Averages And Nonvanishing Of Central Values Of Triple Product L-Functions Via The Relative Trace Formula, 2020 The Graduate Center, City University of New York

#### Averages And Nonvanishing Of Central Values Of Triple Product L-Functions Via The Relative Trace Formula, Bin Guan

*All Dissertations, Theses, and Capstone Projects*

Harris and Kudla (2004) proved a conjecture of Jacquet, that the central value of a triple product L-function does not vanish if and only if there exists a quaternion algebra over which a period integral of three corresponding automorphic forms does not vanish. Moreover, Gross and Kudla (1992) established an explicit identity relating central L-values and period integrals (which are finite sums in their case), when the cusp forms are of prime levels and weight 2. Böcherer, Schulze-Pillot (1996) and Watson (2002) generalized this identity to more general levels and weights, and Ichino (2008) proved an adelic period formula which ...

Quadratic Packing Polynomials On Sectors Of R2, 2020 The Graduate Center, City University of New York

#### Quadratic Packing Polynomials On Sectors Of R2, Kaare S. Gjaldbaek

*All Dissertations, Theses, and Capstone Projects*

A result by Fueter-Pólya states that the only quadratic polynomials that bijectively map the integral lattice points of the first quadrant onto the non-negative integers are the two Cantor polynomials. We study the more general case of bijective mappings of quadratic polynomials from the lattice points of sectors defined as the convex hull of two rays emanating from the origin, one of which falls along the x-axis, the other being defined by some vector. The sector is considered rational or irrational according to whether this vector can be written with rational coordinates or not. We show that the existence of ...

The Distribution Of The Greatest Common Divisor Of Elements In Quadratic Integer Rings, 2020 CUNY Bernard M Baruch College

#### The Distribution Of The Greatest Common Divisor Of Elements In Quadratic Integer Rings, Asimina S. Hamakiotes

*Student Theses*

For a pair of quadratic integers *n* and *m* chosen randomly, uniformly, and independently from the set of quadratic integers of norm *x* or less, we calculate the probability that the greatest common divisor of (*n,m*) is *k*. We also calculate the expected norm of the greatest common divisor (*n,m*) as *x* tends to infinity, with explicit error terms. We determine the probability and expected norm of the greatest common divisor for quadratic integer rings that are unique factorization domains. We also outline a method to determine the probability and expected norm of the greatest common divisor of ...

An In-Depth Look At P-Adic Numbers, 2020 CUNY New York City College of Technology

#### An In-Depth Look At P-Adic Numbers, Xiaona Zhou

*Publications and Research*

In this study, we consider $p$-adic numbers. We will also study the $p$-adic norm representation of real number, which is defined as $\mathbb{Q}_p = \{\sum_{j=m}^{\infty }a_j p^j: a_j \in \mathbb{D}_p, m\in\mathbb{Z}, a_m\neq 0\} \cup \{0\}$, where $p$ is a prime number. We explore properties of the $p$-adics by using examples. In particular, we will show that $\sqrt{6},i \in \mathbb{Q}_5$ and $\sqrt{2} \in \mathbb{Q}_7 $. $p$-adic numbers have a wide range of applicationsnin fields such as string theory, quantum mechanics ...

Combinatorial And Asymptotic Statistical Properties Of Partitions And Unimodal Sequences, 2020 Louisiana State University and Agricultural and Mechanical College

#### Combinatorial And Asymptotic Statistical Properties Of Partitions And Unimodal Sequences, Walter Mcfarland Bridges

*LSU Doctoral Dissertations*

Our main results are asymptotic zero-one laws satisfied by the diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane; the upper boundary is called the shape. For various types of unimodal sequences, we show that, as the number of squares tends to infinity, 100% of shapes are near a certain curve---that is, there is a single limit shape. Similar phenomena have been well-studied for integer partitions, but several technical difficulties arise in the extension of such asymptotic statistical laws to unimodal sequences. We develop a widely applicable method for obtaining these limit ...

Local-Global Principles For Diophantine Equations, 2020 Liberty University

#### Local-Global Principles For Diophantine Equations, Benjamin Barham

*Senior Honors Theses*

The real number field, denoted **ℝ**, is the most well-known extension field of ℚ, the field of rational numbers, but it is not the only one. For each prime *p*, there exists an extension field ℚ* _{p}* of ℚ, and these fields, known as the

*p*-adic fields, have some properties substantially different from

**ℝ**. In this paper, we construct the

*p*-adic numbers from the ground up and discuss the local-global principle, which concerns connections between solutions of equations found in ℚ and in ℚ

*. We state the Hasse-Minkowski theorem, which addresses a type of Diophantine equation to ...*

_{p}Symmetric Presentations And Related Topics, 2020 California State University, San Bernardino

#### Symmetric Presentations And Related Topics, Mayra Mcgrath

*Electronic Theses, Projects, and Dissertations*

In this thesis, we have investigated several permutation and monomialprogenitors for finite images. We have found original symmetric presen-tations for several important non-abelian simple groups, including lineargroups, unitary groups, alternating groups, and sporadic simple groups.We have found a number of finite images, including : L(2,41), PSL(2,11)×2, L(2,8), and L(2,19), as homomorphic images of the permutation progenitors. We have also found PGL(2,16) : 2 =Aut(PSL(2,16)) and PSL(2,16) as homomorphic images of monomial progenitors. We have performed manual double coset enumeration of finte images. In addition, we ...

Collaboration (Reacting To The Past/Math/History/Writing), 2020 California State University, San Bernardino

#### Collaboration (Reacting To The Past/Math/History/Writing), James Hayashi

*Q2S Enhancing Pedagogy*

This is an assignment for a Freshman level course in the College of Natural Science. By the end students will have an understanding of valid research, collaboration and communication skills. Faculty that chooses to use this assignment will be preparing students for an active learning environment, and understanding a “Big Idea”, valid research, technology and communication skills.

Faculty should give an example of what is valid research. As students are completing this assignment mini deadlines (check-ins) shall be set. With the check-ins for this assignment focus on how the group will communicate the check point and the collaboration.

The focus ...

Arithmetic Of Binary Cubic Forms, 2020 The Graduate Center, City University of New York

#### Arithmetic Of Binary Cubic Forms, Gennady Yassiyevich

*All Dissertations, Theses, and Capstone Projects*

The goal of the thesis is to establish composition laws for binary cubic forms. We will describe both the rational law and the integral law. The rational law of composition is easier to describe. Under certain conditions, which will be stated in the thesis, the integral law of composition will follow from the rational law. The end result is a new way of looking at the law of composition for integral binary cubic forms.

On Consecutive Triples Of Powerful Numbers, 2020 Oliver Wendell Holmes Junior High School, Davis, CA

#### On Consecutive Triples Of Powerful Numbers, Edward Beckon

*Rose-Hulman Undergraduate Mathematics Journal*

No abstract provided.

Consecutive Prime And Highly Total Prime Labeling In Graphs, 2020 La Salle University

#### Consecutive Prime And Highly Total Prime Labeling In Graphs, Robert Scholle

*Rose-Hulman Undergraduate Mathematics Journal*

This paper examines the graph-theoretical concepts of consecutive prime labeling and highly total prime labeling. These are variations on prime labeling, introduced by Tout, Dabboucy, and Howalla in 1982. Consecutive prime labeling is defined here for the first time. Consecutive prime labeling requires that the labels of vertices in a graph be relatively prime to the labels of all adjacent vertices as well as all incident edges. We show that all paths, cycles, stars, and complete graphs have a consecutive prime labeling and conjecture that all simple connected graphs have a consecutive prime labeling.

This paper also expands on work ...

Combinatorial Identities On Multinomial Coefficients And Graph Theory, 2020 Montville Township High School

#### Combinatorial Identities On Multinomial Coefficients And Graph Theory, Seungho Lee

*Rose-Hulman Undergraduate Mathematics Journal*

We study combinatorial identities on multinomial coefficients. In particular, we present several new ways to count the connected labeled graphs using multinomial coefficients.

Introduction To Neutroalgebraic Structures And Antialgebraic Structures (Revisited), 2020 University of New Mexico

#### Introduction To Neutroalgebraic Structures And Antialgebraic Structures (Revisited), Florentin Smarandache

*Mathematics and Statistics Faculty and Staff Publications*

In all classical algebraic structures, the Laws of Compositions on a given set are well-defined. But this is a restrictive case, because there are many more situations in science and in any domain of knowledge when a law of composition defined on a set may be only partially-defined (or partially true) and partially-undefined (or partially false), that we call NeutroDefined, or totally undefined (totally false) that we call AntiDefined. Again, in all classical algebraic structures, the Axioms (Associativity, Commutativity, etc.) defined on a set are totally true, but it is again a restrictive case, because similarly there are numerous situations ...

Codes, Cryptography, And The Mceliece Cryptosystem, 2020 Liberty University

#### Codes, Cryptography, And The Mceliece Cryptosystem, Bethany Matsick

*Senior Honors Theses*

Over the past several decades, technology has continued to develop at an incredible rate, and the importance of properly securing information has increased significantly. While a variety of encryption schemes currently exist for this purpose, a number of them rely on problems, such as integer factorization, that are not resistant to quantum algorithms. With the reality of quantum computers approaching, it is critical that a quantum-resistant method of protecting information is found. After developing the proper background, we evaluate the potential of the McEliece cryptosystem for use in the post-quantum era by examining families of algebraic geometry codes that allow ...

An Exploration Of The Use Of The Fibonacci Sequence In Unrelated Mathematics Disciplines, 2020 University of New Hampshire, Durham

#### An Exploration Of The Use Of The Fibonacci Sequence In Unrelated Mathematics Disciplines, Molly E. Boodey

*Honors Theses and Capstones*

No abstract provided.

Solutions To Systems Of Equations Over Finite Fields, 2020 University of Kentucky

#### Solutions To Systems Of Equations Over Finite Fields, Rachel Petrik

*Theses and Dissertations--Mathematics*

This dissertation investigates the existence of solutions to equations over finite fields with an emphasis on diagonal equations. In particular:

- Given a system of equations, how many solutions are there?
- In the case of a system of diagonal forms, when does a nontrivial solution exist?

Many results are known that address (1) and (2), such as the classical Chevalley--Warning theorems. With respect to (1), we have improved a recent result of D.R. Heath--Brown, which provides a lower bound on the total number of solutions to a system of polynomials equations. Furthermore, we have demonstrated that several of our lower ...

Complex Powers Of I Satisfying The Continued Fraction Functional Equation Over The Gaussian Integers, 2019 Illinois Mathematics and Science Academy

#### Complex Powers Of I Satisfying The Continued Fraction Functional Equation Over The Gaussian Integers, Matthew Niemiro '20

*Examples of Student Work*

We investigate and then state the conditions under which* i ^{z} *satisfies the simple continued fraction functional equation for real and then complex

*z*over the Gaussian integers.

Elliptic Curves And Power Residues, 2019 University of Massachusetts Amherst

#### Elliptic Curves And Power Residues, Vy Thi Khanh Nguyen

*Doctoral Dissertations*

Let *E*_{1} x *E*_{2} over **Q** be a fixed product of two elliptic curves over **Q** with complex multiplication. I compute the probability that the *p*th Fourier coefficient of *E*_{1} x *E _{2}*, denoted as

*a*

_{p}(

*E*

_{1}) +

*a*(

_{p}*E*), is a square modulo

_{2}*p*. The results are 1/4, 7/16, and 1/2 for different imaginary quadratic fields, given a technical independence of the twists. The similar prime densities for cubes and 4th power are 19/54, and 1/4, respectively. I also compute the probabilities without the technical assumption on ...

Comparison Of Three Dimensional Selfdual Representations By Faltings-Serre Method, 2019 University of Massachusetts Amherst

#### Comparison Of Three Dimensional Selfdual Representations By Faltings-Serre Method, Lian Duan

*Doctoral Dissertations*

In this thesis, we prove that, a selfdual 3-dimensional Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to 3-dimensional Galois representations with ground field not equal to Q. The proof makes use of the Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups.

Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, 2019 University of Massachusetts Amherst

#### Bruhat-Tits Buildings And A Characteristic P Unimodular Symbol Algorithm, Matthew Bates

*Doctoral Dissertations*

Let k be the finite field with q elements, let F be the field of Laurent series in the variable 1/t with coefficients in k, and let A be the polynomial ring in the variable t with coefficients in k. Let SLn(F) be the ring of nxn-matrices with entries in F, and determinant 1. Given a polynomial g in A, let Gamma(g) subset SLn(F) be the full congruence subgroup of level g. In this thesis we examine the action of Gamma(g) on the Bruhat-Tits building Xn associated to SLn(F) for n equals 2 and ...