On P-Adic Fields And P-Groups, 2017 University of Kentucky

#### On P-Adic Fields And P-Groups, Luis A. Sordo Vieira

*Theses and Dissertations--Mathematics*

The dissertation is divided into two parts. The first part mainly treats a conjecture of Emil Artin from the 1930s. Namely, if f = a_1x_1^d + a_2x_2^d +...+ a_{d^2+1}x^d where the coefficients a_i lie in a finite unramified extension of a rational p-adic field, where p is an odd prime, then f is isotropic. We also deal with systems of quadratic forms over finite fields and study the isotropicity of the system relative to the number of variables. We also study a variant of the classical Davenport constant of finite abelian groups and relate it to ...

Combinatorics Of Compositions, 2017 Georgia Southern University

#### Combinatorics Of Compositions, Meghann M. Gibson

*Electronic Theses & Dissertations*

Integer compositions and related enumeration problems have been extensively studied. The cyclic analogues of such questions, however, have significantly fewer results. In this thesis, we follow the cyclic construction of Flajolet and Soria to obtain generating functions for cyclic compositions and n-color cyclic compositions with various restrictions. With these generating functions we present some statistics and asymptotic formulas for the number of compositions and parts in such compositions. Combinatorial explanations are also provided for many of the enumerative observations presented.

Scaling Of Spectra Of Cantor-Type Measures And Some Number Theoretic Considerations, 2017 University of Central Florida

#### Scaling Of Spectra Of Cantor-Type Measures And Some Number Theoretic Considerations, Isabelle Kraus

*Honors in the Major Theses*

We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number *m* generates a complete or incomplete Fourier basis for a Cantor-type measure with scale *g*.

Mathematics Education From A Mathematicians Point Of View, 2016 University of Tennessee, Knoxville

#### Mathematics Education From A Mathematicians Point Of View, Nan Woodson Simpson

*Masters Theses*

This study has been written to illustrate the development from early mathematical learning (grades 3-8) to secondary education regarding the Fundamental Theorem of Arithmetic and the Fundamental Theorem of Algebra. It investigates the progression of the mathematics presented to the students by the current curriculum adopted by the Rhea County School System and the mathematics academic standards set forth by the State of Tennessee.

Explicit Formulae And Trace Formulae, 2016 The Graduate Center, City University of New York

#### Explicit Formulae And Trace Formulae, Tian An Wong

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

In this thesis, motivated by an observation of D. Hejhal, we show that the explicit formulae of A. Weil for sums over zeroes of Hecke L-functions, via the Maass-Selberg relation, occur in the continuous spectral terms in the Selberg trace formula over various number fields. In Part I, we discuss the relevant parts of the trace formulae classically and adelically, developing the necessary representation theoretic background. In Part II, we show how show the explicit formulae intervene, using the classical formulation of Weil; then we recast this in terms of Weil distributions and the adelic formulation of Weil. As an ...

Explicit Reciprocity Laws For Higher Local Fields, 2016 The Graduate Center, City University of New York

#### Explicit Reciprocity Laws For Higher Local Fields, Jorge Florez

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

In this thesis we generalize to higher dimensional local fields the explicit reciprocity laws of Kolyvagin for the Kummer pairing associated to a formal group. The formulas obtained describe the values of the pairing in terms of multidimensional p-adic differentiation, the logarithm of the formal group, the generalized trace and the norm on Milnor K-groups.

On The Free And G-Saturated Weight Monoids Of Smooth Affine Spherical Varieties For G=Sl(N), 2016 The Graduate Center, City University of New York

#### On The Free And G-Saturated Weight Monoids Of Smooth Affine Spherical Varieties For G=Sl(N), Won Geun Kim

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

Let $X$ be an affine algebraic variety over $\mathbb{C}$ equipped with an action of a connected reductive group $G$. The weight monoid $\Gamma(X)$ of $X$ is the set of isomorphism classes of irreducible representations of $G$ that occur in the coordinate ring $\mathbb{C}[X]$ of $X$. Losev has shown that if $X$ is a smooth affine spherical variety, that is, if $X$ is smooth and $\mathbb{C}[X]$ is multiplicity-free as a representation of $G$, then $\Gamma(X)$ determines $X$ up to equivariant automorphism.

Pezzini and Van Steirteghem have recently obtained a combinatorial characterization of the weight ...

Nullification Of Torus Knots And Links, 2016 Western Kentucky University

#### Nullification Of Torus Knots And Links, Zachary S. Bettersworth

*Masters Theses & Specialist Projects*

Knot nullification is an unknotting operation performed on knots and links that can be used to model DNA recombination moves of circular DNA molecules in the laboratory. Thus nullification is a biologically relevant operation that should be studied.

Nullification moves can be naturally grouped into two classes: coherent nullification, which preserves the orientation of the knot, and incoherent nullification, which changes the orientation of the knot. We define the coherent (incoherent) nullification number of a knot or link as the minimal number of coherent (incoherent) nullification moves needed to unknot any knot or link. This thesis concentrates on the study ...

P-Adic L-Functions And The Geometry Of Hida Families, 2016 Graduate Center, City University of New York

#### P-Adic L-Functions And The Geometry Of Hida Families, Joseph Kramer-Miller

*All Graduate Works by Year: Dissertations, Theses, and Capstone Projects*

A major theme in the theory of $p$-adic deformations of automorphic forms is how $p$-adic $L$-functions over eigenvarieties relate to the geometry of these eigenvarieties. In this talk we explain results in this vein for the ordinary part of the eigencurve (i.e. Hida families). We address how Taylor expansions of one variable $p$-adic $L$-functions varying over families can detect geometric phenomena: crossing components of a certain intersection multiplicity and ramification over the weight space. Our methods involve proving a converse to a result of Vatsal relating congruences between eigenforms to their algebraic special $L ...

Comparing Local Constants Of Ordinary Elliptic Curves In Dihedral Extensions, 2016 College of Saint Benedict/Saint John's University

#### Comparing Local Constants Of Ordinary Elliptic Curves In Dihedral Extensions, Sunil Chetty

*Mathematics Faculty Publications*

We establish, for a substantial class of elliptic curves, that the arithmetic local constants introduced by Mazur and Rubin agree with quotients of analytic root numbers.

Mathematical Reasoning And The Inductive Process: An Examination Of The Law Of Quadratic Reciprocity, 2016 California State University - San Bernardino

#### Mathematical Reasoning And The Inductive Process: An Examination Of The Law Of Quadratic Reciprocity, Nitish Mittal

*Electronic Theses, Projects, and Dissertations*

This project investigates the development of four different proofs of the law of quadratic reciprocity, in order to study the critical reasoning process that drives discovery in mathematics. We begin with an examination of the first proof of this law given by Gauss. We then describe Gauss’ fourth proof of this law based on Gauss sums, followed by a look at Eisenstein’s geometric simplification of Gauss’ third proof. Finally, we finish with an examination of one of the modern proofs of this theorem published in 1991 by Rousseau. Through this investigation we aim to analyze the different strategies used ...

The Evolution Of Cryptology, 2016 California State University - San Bernardino

#### The Evolution Of Cryptology, Gwendolyn Rae Souza

*Electronic Theses, Projects, and Dissertations*

We live in an age when our most private information is becoming exceedingly difficult to keep private. Cryptology allows for the creation of encryptive barriers that protect this information. Though the information is protected, it is not entirely inaccessible. A recipient may be able to access the information by decoding the message. This possible threat has encouraged cryptologists to evolve and complicate their encrypting methods so that future information can remain safe and become more difficult to decode. There are various methods of encryption that demonstrate how cryptology continues to evolve through time. These methods revolve around different areas of ...

Counting Solutions To Discrete Non-Algebraic Equations Modulo Prime Powers, 2016 Rose-Hulman Institute of Technology

#### Counting Solutions To Discrete Non-Algebraic Equations Modulo Prime Powers, Abigail Mann

*Mathematical Sciences Technical Reports (MSTR)*

As society becomes more reliant on computers, cryptographic security becomes increasingly important. Current encryption schemes include the ElGamal signature scheme, which depends on the complexity of the discrete logarithm problem. It is thought that the functions that such schemes use have inverses that are computationally intractable. In relation to this, we are interested in counting the solutions to a generalization of the discrete logarithm problem modulo a prime power. This is achieved by interpolating to p-adic functions, and using Hensel's lemma, or other methods in the case of singular lifting, and the Chinese Remainder Theorem.

Statistical Analysis Of Binary Functional Graphs Of The Discrete Logarithm, 2016 Rose-Hulman Institute of Technology

#### Statistical Analysis Of Binary Functional Graphs Of The Discrete Logarithm, Mitchell Orzech

*Mathematical Sciences Technical Reports (MSTR)*

The increased use of cryptography to protect our personal information makes us want to understand the security of cryptosystems. The security of many cryptosystems relies on solving the discrete logarithm, which is thought to be relatively difficult. Therefore, we focus on the statistical analysis of certain properties of the graph of the discrete logarithm. We discovered the expected value and variance of a certain property of the graph and compare the expected value to experimental data. Our finding did not coincide with our intuition of the data following a Gaussian distribution given a large sample size. Thus, we found the ...

On The Dimension Of Algebraic-Geometric Trace Codes, 2016 College of Saint Benedict/Saint John's University

#### On The Dimension Of Algebraic-Geometric Trace Codes, Phong Le, Sunil Chetty

*Mathematics Faculty Publications*

We study trace codes induced from codes defined by an algebraic curve X. We determine conditions on X which admit a formula for the dimension of such a trace code. Central to our work are several dimension reducing methods for the underlying functions spaces associated to X.

The History And Applications Of Fibonacci Numbers, 2016 University of Nebraska - Lincoln

#### The History And Applications Of Fibonacci Numbers, Cashous W. Bortner, Allan C. Peterson

*UCARE Research Products*

The Fibonacci sequence is arguably the most observed sequence not only in mathematics, but also in nature. As we begin to learn more and more about the Fibonacci sequence and the numbers that make the sequence, many new and interesting applications of the have risen from different areas of algebra to market trading strategies. This poster analyzes not only the history of Leonardo Bonacci, but also the elegant sequence that is now his namesake and its appearance in nature as well as some of its current mathematical and non-mathematical applications.

Cyclic Critical Groups Of Graphs, 2016 Gettysburg College

#### Cyclic Critical Groups Of Graphs, Ryan P. Becker, Darren B. Glass

*Math Faculty Publications*

In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the group exactly.

Arithmetic Local Constants For Abelian Varieties With Extra Endomorphisms, 2016 College of Saint Benedict/Saint John's University

#### Arithmetic Local Constants For Abelian Varieties With Extra Endomorphisms, Sunil Chetty

*Mathematics Faculty Publications*

This work generalizes the theory of arithmetic local constants, introduced by Mazur and Rubin, to better address abelian varieties with a larger endomorphism ring than ℤ. We then study the growth of the p^{∞}- Selmer rank of our abelian variety, and we address the problem of extending the results of Mazur and Rubin to dihedral towers *k* ⊂ *K* ⊂ *F* in which [*F* : *K*] is not a *p*-power extension.

Adinkras And Arithmetical Graphs, 2016 Harvey Mudd College

#### Adinkras And Arithmetical Graphs, Madeleine Weinstein

*HMC Senior Theses*

Adinkras and arithmetical graphs have divergent origins. In the spirit of Feynman diagrams, adinkras encode representations of supersymmetry algebras as graphs with additional structures. Arithmetical graphs, on the other hand, arise in algebraic geometry, and give an arithmetical structure to a graph. In this thesis, we will interpret adinkras as arithmetical graphs and see what can be learned.

Our work consists of three main strands. First, we investigate arithmetical structures on the underlying graph of an adinkra in the specific case where the underlying graph is a hypercube. We classify all such arithmetical structures and compute some of the corresponding ...

Visual Properties Of Generalized Kloosterman Sums, 2016 Pomona College

#### Visual Properties Of Generalized Kloosterman Sums, Paula Burkhardt '16, Alice Zhuo-Yu Chan '14, Gabriel Currier '16, Stephan Ramon Garcia, Florian Luca, Hong Suh '16

*Pomona Faculty Publications and Research*

For a positive integer *m* and a subgroup A of the unit group (*Z*/*mZ*)^{x}, the corresponding *generalized Kloosterman sum* is the function *K*(*a, b, m, A*) = **Σ**_{uEA} *e(**au+bu*^{-1}*/m)*. Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.