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Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, Heather Moore
Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, Heather Moore
University Honors Theses
This thesis presents a surprising result that the difference in a certain sums of constant rotations by the golden mean approaches exactly 1/5. Specifically, we focus on the Birkhoff sums of these rotations, with the number of terms equal to squared Fibonacci numbers. The proof relies on the properties of continued fraction approximants, Vajda's identity and the explicit formula for the Fibonacci numbers.
Pairs Of Quadratic Forms Over P-Adic Fields, John Hall
Pairs Of Quadratic Forms Over P-Adic Fields, John Hall
Theses and Dissertations--Mathematics
Given two quadratic forms $Q_1, Q_2$ over a $p$-adic field $K$ in $n$ variables, we consider the pencil $\mathcal{P}_K(Q_1, Q_2)$, which contains all nontrivial $K$-linear combinations of $Q_1$ and $Q_2$. We define $D$ to be the maximal dimension of a subspace in $K^n$ on which $Q_1$ and $Q_2$ both vanish. We define $H$ to be the maximal number of hyperbolic planes that a form in $\mathcal{P}_K(Q_1, Q_2)$ splits off over $K$. We will determine which values for $(D, H)$ are possible for a nonsingular pair of quadratic forms over a $p$-adic field $K$.
Coefficients Of Modular Forms And Applications To Partition Theory, Michael Anthony Hanson
Coefficients Of Modular Forms And Applications To Partition Theory, Michael Anthony Hanson
Doctoral Dissertations
We begin with an overview of the theory of modular forms as well as some relevant sub-topics in order to discuss three results: the first result concerns positivity of self-conjugate t-core partitions under the assumption of the Generalized Riemann Hypothesis; the second result bounds certain types of congruences called "Ramanujan congruences" for an infinite class of eta-quotients - this has an immediate application to a certain restricted partition function whose congruences have been studied in the past; the third result strengthens a previous result that relates weakly holomorphic modular forms to newforms via p-adic limits.
Lattice Extensions And Zeros Of Multilinear Polynomials, Maxwell Forst
Lattice Extensions And Zeros Of Multilinear Polynomials, Maxwell Forst
CGU Theses & Dissertations
We treat several problems related to the existence of lattice extensions preserving certain geometric properties and small-height zeros of various multilinear polynomials. An extension of a Euclidean lattice $L_1$ is a lattice $L_2$ of higher rank containing $L_1$ so that the intersection of $L_2$ with the subspace spanned by $L_1$ is equal to $L_1$. Our first result provides a counting estimate on the number of ways a primitive collection of vectors in a lattice can be extended to a basis for this lattice. Next, we discuss the existence of lattice extensions with controlled determinant, successive minima and covering radius. In …
Mathematical Structure Of Musical Tuning Systems, Shay Joel Francis Spitzer
Mathematical Structure Of Musical Tuning Systems, Shay Joel Francis Spitzer
Senior Projects Spring 2023
Over the course of history, western music has created a unique mathematical problem for itself. From acoustics, we know that two notes sound good together when they are related by simple ratios consisting of low primes. The problem arises when we try to build a finite set of pitches, like the 12 notes on a piano, that are all related by such ratios. We approach the problem by laying out definitions and axioms that seek to identify and generalize desirable properties. We can then apply these ideas to a broadened algebraic framework. Rings in which low prime integers can be …
Elliptic Functions And Iterative Algorithms For Π, Eduardo Jose Evans
Elliptic Functions And Iterative Algorithms For Π, Eduardo Jose Evans
UNF Graduate Theses and Dissertations
Preliminary identities in the theory of basic hypergeometric series, or `q-series', are proven. These include q-analogues of the exponential function, which lead to a fairly simple proof of Jacobi's celebrated triple product identity due to Andrews. The Dedekind eta function is introduced and a few identities of it derived. Euler's pentagonal number theorem is shown as a special case of Ramanujan's theta function and Watson's quintuple product identity is proved in a manner given by Carlitz and Subbarao. The Jacobian theta functions are introduced as special kinds of basic hypergeometric series and various relations between them derived using the triple …
On The Iwasawa Invariants Of Nonordinary Modular Forms, Rylan J. Gajek-Leonard
On The Iwasawa Invariants Of Nonordinary Modular Forms, Rylan J. Gajek-Leonard
Doctoral Dissertations
We extend known results on the behavior of Iwasawa invariants attached to Mazur-Tate elements for p-nonordinary modular forms of weight k=2 to higher weight modular forms with a_p=0. This is done by using a decomposition of the p-adic L-function due to R. Pollack in order to construct explicit lifts of Mazur-Tate elements to the full Iwasawa algebra. We then study the behavior of Iwasawa invariants upon projection to finite layers, allowing us to express the invariants of Mazur-Tate elements in terms of those coming from plus/minus p-adic L-functions. Our results combine with work of Pollack and Weston to relate the …
The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles
The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles
Electronic Theses, Projects, and Dissertations
This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i2 = 3, j2 = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL2(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This …
Some Properties And Applications Of Spaces Of Modular Forms With Eta-Multiplier, Cuyler Daniel Warnock
Some Properties And Applications Of Spaces Of Modular Forms With Eta-Multiplier, Cuyler Daniel Warnock
Theses and Dissertations
This dissertation considers two topics. In the first part of the dissertation, we prove the existence of fourteen congruences for the $p$-core partition function of the form given by Garvan in \cite{G1}. Different from the congruences given by Garvan, each of the congruences we give yield infinitely many congruences of the form $$a_p(\ell\cdot p^{t+1} \cdot n + p^t \cdot k - \delta_p) \equiv 0 \pmod \ell.$$ For example, if $t \geq 0$ and $\sfrac{m}{n}$ is the Jacobi symbol, then we prove $$a_7(7^t \cdot n - 2) \equiv 0 \pmod 5, \text{ \ \ if $\bfrac{n}{5} = 1$ and $\bfrac{n}{7} = …
Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), Marcus L. Shell
Streaming Down The Stern-Brocot Tree: Finding And Expressing Solutions To Pell's Equation In Sl(2,Z), Marcus L. Shell
Theses
This paper explores and elaborates on a method of solving Pell’s equation as introduced by Norman Wildberger. In the first chapters of the paper, foundational topics are introduced in expository style including an explanation of Pell’s equation. An explanation of continued fractions and their ability to express quadratic irrationals is provided as well as a connection to the Stern-Brocot tree and a convenient means of representation for each in terms of 2×2 matrices with integer elements. This representation will provide a useful way of navigating the Stern-Brocot tree computationally and permit us a means of computing continued fractions without the …
Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang
Zeta Function Regularization And Its Relationship To Number Theory, Stephen Wang
Electronic Theses and Dissertations
While the "path integral" formulation of quantum mechanics is both highly intuitive and far reaching, the path integrals themselves often fail to converge in the usual sense. Richard Feynman developed regularization as a solution, such that regularized path integrals could be calculated and analyzed within a strictly physics context. Over the past 50 years, mathematicians and physicists have retroactively introduced schemes for achieving mathematical rigor in the study and application of regularized path integrals. One such scheme was introduced in 2007 by the mathematicians Klaus Kirsten and Paul Loya. In this thesis, we reproduce the Kirsten and Loya approach to …
On The Local Theory Of Profinite Groups, Mohammad Shatnawi
On The Local Theory Of Profinite Groups, Mohammad Shatnawi
Dissertations
Let G be a finite group, and H be a subgroup of G. The transfer homomorphism emerges from the natural action of G on the cosets of H. The transfer was first introduced by Schur in 1902 [22] as a construction in group theory, which produce a homomorphism from a finite group G into H/H' an abelian group where H is a subgroup of G and H' is the derived group of H. One important first application is Burnside’s normal p-complement theorem [5] in 1911, although he did not use the transfer homomorphism explicitly to prove it. …
Classification Of Torsion Subgroups For Mordell Curves, Zachary Porat
Classification Of Torsion Subgroups For Mordell Curves, Zachary Porat
Honors Theses
Elliptic curves are an interesting area of study in mathematics, laying at the intersection of algebra, geometry, and number theory. They are a powerful tool, having applications in everything from Andrew Wiles’ proof of Fermat’s Last Theorem to cybersecurity. In this paper, we first provide an introduction to elliptic curves by discussing their geometry and associated group structure. We then narrow our focus, further investigating the torsion subgroups of elliptic curves. In particular, we will examine two methods used to classify these subgroups. We finish by employing these methods to categorize the torsion subgroups for a specific family of elliptic …
Inquiry In Inquiry: A Classification Of The Learning Theories Underlying Inquiry-Based Undergraduate Number Theory Texts, Rebecca L. Butler
Inquiry In Inquiry: A Classification Of The Learning Theories Underlying Inquiry-Based Undergraduate Number Theory Texts, Rebecca L. Butler
Honors Projects
While undergraduate inquiry-based texts in number theory share similar approaches with respect to learning as the embodiment of professional practice, this does not entail that these texts all operate from the same fundamental understanding of what it means to learn mathematics. In this paper, the instructional design of several texts of the aforementioned types are analyzed to assess the theory of learning under which they operate. From this understanding of the different theories of learning employed in an inquiry-based mathematical setting, one can come to understand the popular model of what it is to learn number theory in a meaningful …
Dedekind Sums: Properties And Applications To Number Theory And Lattice Point Enumeration, Oliver Meldrum
Dedekind Sums: Properties And Applications To Number Theory And Lattice Point Enumeration, Oliver Meldrum
Honors Papers
In this paper, we will explore the origins, applications, and properties of these sums and one of their generalizations. We seek to explain what these sums represent and how they behave by exploring some of their arithmetic properties. In addition, we hope to show the reader why one should care about these sums. We do this by presenting two important areas in which these sums appear: number theory and the study of enumerating lattice points inside of polytopes.
Identities For Partitions Of N With Parts From A Finite Set, Acadia Larsen
Identities For Partitions Of N With Parts From A Finite Set, Acadia Larsen
Theses and Dissertations
We show for a prime power number of parts m that the first differences of partitions into at most m parts can be expressed as a non-negative linear combination of partitions into at most m – 1 parts. To show this relationship, we combine a quasipolynomial construction of p(n,m) with a new partition identity for a finite number of parts. We prove these results by providing combinatorial interpretations of the quasipolynomial of p(n,m) and the new partition identity. We extend these results by establishing conditions for when partitions of n with parts coming from …
Vector Partitions, Jennifer French
Vector Partitions, Jennifer French
Electronic Theses and Dissertations
Integer partitions have been studied by many mathematicians over hundreds of years. Many identities exist between integer partitions, such as Euler’s discovery that every number has the same amount of partitions into distinct parts as into odd parts. These identities can be proven using methods such as conjugation or generating functions. Over the years, mathematicians have worked to expand partition identities to vectors. In 1963, M. S. Cheema proved that every vector has the same number of partitions into distinct vectors as into vectors with at least one component odd. This parallels Euler’s result for integer partitions. The primary purpose …
Problem Book On Higher Algebra And Number Theory, Ryanto Putra
Problem Book On Higher Algebra And Number Theory, Ryanto Putra
Theses and Dissertations
This book is an attempt to provide relevant end-of-section exercises, together with their step-by-step solutions, to Dr. Zieschang's classic class notes Higher Algebra and Number Theory. It's written under the notion that active hands-on working on exercises is an important part of learning, whereby students would see the nuance and intricacies of a math concepts which they may miss from passive reading. The problems are selected here to provide background on the text, examples that illuminate the underlying theorems, as well as to fill in the gaps in the notes.
On P-Adic Fields And P-Groups, Luis A. Sordo Vieira
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 the isotropicity of diagonal …
Realizing Tournaments As Models For K-Majority Voting, Gina Marie Cheney
Realizing Tournaments As Models For K-Majority Voting, Gina Marie Cheney
Electronic Theses, Projects, and Dissertations
A k-majority tournament is a directed graph that models a k-majority voting scenario, which is realized by 2k - 1 rankings, called linear orderings, of the vertices in the tournament. Every k-majority voting scenario can be modeled by a tournament, but not every tournament is a model for a k-majority voting scenario. In this thesis we show that all acyclic tournaments can be realized as 2-majority tournaments. Further, we develop methods to realize certain quadratic residue tournaments as k-majority tournaments. Thus, each tournament within these classes of tournaments is a model for a k …
A Computational And Theoretical Exploration Of The St. Petersburg Paradox, Alexander Olivero
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.
Integer Generalized Splines On The Diamond Graph, Emmet Reza Mahdavi
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.
A Partition Function Connected With The Göllnitz-Gordon Identities, Nicolas A. Smoot
A Partition Function Connected With The Göllnitz-Gordon Identities, Nicolas A. Smoot
Electronic Theses and 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 …
Characterizing The Properties Of Specific Binomial Coefficients In Congruence Relations, Tyler Robert Russ
Characterizing The Properties Of Specific Binomial Coefficients In Congruence Relations, Tyler Robert Russ
Master's Theses and Doctoral Dissertations
The number theoretic conjecture we examine in this paper originates when trying to construct a characterizable generating set for the complex cobordism polynomial ring. To date there is no efficient, universal method for characterizing such a generating set. Wilfong conjectures that smooth projective toric varieties can act as these generators [7]. Toric varieties are related to polytopes by a bijective correspondence. Studying the combinatorial structure of these polytopes is much more manageable than studying properties of toric varieties directly. This gives rise to the number theoretic conjecture considered here. A proof of this number theoretic conjecture would in turn prove …
Commutative N-Ary Arithmetic, Aram Bingham
Commutative N-Ary Arithmetic, Aram Bingham
University of New Orleans Theses and Dissertations
Motivated by primality and integer factorization, this thesis introduces generalizations of standard binary multiplication to commutative n-ary operations based upon geometric construction and representation. This class of operations are constructed to preserve commutativity and identity so that binary multiplication is included as a special case, in order to preserve relationships with ordinary multiplicative number theory. This leads to a study of their expression in terms of elementary symmetric polynomials, and connections are made to results from the theory of polyadic (n-ary) groups. Higher order operations yield wider factorization and representation possibilities which correspond to reductions in the set of primes …
Explorations Of The Collatz Conjecture (Mod M), Glenn Micah Jackson Jr
Explorations Of The Collatz Conjecture (Mod M), Glenn Micah Jackson Jr
Honors College Theses
The Collatz Conjecture is a deceptively difficult problem recently developed in mathematics. In full, the conjecture states: Begin with any positive integer and generate a sequence as follows: If a number is even, divide it by two. Else, multiply by three and add one. Repetition of this process will eventually reach the value 1. Proof or disproof of this seemingly simple conjecture have remained elusive. However, it is known that if the generated Collatz Sequence reaches a cycle other than 4, 2, 1, the conjecture is disproven. This fact has motivated our search for occurrences of 4, 2, 1, and …
On The Characterization Of Prime Sets Of Polynomials By Congruence Conditions, Arvind Suresh
On The Characterization Of Prime Sets Of Polynomials By Congruence Conditions, Arvind Suresh
CMC Senior Theses
This project is concerned with the set of primes modulo which some monic, irreducible polynomial over the integers has a root, called the Prime Set of the polynomial. We completely characterise these sets for degree 2 polynomials, and develop sufficient machinery from algebraic number theory to show that if the Galois group of a monic, irreducible polynomial over the integers is abelian, then its Prime Set can be written as the union of primes in some congruence classes modulo some integer.
On Cubic Multisections, Andrew Alaniz
On Cubic Multisections, Andrew Alaniz
Theses and Dissertations - UTB/UTPA
In this thesis, a systematic procedure is given for generating cubic multi-sections of Eisenstein series. The relevant series are determined from Fourier expansions for Eisenstein series by restricting the congruence class of the summation index modulo three. The resulting series are shown to be rational functions of the Dedekind eta function. A more general treatment of cubic dissection formulas is given by describing the dissection operators in terms of linear transformations.
An Algebraic Approach To Number Theory Using Unique Factorization, Mark Sullivan
An Algebraic Approach To Number Theory Using Unique Factorization, Mark Sullivan
Honors Theses
Though it may seem non-intuitive, abstract algebra is often useful in the study of number theory. In this thesis, we explore some uses of abstract algebra to prove number theoretic statements. We begin by examining the structure of unique factorization domains in general. Then we introduce number fields and their rings of algebraic integers, whose structures have characteristics that are analogous to some of those of the rational numbers and the rational integers. Next we discuss quadratic fields, a special case of number fields that have important applications to number theoretic problems. We will use the structures that we introduce …
Comparing The Algebraic And Analytical Properties Of P-Adic Numbers With Real Numbers, Joseph Colton Wilson
Comparing The Algebraic And Analytical Properties Of P-Adic Numbers With Real Numbers, Joseph Colton Wilson
Theses Digitization Project
This study will provide a glimpse into the world of p-adic numbers, which encompasses a different way to measure the distance between rational numbers. Simple calculations and surprising results are examined to help familiarize the reader to the new space.