This thesis explores the area of Ramanujan graphs, gives details of most of the known constructions, and explores results related to Ramanujan graphs including Generalized Ramanujan graphs. In addition, it also describes programming many different Ramanujan graph constructions.

This thesis explores the area of Ramanujan graphs, gives details of most of the known constructions, and explores results related to Ramanujan graphs including Generalized Ramanujan graphs. In addition, it also describes programming many different Ramanujan graph constructions.

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The theory of hypergeometric functions over finite fields was developed in the mid- 1980s by Greene. Since that time, connections between these functions and elliptic curves and modular forms have been investigated by mathematicians such as Ahlgren, Frechette, Koike, Ono, and Papanikolas. In this dissertation, we begin by giving a survey of these results and introducing hypergeometric functions over finite fields. We then focus on a particular family of elliptic curves whose j-invariant gives an automorphism of P1. We present an explicit relationship between the number of points on this family over Fp and the values of a particular hypergeometric function over Fp. Then, we use the same family of elliptic curves to construct a formula for the traces of Hecke operators on cusp forms in level 1, utilizing results of Hijikata and Schoof. This leads to formulas for Ramanujan’s -function in terms of hypergeometric functions.

The expanding constant is a measure of graph connectivity that is important for certain applications. This paper discusses the mathematical foundations for the construction of Winnie Li's graphs and for the proof that Winnie Li's graphs are Ramanujan. The paper also establishes the implications of the Ramanujan property for the expanding constant. / Master of Science

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

Dissertação (mestrado)—Universidade de Brasília, Departamento de Matemática, 2013. / Submitted by Alaíde Gonçalves dos Santos (alaide@unb.br) on 2013-07-16T14:10:41Z No. of bitstreams: 1 2013_MatheusBernardinideSouza.pdf: 449129 bytes, checksum: 84b41aaa9be182b0ab45d842af511738 (MD5) / Approved for entry into archive by Leandro Silva Borges(leandroborges@bce.unb.br) on 2013-07-16T17:14:02Z (GMT) No. of bitstreams: 1 2013_MatheusBernardinideSouza.pdf: 449129 bytes, checksum: 84b41aaa9be182b0ab45d842af511738 (MD5) / Made available in DSpace on 2013-07-16T17:14:02Z (GMT). No. of bitstreams: 1 2013_MatheusBernardinideSouza.pdf: 449129 bytes, checksum: 84b41aaa9be182b0ab45d842af511738 (MD5) / O objetivo deste trabalho é mostrar algumas técnicas para resolução de equações diofantinas. Métodos algébricos são ferramentas de grande utilidade para a resolução da equação equation x2 + 7 = yn, em que y = 2 ou Y é ímpar. O uso do método hipergeométrico traz um resultado recente (de 2008) no estudo da equação x2 + 7 =2n. m e técnicas algébricas garantem uma condição necessária para que essa última equação tenha solução. _______________________________________________________________________________________ ABSTRACT / The objective of this work is to show some techniques for solving Diophantine equations. Algebraic methods are useful tools for solving the equation x2 + 7 = yn, where y = 2 or y is odd. The use of the hypergeometric method brings a recent result (from 2008) in the study of the equation x2 + 7 = 2n.m and algebraic techniques ensure a necessary condition for the last equation to have a solution.

Une partition d'un entier positif est une façon d'écrire ce nombre comme une somme d'entiers strictement positifs où l'ordre des termes ne compte pas. Plusieurs généralisations des partitions ont été étudiées, parmi lesquelles les surpartitions, qui sont des partitions où l'on peut surligner la dernière occurrence d'un nombre, les paires de surpartitions ou encore les partitions n-colorées, qui sont liées à un modèle de physique statistique. Dans cette thèse, on généralise aux paires de surpartitions les identités d'Andrews-Gordon, qui sont une extension d'un résultat classique de la théorie des partitions : les identités de Rogers-Ramanujan. Pour cela, on définit deux classes de séries hypergéométriques basiques et on montre que ce sont les séries génératrices des paires de surpartitions vérifiant différents types de conditions (multiplicités, rangs successifs, dissection de Durfee) et de certains chemins du plan. On montre également que pour certaines valeurs des paramètres, ces séries peuvent s'écrire comme des produits infinis, ce qui conduit à plusieurs identités de type Rogers-Ramanujan. La démonstration utilise diverses méthodes combinatoires et analytiques. On définit enfin une généralisation des partitions n-colorées, les surpartitions n-colorées, et on les utilise pour interpréter combinatoirement certaines séries multiples et démontrer d'autres identités de type Rogers-Ramanujan.

Modular forms are important in different areas of mathematics and theoretical physics. The theory is well known for the modular group PSL(2,Z), but is also of interest for other Fuchsian groups. In this thesis we will be interested in triangle groups with a cusp. We review some theory about mapping of hyperbolic triangles in order to derive an expression for the Hauptmodul of a triangle group, and use this to write a SageMath-program that calculates the Fourier series of the Hauptmodul. We then review some of the results presented in [4] that describe generalizations of well known concepts such as the Eisenstein series, the Serre derivative and some general results about the algebra of modular forms for triangle groups with a cusp. We correct some of the mistakes made in [4] and prove some further properties of the generators of the algebra of modular forms in the case of Hecke groups. Then we use the results from [4] to write a SageMath-program that calculates the Fourier series of the generators of the algebra of modular forms for triangle groups with a cusp and that also finds the relations between the generators in the special case of Hecke groups. Using the results from this program, we present some conjectures concerning the generators of the algebra of modular forms for a Hecke group, which, if proven to be true, give us a generalization of some of the Ramanujan equations. We conclude by explicitly calculating the generalized Ramanujan equations for the first few Hecke groups.

<p>The main focus of this dissertation is to find and explain the periodic points of certain algebraic functions that are related to some modular functions, which themselves can be represented by continued fractions. Some of these continued fractions are first explored by Srinivasa Ramanujan in early 20th century. Later on, much work has been done in terms of studying the continued fractions, and proving several relations, identities, and giving different representations for them.</p> <p><br></p> <p>The layout of this report is as follows. Chapter 1 has all the basic background knowledge and ingredients about algebraic number theory, class field theory, Ramanujan’s theta functions, etc. In Chapter 2, we look at the Ramanujan-Göllnitz-Gordon continued fraction that we call v(τ) and evaluate it at certain arguments in the field K = Q(√−d), with −d ≡ 1 (mod 8), in which the ideal (2) = ℘₂℘′₂  is a product of two prime ideals. We prove several identities related to itself and with other modular functions. Some of these are new, while some of them are known but with different proofs. These values of v(τ) are shown to generate the inertia field of ℘₂ or ℘′₂ in an extended ring class field over the field K. The conjugates over Q of these same values, together with 0, −1 ± √2, are shown to form the exact set of periodic points of a fixed algebraic function ˆF(x), independent of d. These are analogues of similar results for the Rogers-Ramanujan continued fraction. See [1] and [2]. This joint work with my advisor Dr. Morton, is submitted for publication to the New York Journal.</p> <p><br> In Chapters 3 and 4, we take a similar approach in studying two more continued fractions c(τ) and u(τ), the first of which is more commonly known as the Ramanujan’s cubic continued fraction. We show what fields a value of this continued fraction generates over Q, and we describe how the periodic points for described functions arise as values of these continued fractions. Then in the last chapter, we summarise all these results, give some possible directions for future research as well as mentioning some conjectures.</p>