Browsing by Subject "Number theory"
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Item A construction of arithmetic progression-free sequences(Texas Tech University, 2004-12) Miller, Brian LWe describe a particular greedy construction of an arithmetic progression-free sequence from a finite composition. We also give an analysis on the properties of the resulting sequence.Item Combinatorial and probabilistic techniques in harmonic analysis(2012-05) Lewko, Mark J., 1983-; Vaaler, Jeffrey D.; Beckner, William; Pavlovic, Natasa; Rodriguez-Villegas, Fernando; Zuckerman, DavidWe prove several theorems in the intersection of harmonic analysis, combinatorics, probability and number theory. In the second section we use combinatorial methods to construct various sets with pathological combinatorial properties. In particular, we answer a question of P. Erdos and V. Sos regarding unions of Sidon sets. In the third section we use incidence bounds and bilinear methods to prove several new endpoint restriction estimates for the Paraboloid over finite fields. In the fourth and fifth sections we study a variational maximal operators associated to orthonormal systems. Here we use probabilistic techniques to construct well-behaved rearrangements and base changes. In the sixth section we apply our variational estimates to a problem in sieve theory. In the seventh section, motivated by applications to sieve theory, we disprove a maximal inequality related to multiplicative characters.Item Continued fractions(2011-08) Hannsz, Baron Kurt; Armendáriz, Efraim P.; Daniels, MarkThis report examines the theory of continued fractions and how their use enhances the secondary mathematics curriculum. The use of continued fractions to calculate best approximants of real numbers is justified geometrically, and famous irrational numbers are described as continued fractions. Periodic continued fractions and other applications of continued fractions are also discussed.Item Dirichlet's theorem on primes in an arithmetic progression(Texas Tech University, 1966-05) Chenault, Thelma AnnNot availableItem Elliptic curves(2010-08) Jensen, Crystal Dawn; Daniels, Mark L.; Armendáriz, Efraim P.This report discusses the history, use, and future of elliptic curves. Uses of elliptic curves in various number theory settings are presented. Fermat’s Last Proof is shown to be proven with elliptic curves. Finally, the future of elliptic curves with respect to cryptography and primality is shown.Item Item Heights and infinite algebraic extensions of the rationals(2014-05) Grizzard, Robert Vernon Lees; Vaaler, Jeffrey D.This dissertation contains a number of results on properties of infinite algebraic extensions of the rational field, all of which have a view toward the study of heights in diophantine geometry. We investigate whether subextensions of extensions generated by roots of polynomials of a given degree are themselves generated by polynomials of small degree, a problem motivated by the study of heights. We discuss a relative version of the Bogomolov property (the absence of small points) for extensions of fields of algebraic numbers. We describe the relationship between the Bogomolov property and the structure of the multiplicative group. Finally, we describe some results on height lower bounds which can be interpreted as diophantine approximation results in the multiplicative group.Item I.M. Vinogradov's representation of Goldbach's problem for odd integers(Texas Tech University, 1966-05) Rexrode, Doyle DanielNot availableItem Multiplicative and dynamical analysis on idèles and idèle class groups(2016-05) Hughes, Adam Miles; Vaaler, Jeffrey D.; Ciperiani, Mirela, 1976-; Mohammadi, Amir; Allcock, Daniel; Sinclair, Christopher; Widmer, MartinWe prove an extension of a result due to Allcock and Vaaler from 2009. In the main theorem we show that an idèle group associated to Q-bar is naturally dense in a Banach algebra normed by the Weil height. We establish bounds for the dynamics of generic idèlic points of a field modulo the diagonally-embedded multiplicative groups of the associated fields.Item Obstructions to the integral Hasse principle for generalized affine Chatelet surfaces(2016-05) Berg, Jennifer Sara; Voloch, José Felipe; Ciperiani, Mirela; Allcock, Daniel; Keel, Sean; Varilly-Alvarado, AnthonyThis dissertation contains results on the integral Hasse principle and strong approximation for generalized affine Chatelet surfaces defined over a number field k by x^2 −ay^2 =P(t) inside affine 3 space, with P(t) in k[t] a separable polynomial in one variable. The first portion of this dissertation is devoted to enumerating the isomorphism types of the Brauer groups of such surfaces, under certain conditions on the Galois groups of the polynomial P(t). We then provide an approach for constructing explicit Brauer classes for a larger class of varieties known as norm form varieties, and use this approach to compute generators of the Brauer groups of the generalized affine Chatelet surfaces. We provide an explicit method for computing the Brauer-Manin set for the non-cyclic algebras generating these Brauer groups. Finally, we use the results of the previous chapters to provide counterexamples to strong approximation explained by a Brauer-Manin obstruction.Item Practicality of algorithmic number theory(2013-08) Taylor, Ariel Jolishia; Luecke, John EdwinThis report discusses some of the uses of algorithms within number theory. Topics examined include the applications of algorithms in the study of cryptology, the Euclidean Algorithm, prime generating functions, and the connections between algorithmic number theory and high school algebra.Item Some applications of classical modular forms to number theory(2005) Masri, Riad Mohamad; Rodriguez-Villegas, FernandoItem Some aspects of unique factorization in algebraic number fields(Texas Tech University, 1969-12) Clark, William EarlNot available