On Sums Of Binary Hermitian Forms, 2016 The Graduate Center, City University of New York

#### On Sums Of Binary Hermitian Forms, Cihan Karabulut

*Dissertations, Theses, and Capstone Projects*

In one of his papers, Zagier defined a family of functions as sums of powers of quadratic polynomials. He showed that these functions have many surprising properties and are related to modular forms of integral weight and half integral weight, certain values of Dedekind zeta functions, Diophantine approximation, continued fractions, and Dedekind sums. He used the theory of periods of modular forms to explain the behavior of these functions. We study a similar family of functions, defining them using binary Hermitian forms. We show that this family of functions also have similar properties.

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

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

*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 ...

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

*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

*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 ...

Cayley Graphs Of Semigroups And Applications To Hashing, 2016 Graduate Center, City University of New York

#### Cayley Graphs Of Semigroups And Applications To Hashing, Bianca Sosnovski

*Dissertations, Theses, and Capstone Projects*

In 1994, Tillich and Zemor proposed a scheme for a family of hash functions that uses products of matrices in groups of the form $SL_2(F_{2^n})$. In 2009, Grassl et al. developed an attack to obtain collisions for palindromic bit strings by exploring a connection between the Tillich-Zemor functions and maximal length chains in the Euclidean algorithm for polynomials over $F_2$.

In this work, we present a new proposal for hash functions based on Cayley graphs of semigroups. In our proposed hash function, the noncommutative semigroup of linear functions under composition is considered as platform for the scheme ...

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.

General Multi-Sum Transformations And Some Implications, 2016 West Chester University of Pennsylvania

#### General Multi-Sum Transformations And Some Implications, James Mclaughlin

*Mathematics Faculty Publications*

We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence {g(k)}), to be reduced to an infinite q-product times a single basic hypergeometric sum. Various applications are given, including summation formulae for some q orthogonal polynomials, and various multisums that are expressible as infinite products.

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.

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 ...

Refinements Of Some Partition Inequalities, 2016 West Chester University of Pennsylvania

#### Refinements Of Some Partition Inequalities, James Mclaughlin

*Mathematics Faculty Publications*

In the present paper we initiate the study of a certain kind of partition inequality, by showing, for example, that if M ≥ 5 is an integer and the integers a and b are relatively prime to M and satisfy 1 ≤ a < b < M/2, and the c(m, n) are defined by 1 (sqa, sqM−a; qM)∞ − 1 (sqb , sqM−b ; qM)∞ := X m,n≥0 c(m, n)s mq n , then c(m, Mn) ≥ 0 for all integers m ≥ 0, n ≥ 0. A similar result is proved for the integers d(m, n) defined by (−sqa , −sqM−a ; q M)∞ − (−sqb , −sqM−b ; q M)∞ := X m,n≥0 d(m, n)s mq n . In each case there are obvious interpretations in terms of integer partitions. For example, if p1,5(m, n) (respectively p2,5(m, n)) denotes the number of partitions of n into exactly m parts ≡ ±1( mod 5) (respectively ≡ ±2( mod 5)), then for each integer n ≥ 1, p1,5(m, 5n) ≥ p2,5(m, 5n), 1 ≤ m ≤ 5n.

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.

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.