Thesis (Ph.D.)--University of Central Florida, 2009. / Adviser: Juli K. Dixon. Includes bibliographical references (p. 212-224).

In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems.

Syftet med studien är att bidra till ökad förståelse av och fördjupad kunskap om taluppfattningens och skriftliga räknemetoders betydelse för hinder i elevers matematikutveckling, särskilt avseende addition och subtraktion, samt undersöka hur lärare arbetar med dessa områden för att förebygga och möta hinder för matematikutveckling. Elevers matematikkunskaper sjunker och på senare år har brister i taluppfattning uppmärksammats som en möjlig orsak. Denna studie med kvalitativ ansats har intervjuer och skriftliga dokument som datainsamlingsmetod. Hur uppfattar nio lärare som undervisar i år 1-6 nödvändiga kunskaper i taluppfattning för att hantera skriftliga räknemetoder i addition och subtraktion och vilka förklaringar till brister lyfter de. Vilka verktyg används för att få kännedom om elevers kunskaper i matematik vad gäller taluppfattning och skriftliga räknemetoder? Av resultatet av studien framkommer att det finns variationer i uppfattningar om nödvändiga kunskaper och undervisning om taluppfattning och skriftliga räknemetoder. Resultaten på Nationella prov vad gäller de båda studerade områdena visar på ett bättre resultat då det gäller taluppfattning jämfört med skriftliga räknemetoder. Detta kan bero på den komplexitet som det sociala samspelet mellan olika strukturer i samhället innebär.

This project is concerned with the set of primes modulo which some monic, irreducible polynomial over the integers has a root, called the Prime Set of the polynomial. We completely characterise these sets for degree 2 polynomials, and develop sufficient machinery from algebraic number theory to show that if the Galois group of a monic, irreducible polynomial over the integers is abelian, then its Prime Set can be written as the union of primes in some congruence classes modulo some integer.

<p>Given a positive integer k, Conrey, Farmer, Keating, Rubinstein and Snaith conjectured a formula for the asymptotics of the k-th moments of the central values of quadratic Dirichlet L-functions. The conjectured formula for the moments is expressed as sum of a k(k+1)/2 degree polynomial in log |d|. In the sum, d varies over the set of fundamental discriminants. This polynomial, called the moment polynomial, is given as a k-fold residue. In Part I of this thesis, we derive explicit formulae for first k lower order terms of the moment polynomial.</p> <p> In Part II, we present a formula bounding the average of S(t,f), the remainder term in the formula for the number of zeros of an L-function, L(s,f), where f is a newform of weight k and level N. This is Turing's method applied to cuspforms. We carry out the improvements to Turing's original method including using techniques of Booker and Trudgian. These improvements have application to the numerical verification of the Riemann Hypothesis.</p>

In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems.

Thesis (M. Sc.)--University of Adelaide, 1987. / Includes bibliographical references (leaves 90-93).

Thesis (Ph.D.) - Carleton University, 2005. / Includes bibliographical references (p. 295-301). Also available in electronic format on the Internet.

No description available.

abstract: Since the seminal work of Tur ́an, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n;F) be the smallest number m such that any graph on n vertices with m edges contains F as a subgraph. Then the forbidden subgraph problem asks to find ex(n; F ) for various graphs F . The question can be further generalized by asking for the extreme values of other graph parameters like minimum degree, maximum degree, or connectivity. We call this type of question a Tura ́n-type problem. In this thesis, we will study Tura ́n-type problems and their variants for graphs and hypergraphs. Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs. In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures. Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2018