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On The Iwasawa Invariants Of Nonordinary Modular Forms, Rylan J. Gajek-Leonard 2022 University of Massachusetts Amherst

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 ...


Unomaha Problem Of The Week (2021-2022 Edition), Brad Horner, Jordan M. Sahs 2022 University of Nebraska at Omaha

Unomaha Problem Of The Week (2021-2022 Edition), Brad Horner, Jordan M. Sahs

Student Research and Creative Activity Fair

The University of Omaha math department's Problem of the Week was taken over in Fall 2019 from faculty by the authors. The structure: each semester (Fall and Spring), three problems are given per week for twelve weeks, with each problem worth ten points - mimicking the structure of arguably the most well-regarded university math competition around, the Putnam Competition, with prizes awarded to top-scorers at semester's end. The weekly competition was halted midway through Spring 2020 due to COVID-19, but relaunched again in Fall 2021, with massive changes.

Now there are three difficulty tiers to POW problems, roughly corresponding ...


On Isomorphic K-Rational Groups Of Isogenous Elliptic Curves Over Finite Fields, Ben Kuehnert, Geneva Schlafly, Zecheng Yi 2022 University of Rochester

On Isomorphic K-Rational Groups Of Isogenous Elliptic Curves Over Finite Fields, Ben Kuehnert, Geneva Schlafly, Zecheng Yi

Rose-Hulman Undergraduate Mathematics Journal

It is well known that two elliptic curves are isogenous if and only if they have same number of rational points. In fact, isogenous curves can even have isomorphic groups of rational points in certain cases. In this paper, we consolidate all the current literature on this relationship and give a extensive classification of the conditions in which this relationship arises. First we prove two ordinary isogenous elliptic curves have isomorphic groups of rational points when they have the same $j$-invariant. Then, we extend this result to certain isogenous supersingular elliptic curves, namely those with equal $j$-invariant of ...


An Overview Of Monstrous Moonshine, Catherine E. Riley 2022 Cedarville University

An Overview Of Monstrous Moonshine, Catherine E. Riley

Channels: Where Disciplines Meet

The Conway-Norton monstrous moonshine conjecture set off a quest to discover the connection between the Monster and the J-function. The goal of this paper is to give an overview of the components of the conjecture, the conjecture itself, and some of the ideas that led to its solution. Special focus is given to Klein's J-function.


John Horton Conway: The Man And His Knot Theory, Dillon Ketron 2022 East Tennessee State University

John Horton Conway: The Man And His Knot Theory, Dillon Ketron

Electronic Theses and Dissertations

John Horton Conway was a British mathematician in the twentieth century. He made notable achievements in fields such as algebra, number theory, and knot theory. He was a renowned professor at Cambridge University and later Princeton. His contributions to algebra include his discovery of the Conway group, a group in twenty-four dimensions, and the Conway Constellation. He contributed to number theory with his development of the surreal numbers. His Game of Life earned him long-lasting fame. He contributed to knot theory with his developments of the Conway polynomial, Conway sphere, and Conway notation.


The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles 2022 California State University - San Bernardino

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 list of ...


Symmetric Presentations Of Finite Groups And Related Topics, Samar Mikhail Kasouha 2022 California State University - San Bernardino

Symmetric Presentations Of Finite Groups And Related Topics, Samar Mikhail Kasouha

Electronic Theses, Projects, and Dissertations

A progenitor is an infinite semi-direct product of the form m∗n : N, where N ≤ Sn and m∗n : N is a free product of n copies of a cyclic group of order m. A progenitor of this type, in particular 2∗n : N, gives finite non-abelian simple groups and groups involving these, including alternating groups, classical groups, and the sporadic group. We have conducted a systematic search of finite homomorphic images of numerous progenitors. In this thesis we have presented original symmetric presentations of the sporadic simple groups, M12, J1 as homomorphic images of the progenitor 2∗12 : (2 ...


A Study Of Semiprime Arithmetic Sequences, Sam C. Uselton 2022 Belmont University

A Study Of Semiprime Arithmetic Sequences, Sam C. Uselton

Honors Theses

Semiprimes are positive integers of the form n = p*q where p and q are primes that are not necessarily distinct. Semiprimes can occur in arithmetic sequence, and the upper bound changes depending on the distance, d, between semiprimes in the sequence. The goal of this work is to present and prove theorems for different cases of semiprime arithmetic sequences, including d = 2k+1 where k is greater than or equal to 0 and d=2k where k is positive and not a multiple of 3, as well as a general case for finding upper bounds for varying distances. The ...


A Strange Attractor Of Primes, Alexander Hare 2022 Ohio Northern University

A Strange Attractor Of Primes, Alexander Hare

ONU Student Research Colloquium

The greatest prime factor sequences (GPF sequences), born at ONU in 2005, are integer sequences satisfying recursions in which every term is the greatest prime factor of a linear combination with positive integer coefficients of the preceding k terms (where k is the order of the sequence), possibly including a positive constant. The very first GPF sequence that was introduced satisfies the recursion x(n+1)=GPF(2*x(n)+1). In 2005 it was conjectured that no matter the seed, this particular GPF sequence enters the limit cycle (attractor) 3-7-5-11-23-47-19-13. In our current work, of a computational nature, we ...


An Introduction To Number Theory, J. J. P. Veerman 2022 Portland State University

An Introduction To Number Theory, J. J. P. Veerman

PDXOpen: Open Educational Resources

These notes are intended for a graduate course in Number Theory. No prior familiarity with number theory is assumed.

Chapters 1-14 represent almost 3 trimesters of the course. Eventually we intend to publish a full year (3 trimesters) course on number theory. The current content represents courses the author taught in the academic years 2020-2021 and 2021-2022.

It is a work in progress. If you have questions or comments, please contact Peter Veerman (veerman@pdx.edu).


Provably Weak Instances Of Plwe Revisited, Again, Katherine Mendel 2022 College of Saint Benedict/Saint John's University

Provably Weak Instances Of Plwe Revisited, Again, Katherine Mendel

CSBSJU Distinguished Thesis

Learning with Errors has emerged as a promising possibility for postquantum cryptography. Variants known as RLWE and PLWE have been shown to be more efficient, but the increased structure can leave them vulnerable to attacks for certain instantiations. This work aims to identify specific cases where proposed cryptographic schemes based on PLWE work particularly poorly under a specific attack.


Plane Figurate Number Proofs Without Words Explained With Pattern Blocks, Gunhan Caglayan 2022 New Jersey City University

Plane Figurate Number Proofs Without Words Explained With Pattern Blocks, Gunhan Caglayan

Journal of Humanistic Mathematics

This article focuses on an artistic interpretation of pattern block designs with primary focus on the connection between pattern blocks and plane figurate numbers. Through this interpretation, it tells the story behind a handful of proofs without words (PWWs) that are inspired by such pattern block designs.


Generating B-Nomial Numbers, Ji Young Choi 2022 Shippensburg University of Pennsylvania

Generating B-Nomial Numbers, Ji Young Choi

Communications on Number Theory and Combinatorial Theory

This paper presents three new ways to generate each type of b-nomial numbers: We develop ordinary generating functions, we find a whole new set of recurrence relations, and we identify each b-nomial number as a single binomial coefficient or as an alternating sum of products of two binomial coefficients.


An Integration Of Art And Mathematics, Henry Jaakola 2022 Central Washington University

An Integration Of Art And Mathematics, Henry Jaakola

Undergraduate Honors Theses

Mathematics and art are seemingly unrelated fields, requiring different skills and mindsets. Indeed, these disciplines may be difficult to understand for those not immersed in the field. Through art, math can be more relatable and understandable, and with math, art can be imbued with a different kind of order and structure. This project explores the intersection and integration of math and art, and culminates in a physical interdisciplinary product. Using the Padovan Sequence of numbers as a theoretical basis, two artworks are created with different media and designs, yielding unique results. Through these pieces, the order and beauty of number ...


A Proof Of A Generalization Of Niven's Theorem Using Algebraic Number Theory, Caroline Nunn 2021 University of Maryland, College Park

A Proof Of A Generalization Of Niven's Theorem Using Algebraic Number Theory, Caroline Nunn

Rose-Hulman Undergraduate Mathematics Journal

Niven’s theorem states that the sine, cosine, and tangent functions are rational for only a few rational multiples of π. Specifically, for angles θ that are rational multiples of π, the only rational values of sin(θ) and cos(θ) are 0, ±½, and ±1. For tangent, the only rational values are 0 and ±1. We present a proof of this fact, along with a generalization, using the structure of ideals in imaginary quadratic rings. We first show that the theorem holds for the tangent function using elementary properties of Gaussian integers, before extending the approach to other imaginary ...


A Study In Applications Of Continued Fractions, Karen Lynn Parrish 2021 California State University, San Bernardino

A Study In Applications Of Continued Fractions, Karen Lynn Parrish

Electronic Theses, Projects, and Dissertations

This is an expository study of continued fractions collecting ideas from several different sources including textbooks and journal articles. This study focuses on several applications of continued fractions from a variety of levels and fields of mathematics. Studies begin with looking at a number of properties that pertain to continued fractions and then move on to show how applications of continued fractions is relevant to high school level mathematics including approximating irrational numbers and developing new ideas for understanding and solving quadratics equations. Focus then continues to more advanced applications such as those used in the studies of number theory ...


Congruences Between Coefficients Of A Class Of Eta-Quotients And Their Applications To Combinatorics, Shashika Petta Mestrige 2021 Louisiana State University and Agricultural and Mechanical College

Congruences Between Coefficients Of A Class Of Eta-Quotients And Their Applications To Combinatorics, Shashika Petta Mestrige

LSU Doctoral Dissertations

Ramanujan in $1920$s discovered remarkable congruence properties of the partition function $p(n)$. Later, Watson and Atkin proved these congruences using the theory of modular forms. Atkin, Gordon, and Hughes extended these works to $k$-colored partition functions. In $2010$, Folsom-Kent-Ono and Boylan-Webb proved the congruences of $p(n)$ by studying a $\ell$-adic module associated with a certain sequence of modular functions which are related to $p(n)$.

Primary goal of this thesis is to generalize the work of Atkin, Gordon, Hughes, Folsom-Kent-Ono, and Boylan-Webb about the partition function to a larger class of partition functions. For this ...


Distribution Of The P-Torsion Of Jacobian Groups Of Regular Matroids, Sergio R. Zapata Ceballos 2021 The University of Western Ontairo

Distribution Of The P-Torsion Of Jacobian Groups Of Regular Matroids, Sergio R. Zapata Ceballos

Electronic Thesis and Dissertation Repository

Given a regular matroid $M$ and a map $\lambda\colon E(M)\to \N$, we construct a regular matroid $M_\lambda$. Then we study the distribution of the $p$-torsion of the Jacobian groups of the family $\{M_\lambda\}_{\lambda\in\N^{E(M)}}$. We approach the problem by parameterizing the Jacobian groups of this family with non-trivial $p$-torsion by the $\F_p$-rational points of the configuration hypersurface associated to $M$. In this way, we reduce the problem to counting points over finite fields. As a result, we obtain a closed formula for the proportion of these groups ...


Introduction To Discrete Mathematics: An Oer For Ma-471, Mathieu Sassolas 2021 CUNY Queensborough Community College

Introduction To Discrete Mathematics: An Oer For Ma-471, Mathieu Sassolas

Open Educational Resources

The first objective of this book is to define and discuss the meaning of truth in mathematics. We explore logics, both propositional and first-order , and the construction of proofs, both formally and human-targeted. Using the proof tools, this book then explores some very fundamental definitions of mathematics through set theory. This theory is then put in practice in several applications. The particular (but quite widespread) case of equivalence and order relations is studied with detail. Then we introduces sequences and proofs by induction, followed by number theory. Finally, a small introduction to combinatorics is given.


Genus Bounds For Some Dynatomic Modular Curves, Andrew W. Herring 2021 The University of Western Ontario

Genus Bounds For Some Dynatomic Modular Curves, Andrew W. Herring

Electronic Thesis and Dissertation Repository

We prove that for every $n \ge 10$ there are at most finitely many values $c \in \mathbb{Q} $ such that the quadratic polynomial $x^2 + c$ has a point $\alpha \in \mathbb{Q} $ of period $n$. We achieve this by proving that for these values of $n$, every $n$-th dynatomic modular curve has genus at least two.


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