. Furthermore, we re ne one of the theorems of George Andrews on partitions with initial repetitions. In the same pursuit, we construct a non-diagram version of the Keith's bijection that not only proves the theorem, but also provides a clear proof of the re nement. Various directions in the spirit of partitions with initial repetitions are discussed and results enumerated. In one case, an identity of the Euler-Pentagonal type is presented and its analytic proof given. / M T 2016

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.

The present thesis has three main topics: geometry of coamoebas, hypergeometric functions, and geometry of zeros. First, we study the coamoeba of a Laurent polynomial f in n complex variables. We define a simpler object, which we call the lopsided coamoeba, and associate to the lopsided coamoeba an order map. That is, we give a bijection between the set of connected components of the complement of the closed lopsided coamoeba and a finite set presented as the intersection of an affine lattice and a certain zonotope. Using the order map, we then study the topology of the coamoeba. In particular, we settle a conjecture of M. Passare concerning the number of connected components of the complement of the closed coamoeba in the case when the Newton polytope of f has at most n+2 vertices. In the second part we study hypergeometric functions in the sense of Gel'fand, Kapranov, and Zelevinsky. We define Euler-Mellin integrals, a family of Euler type hypergeometric integrals associated to a coamoeba. As opposed to previous studies of hypergeometric integrals, the explicit nature of Euler-Mellin integrals allows us to study in detail the dependence of A-hypergeometric functions on the homogeneity parameter of the A-hypergeometric system. Our main result is a complete description of this dependence in the case when A represents a toric projective curve. In the last chapter we turn to the theory of real univariate polynomials. The famous Descartes' rule of signs gives necessary conditions for a pair (p,n) of integers to represent the number of positive and negative roots of a real polynomial. We characterize which pairs fulfilling Descartes' conditions are realizable up to degree 7, and we provide restrictions valid in arbitrary degree.

A family of parameterized Thue equations is defined as F_{t,s,...}(X, Y ) = m, m ∈ Z where F_{t,s,...}(X,Y) is a form in X and Y with degree greater than or equal to 3 and integer coefficients that are parameterized by t, s, . . . ∈ Z. A variety of these families have been studied by different authors. In this thesis, we study the following families of Thue inequalities |sx3 −tx2y−(t+3s)xy2 −sy3|≤2t+3s, |sx4 −tx3y−6sx2y2 +txy3 +sy4|≤6t+7s, |sx6 − 2tx5y − (5t + 15s)x4y2 − 20sx3y3 + 5tx2y4 +(2t + 6s)xy5 + sy6| ≤ 120t + 323s, where s and t are integers. The forms in question are “simple”, in the sense that the roots of the underlying polynomials can be permuted transitively by automorphisms. With this nice property and the hypergeometric functions, we construct sequences of good approximations to the roots of the underlying polynomials. We can then prove that under certain conditions on s and t there are upper bounds for the number of integer solutions to the above Thue inequalities.

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.

Depuis les années 90, la méthode de création télescopique de Zeilberger a joué un rôle important dans la preuve automatique d'identités mettant en jeu des fonctions spéciales. L'objectif de long terme que nous attaquons dans ce travail est l'obtension d'algorithmes et d'implantations rapides pour l'intégration et la sommation définies dans le cadre de cette création télescopique. Nos contributions incluent de nouveaux algorithmes pratiques et des critères théoriques pour tester la terminaison d'algorithmes existants. Sur le plan pratique, nous nous focalisons sur la construction de télescopeurs minimaux pour les fonctions rationnelles en deux variables, laquelle a de nombreuses applications en lien avec les fonctions algébriques et les diagonales de séries génératrices rationnelles. En considérant cette classe d'entrées contraintes, nous parvenons à mâtiner la méthode générale de création télescopique avec réduction bien connue d'Hermite, issue de l'intégration symbolique. En outre, nous avons obtenu pour cette sous-classe quelques améliorations des algorithmes classiques d'Almkvist et Zeilberger. Nos résultats expérimentaux ont montré que les algorithmes à base de réduction d'Hermite battent tous les autres algorithmes connus, à la fois en ce qui concerne la complexité au pire et en ce qui concerne les mesures de temps sur nos implantations. Sur le plan théorique, notre premier résultat est motivé par la conjecture de Wilf et Zeilberger au sujet des fonctions hyperexponentielles-hypergéométriques holonomes. Nous présentons un théorème de structure pour les fonctions hyperexponentielles-hypergéométriques de plusieurs variables, indiquant qu'une telle fonction peut s'écrire comme le produit de fonctions usuelles. Ce théorème étend à la fois le théorème d'Ore et Sato pour les termes hypergéométriques en plusieurs variables et le résultat récent par Feng, Singer et Wu. Notre second résultat est relié au problème de l'existence de télescopeurs. Dans le cas discret à deux variables, Abramov a obtenu un critère qui indique quand un terme hypergéométrique a un télescopeur. Des résultats similaires ont été obtenus pour le $q$-décalage par Chen, Hou et Mu. Ces résultats sont fondamentaux pour la terminaison des algorithmes s'inspirant de celui de Zeilberger. Dans les autres cas mixtes continus/discrets, nous avons obtenu deux critères pour l'existence de télescopeurs pour des fonctions hyperexponentielles-hypergéométriques en deux variables. Nos critères s'appuient sur une représentation standard des fonctions hyperexponentielles-hypergéométriques en deux variables, sur sur deux décompositions additives.

No description available.

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