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 ...

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 ...

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.

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 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.

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 ...

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.

Kronecker's Theory Of Binary Bilinear Forms With Applications To Representations Of Integers As Sums Of Three Squares, 2016 University of Kentucky

#### Kronecker's Theory Of Binary Bilinear Forms With Applications To Representations Of Integers As Sums Of Three Squares, Jonathan A. Constable

*Theses and Dissertations--Mathematics*

In 1883 Leopold Kronecker published a paper containing “a few explanatory remarks” to an earlier paper of his from 1866. His work loosely connected the theory of integral binary bilinear forms to the theory of integral binary quadratic forms. In this dissertation we discover the statements within Kronecker's paper and offer detailed arithmetic proofs. We begin by developing the theory of binary bilinear forms and their automorphs, providing a classification of integral binary bilinear forms up to equivalence, proper equivalence and complete equivalence.

In the second chapter we introduce the class number, proper class number and complete class number ...

Congruence Subarragements Of The Schmidt Arrangement, 2016 University of Colorado, Boulder

#### Congruence Subarragements Of The Schmidt Arrangement, Evan Oliver

*Undergraduate Honors Theses*

We begin with background in algebraic number theory, specifically studying quadratic fields K and rings of integers inside those fields O_K. From there, we study properties of the matrix group PSL_2(O_K) and its congruence subgroup PSL_2(p) for some prime ideal p. The Schmidt arrangement is the orbit of the real line under Mobius transformations described by the matrices in PSL_2(O_K). We then examine how these arrangements are affected when the transformations are limited to the matrices in PSL_2(p); this forms what we will call a congruence-p subarrangement of the full arrangement. We prove some tangency properties ...

A Measure Theoretic Approach To Problems Of Number Theory With Applications To The Proof Of The Prime Number Theorem, 2016 Minnesota State University Mankato

#### A Measure Theoretic Approach To Problems Of Number Theory With Applications To The Proof Of The Prime Number Theorem, Russell Lee Jahn

*All Theses, Dissertations, and Other Capstone Projects*

In this paper we demonstrate how the principles of measure theory can be applied effectively to problems of number theory. Initially, necessary concepts from number theory will be presented. Next, we state standard concepts and results from measure theory to which we will need to refer. We then develop our repertoire of measure theoretic machinery by constructing the needed measures and defining a generalized version of the multiplicative convolution of measures. A suitable integration by parts formula, one that is general enough to handle various combinations of measures, will then be derived. At this juncture we will be ready to ...

Mckay Graphs And Modular Representation Theory, 2016 Bard College

#### Mckay Graphs And Modular Representation Theory, Polina Aleksandrovna Vulakh

*Senior Projects Spring 2016*

Ordinary representation theory has been widely researched to the extent that there is a well-understood method for constructing the ordinary irreducible characters of a finite group. In parallel, John McKay showed how to associate to a finite group a graph constructed from the group's irreducible representations. In this project, we prove a structure theorem for the McKay graphs of products of groups as well as develop formulas for the graphs of two infinite families of groups. We then study the modular representations of these families and give conjectures for a modular version of the McKay graphs.

Integer Generalized Splines On The Diamond Graph, 2016 Bard College

#### Integer Generalized Splines On The Diamond Graph, Emmet Reza Mahdavi

*Senior Projects Spring 2016*

In this project we extend previous research on integer splines on graphs, and we use the methods developed on n-cycles to characterize integer splines on the diamond graph. First, we find an explicit module basis consisting of flow-up classes. Then we develop a determinantal criterion for when a given set of splines forms a basis.

The Schur Factorization Property As It Applies To Subsets Of The General Laguerre Polynomials, 2016 Bard College

#### The Schur Factorization Property As It Applies To Subsets Of The General Laguerre Polynomials, Christopher A. Gunnell

*Senior Projects Spring 2016*

Senior Project submitted to The Division of Science, Mathematics and Computing of Bard College.

A Computational And Theoretical Exploration Of The St. Petersburg Paradox, 2016 Butler University

#### A Computational And Theoretical Exploration Of The St. Petersburg Paradox, Alexander Olivero

*Undergraduate Honors Thesis Collection*

This thesis displays a sample distribution, generated from both a simulation (for large n) by computer program and explicitly calculated (for smaller n), that is not governed by the Central Limit Theorem and, in fact seems to display chaotic behavior. To our knowledge, the explicit calculation of the sample distribution function is new. This project outlines the results that have found a relation to number theory in a probabilistic game that has perplexed mathematicians for hundreds of years.

Automated Conjecturing Approach To The Discrete Riemann Hypothesis, 2016 Virginia Commonwealth University

#### Automated Conjecturing Approach To The Discrete Riemann Hypothesis, Alexander Bradford

*Theses and Dissertations*

This paper is a study on some upper bounds of the Mertens function, which is often considered somewhat of a ``mysterious" function in mathematics and is closely related to the Riemann Hypothesis. We discuss some known bounds of the Mertens function, and also seek new bounds with the help of an automated conjecture-making program named CONJECTURING, which was created by C. Larson and N. Van Cleemput, and inspired by Fajtowicz's Dalmatian Heuristic. By utilizing this powerful program, we were able to form, validate, and disprove hypotheses regarding the Mertens function and how it is bounded.

A Partition Function Connected With The Göllnitz-Gordon Identities, 2016 Georgia Southern University

#### A Partition Function Connected With The Göllnitz-Gordon Identities, Nicolas A. Smoot

*Electronic Theses & Dissertations*

Nearly a century ago, the mathematicians Hardy and Ramanujan established their celebrated circle method to give a remarkable asymptotic expression for the unrestricted partition function. Following later improvements by Rademacher, the method was utilized by Niven, Lehner, Iseki, and others to develop rapidly convergent series representations of various restricted partition functions. Following in this tradition, we use the circle method to develop formulas for counting the restricted classes of partitions that arise in the Gollnitz-Gordon identities. We then show that our results are strongly supported by numerical tests. As a side note, we also derive and compare the asymptotic behavior ...

An Exposition Of The Eisenstein Integers, 2016 Eastern Illinois University

#### An Exposition Of The Eisenstein Integers, Sarada Bandara

*Masters Theses*

In this thesis, we will give a brief introduction to number theory and prime numbers. We also provide the necessary background to understand how the imaginary ring of quadratic integers behaves.

An example of said ring are complex numbers of the form ℤ[*ω*] = {*a*+*bω* ∣ *a*, *b* ∈ ℤ} where *ω*^{2} + *ω* + 1 = 0. These are known as the Eisenstein integers, which form a triangular lattice in the complex plane, in contrast with the Gaussian integers, ℤ[*i*] = {*a* + *bi* ∣ *a*, *b* ∈ ℤ} which form a square lattice in the complex plane. The Gaussian moat problem, first posed by ...

Tabulating Pseudoprimes And Tabulating Liars, 2016 Illinois Wesleyan University

#### Tabulating Pseudoprimes And Tabulating Liars, Andrew Shallue

*Scholarship*

This paper explores the asymptotic complexity of two problems related to the Miller-Rabin-Selfridge primality test. The first problem is to tabulate strong pseudoprimes to a single fixed base $a$. It is now proven that tabulating up to $x$ requires $O(x)$ arithmetic operations and $O(x\log{x})$ bits of space.The second problem is to find all strong liars and witnesses, given a fixed odd composite $n$.This appears to be unstudied, and a randomized algorithm is presented that requires an expected $O((\log{n})^2 + |S(n)|)$ operations (here $S(n)$ is the set of strong liars).Although ...