On The Spectrum Of Quaquaversal Operators,
2023
CUNY Graduate Center
On The Spectrum Of Quaquaversal Operators, Josiah Sugarman
Dissertations, Theses, and Capstone Projects
In 1998 Charles Radin and John Conway introduced the Quaquaversal Tiling. A three dimensional hierarchical tiling with the property that the orientations of its tiles approach a uniform distribution faster than what is possible for hierarchical tilings in two dimensions. The distribution of orientations is controlled by the spectrum of a certain Hecke operator, which we refer to as the Quaquaversal Operator. For example, by showing that the largest eigenvalue has multiplicity equal to one, Charles Radin and John Conway showed that the orientations of this tiling approach a uniform distribution. In 2008, Bourgain and Gamburd showed that this operator …
Approaches To The Erdős–Straus Conjecture,
2023
CUNY New York City College of Technology
Approaches To The Erdős–Straus Conjecture, Ivan V. Morozov
Publications and Research
The Erdős–Straus conjecture, initially proposed in 1948 by Paul Erdős and Ernst G. Straus, asks whether the equation 4/n = 1/x + 1/y + 1/z is solvable for all n ∈ N and some x, y, z ∈ N. This problem touches on properties of Egyptian fractions, which had been used in ancient Egyptian mathematics. There exist many partial solutions, mainly in the form of arithmetic progressions and therefore residue classes. In this work we explore partial solutions and aim to expand them.
Algebraic And Integral Closure Of A Polynomial Ring In Its Power Series Ring,
2023
Clemson University
Algebraic And Integral Closure Of A Polynomial Ring In Its Power Series Ring, Joseph Swanson
All Dissertations
Let R be a domain. We look at the algebraic and integral closure of a polynomial ring, R[x], in its power series ring, R[[x]]. A power series α(x) ∈ R[[x]] is said to be an algebraic power series if there exists F (x, y) ∈ R[x][y] such that F (x, α(x)) = 0, where F (x, y) ̸ = 0. If F (x, y) is monic, then α(x) is said to be an integral power series. We characterize the units of algebraic and integral power series. We show that the only algebraic power series with infinite radii of convergence are …
Zeros Of Modular Forms,
2023
Clemson University
Zeros Of Modular Forms, Daozhou Zhu
All Dissertations
Let $E_k(z)$ be the normalized Eisenstein series of weight $k$ for the full modular group $\text{SL}(2, \mathbb{Z})$. It is conjectured that all the zeros of the weight $k+\ell$ cusp form $E_k(z)E_\ell(z)-E_{k+\ell}(z)$ in the standard fundamental domain lie on the boundary. Reitzes, Vulakh and Young \cite{Reitzes17} proved that this statement is true for sufficiently large $k$ and $\ell$. Xue and Zhu \cite{Xue} proved the cases when $\ell=4,6,8$ with $k\geq\ell$, all the zeros of $E_k(z)E_\ell(z)-E_{k+\ell}(z)$ lie on the arc $|z|=1$. For all $k\geq\ell\geq10$, we will use the same method as \cite{Reitzes17} to locate these zeros on the arc $|z|=1$, and improve the …
The Spectrum Of Nim-Values For Achievement Games For Generating Finite Groups,
2023
College of Saint Benedict/Saint John's University
The Spectrum Of Nim-Values For Achievement Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben
Mathematics Faculty Publications
We study an impartial achievement game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The game ends when the jointly selected elements generate the group. The last player able to make a move is the winner of the game. We prove that the spectrum of nim-values of these games is {0, 1, 2, 3, 4}. This positively answers two conjectures from a previous paper by the last two authors.
One Formula For Non-Prime Numbers: Motivations And Characteristics,
2023
Department of Basic Science, Faculty of Engineering, The British University in Egypt
One Formula For Non-Prime Numbers: Motivations And Characteristics, Mahmoud Mansour, Kamal Hassan Prof.
Basic Science Engineering
Primes are essential for computer encryption and cryptography, as they are fundamental units of whole numbers and are of the highest importance due to their mathematical qualities. However, identifying a pattern of primes is not easy. Thinking in a different way may get benefits, by considering the opposite side of the problem which means focusing on non-prime numbers. Recently, researchers introduced, the pattern of non-primes in two maximal sets while in this paper, non-primes are presented in one formula. Getting one-way formula for non-primes may pave the way for further applications based on the idea of primes.
Lagrange’S Study Of Wilson’S Theorem,
2023
Ursinus College
Lagrange’S Study Of Wilson’S Theorem, Carl Lienert
Number Theory
No abstract provided.
Lagrange’S Proof Of Wilson’S Theorem—And More!,
2023
Ursinus College
Lagrange’S Proof Of Wilson’S Theorem—And More!, Carl Lienert
Number Theory
No abstract provided.
Lagrange’S Proof Of The Converse Of Wilson’S Theorem,
2023
Ursinus College
Lagrange’S Proof Of The Converse Of Wilson’S Theorem, Carl Lienert
Number Theory
No abstract provided.
Lagrange’S Alternate Proof Of Wilson’S Theorem,
2023
Ursinus College
Lagrange’S Alternate Proof Of Wilson’S Theorem, Carl Lienert
Number Theory
No abstract provided.
Some Thoughts On The 3 × 3 Magic Square Of Squares Problem,
2023
Case Western Reserve University
Some Thoughts On The 3 × 3 Magic Square Of Squares Problem, Desmond Weisenberg
Rose-Hulman Undergraduate Mathematics Journal
A magic square is a square grid of numbers where each row, column, and long diagonal has the same sum (called the magic sum). An open problem popularized by Martin Gardner asks whether there exists a 3×3 magic square of distinct positive square numbers. In this paper, we expand on existing results about the prime factors of elements of such a square, and then provide a full list of the ways a prime factor could appear in one. We also suggest a separate possible computational approach based on the prime signature of the center entry of the square.
(R2050) Dual Quaternion Matrices And Matlab Applications,
2023
Beykent University
(R2050) Dual Quaternion Matrices And Matlab Applications, Kemal Gökhan Nalbant, Salim Yüce
Applications and Applied Mathematics: An International Journal (AAM)
There are many studies in the literature on real quaternions and real quaternion matrices. There are few studies in the literature on dual quaternions. Definitions of the matrices of dual quaternions used in this study will be given. The originality of our research, the set of dual quaternion matrix we studied, will be defined for the first time in this study, and its properties will be given. Moreover, this study is critical because it is an applied study related to dual quaternion matrices. It will be easier to solve examples with large matrix sizes with MATLAB. People who use different …
Decomposition Of Beatty And Complementary Sequences,
2023
Smith College
Decomposition Of Beatty And Complementary Sequences, Geremías Polanco
Mathematics Sciences: Faculty Publications
In this paper we express the difference of two complementary Beatty sequences, as the sum of two Beatty sequences closely related to them. In the process we introduce a new Algorithm that generalizes the well known Minimum Excluded algorithm and provides a method to generate combinatorially any pair of complementary Beatty sequences.
The Sharp Bounds Of A Quasi-Isometry Of P-Adic Numbers In A Subset Real Plane,
2023
Bellarmine University
The Sharp Bounds Of A Quasi-Isometry Of P-Adic Numbers In A Subset Real Plane, Kathleen Zopff
Undergraduate Theses
P-adic numbers are numbers valued by their divisibility by high powers of some prime, p. These numbers are an important concept in number theory that are used in major ideas such as the Reimann Hypothesis and Andrew Wiles’ proof of Fermat’s last theorem, and also have applications in cryptography. In this project, we will explore various visualizations of p-adic numbers. In particular, we will look at a mapping of p-adic numbers into the real plane which constructs a fractal similar to a Sierpinski p-gon. We discuss the properties of this map and give formulas for the sharp bounds of its …
Structure Of Extremal Unit Distance Graphs,
2023
University of South Carolina - Columbia
Structure Of Extremal Unit Distance Graphs, Kaylee Weatherspoon
Senior Theses
This thesis begins with a selective overview of problems in geometric graph theory, a rapidly evolving subfield of discrete mathematics. We then narrow our focus to the study of unit-distance graphs, Euclidean coloring problems, rigidity theory and the interplay among these topics. After expounding on the limitations we face when attempting to characterize finite, separable edge-maximal unit-distance graphs, we engage an interesting Diophantine problem arising in this endeavor. Finally, we present a novel subclass of finite, separable edge-maximal unit distance graphs obtained as part of the author's undergraduate research experience.
Unique Factorization In The Rings Of Integers Of Quadratic Fields: A Method Of Proof,
2023
Liberty University
Unique Factorization In The Rings Of Integers Of Quadratic Fields: A Method Of Proof, Zachary Warren
Senior Honors Theses
It is a well-known property of the integers, that given any nonzero a ∈ Z, where a is not a unit, we are able to write a as a unique product of prime numbers. This is because the Fundamental Theorem of Arithmetic (FTA) holds in the integers and guarantees (1) that such a factorization exists, and (2) that it is unique. As we look at other domains, however, specifically those of the form O(√D) = {a + b√D | a, b ∈ Z, D a negative, squarefree integer}, we find that …
Number Theoretic Arithmetic Functions And Dirichlet Series,
2023
CUNY New York City College of Technology
Number Theoretic Arithmetic Functions And Dirichlet Series, Ivan V. Morozov
Publications and Research
In this study, we will study number theoretic functions and their associated Dirichlet series. This study lay the foundation for deep research that has applications in cryptography and theoretical studies. Our work will expand known results and venture into the complex plane.
Ramanujan–Sato Series For 1/Π,
2023
The University of Texas Rio Grande Valley
Ramanujan–Sato Series For 1/Π, Timothy Huber, Daniel Schultz, Dongxi Ye
School of Mathematical & Statistical Sciences Faculty Publications
We compute Ramanujan–Sato series systematically in terms of Thompson series and their modular equations. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived.
Counting Elliptic Curves With A Cyclic M-Isogeny Over Q,
2023
Dartmouth College
Counting Elliptic Curves With A Cyclic M-Isogeny Over Q, Grant S. Molnar
Dartmouth College Ph.D Dissertations
Using methods from analytic number theory, for m > 5 and for m = 4, we obtain asymptotics with power-saving error terms for counts of elliptic curves with a cyclic m-isogeny up to quadratic twist over the rational numbers. For m > 5, we then apply a Tauberian theorem to achieve asymptotics with power saving error for counts of elliptic curves with a cyclic m-isogeny up to isomorphism over the rational numbers.
Euler Archive Spotlight,
2023
University of Washington, Tacoma
Euler Archive Spotlight, Erik R. Tou
Euleriana
A survey of two translations posted to the Euler Archive in 2022.
