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Full-Text Articles in Mathematics

An Enticing Study Of Prime Numbers Of The Shape 𝑝 = 𝑥^2 + 𝑦^2, Xiaona Zhou Dec 2020

An Enticing Study Of Prime Numbers Of The Shape 𝑝 = 𝑥^2 + 𝑦^2, Xiaona Zhou

Publications and Research

We will study and prove important results on primes of the shape 𝑥2 + 𝑦2 using number theoretic techniques. Our analysis involves maps, actions over sets, fixed points and involutions. This presentation is readily accessible to an advanced undergraduate student and lay the groundwork for future studies.


Get The News Out Loudly And Quickly: Modeling The Influence Of The Media On Limiting Infectious Disease, Anna Mummert, Howard Weiss Aug 2020

Get The News Out Loudly And Quickly: Modeling The Influence Of The Media On Limiting Infectious Disease, Anna Mummert, Howard Weiss

Mathematics Faculty Research

During outbreaks of infectious diseases with high morbidity and mortality, individuals closely follow media reports of the outbreak. Many will attempt to minimize contacts with other individuals in order to protect themselves from infection and possibly death. This process is called social distancing. Social distancing strategies include restricting socializing and travel, and using barrier protections. We use modeling to show that for short-term outbreaks, social distancing can have a large influence on reducing outbreak morbidity and mortality. In particular, public health agencies working together with the media can significantly reduce the severity of an outbreak by providing timely accounts of ...


Some Examples Of The Liouville Integrability Of The Banded Toda Flows, Zachary Youmans Aug 2020

Some Examples Of The Liouville Integrability Of The Banded Toda Flows, Zachary Youmans

All Graduate Theses and Dissertations

The Toda lattice is a famous integrable system studied by Toda in the 1960s. One can study the Toda lattice using a matrix representation of the system. Previous results have shown that this matrix of dimension n with 1 band and n−1 bands is Liouville integrable. In this paper, we lay the foundation for proving the general case of the Toda lattice, where we consider the matrix representation with dimension n and a partially filled lower triangular part. We call this the banded Toda flow. The main theorem is that the banded Toda flow up to dimension 10 is ...


"A Comparison Of Variable Selection Methods Using Bootstrap Samples From Environmental Metal Mixture Data", Paul-Yvann Djamen 4785403, Paul-Yvann Djamen Jul 2020

"A Comparison Of Variable Selection Methods Using Bootstrap Samples From Environmental Metal Mixture Data", Paul-Yvann Djamen 4785403, Paul-Yvann Djamen

Mathematics & Statistics ETDs

In this thesis, I studied a newly developed variable selection method SODA, and three customarily used variable selection methods: LASSO, Elastic net, and Random forest for environmental mixture data. The motivating datasets have neuro-developmental status as responses and metal measurements and demographic variables as covariates. The challenges for variable selections include (1) many measured metal concentrations are highly correlated, (2) there are many possible ways of modeling interactions among the metals, (3) the relationships between the outcomes and explanatory variables are possibly nonlinear, (4) the signal to noise ratio in the real data may be low. To compare these methods ...


At The Interface Of Algebra And Statistics, Tai-Danae Bradley Jun 2020

At The Interface Of Algebra And Statistics, Tai-Danae Bradley

All Dissertations, Theses, and Capstone Projects

This thesis takes inspiration from quantum physics to investigate mathematical structure that lies at the interface of algebra and statistics. The starting point is a passage from classical probability theory to quantum probability theory. The quantum version of a probability distribution is a density operator, the quantum version of marginalizing is an operation called the partial trace, and the quantum version of a marginal probability distribution is a reduced density operator. Every joint probability distribution on a finite set can be modeled as a rank one density operator. By applying the partial trace, we obtain reduced density operators whose diagonals ...


Quadratic Packing Polynomials On Sectors Of R2, Kaare S. Gjaldbaek Jun 2020

Quadratic Packing Polynomials On Sectors Of R2, Kaare S. Gjaldbaek

All Dissertations, Theses, and Capstone Projects

A result by Fueter-Pólya states that the only quadratic polynomials that bijectively map the integral lattice points of the first quadrant onto the non-negative integers are the two Cantor polynomials. We study the more general case of bijective mappings of quadratic polynomials from the lattice points of sectors defined as the convex hull of two rays emanating from the origin, one of which falls along the x-axis, the other being defined by some vector. The sector is considered rational or irrational according to whether this vector can be written with rational coordinates or not. We show that the existence of ...


Translation Distance And Fibered 3-Manifolds, Alexander J. Stas Jun 2020

Translation Distance And Fibered 3-Manifolds, Alexander J. Stas

All Dissertations, Theses, and Capstone Projects

A 3-manifold is said to be fibered if it is homeomorphic to a surface bundle over the circle. For a cusped, hyperbolic, fibered 3-manifold M, we study an invariant of the mapping class of a surface homeomorphism called the translation distance in the arc complex and its relation with essential surfaces in M. We prove that the translation distance of the monodromy of M can be bounded above by the Euler characteristic of an essential surface. For one-cusped, hyperbolic, fibered 3-manifolds, the monodromy can also be bounded above by a linear function of the genus of an essential surface.

We ...


Convexity And Curvature In Hierarchically Hyperbolic Spaces, Jacob Russell-Madonia Jun 2020

Convexity And Curvature In Hierarchically Hyperbolic Spaces, Jacob Russell-Madonia

All Dissertations, Theses, and Capstone Projects

Introduced by Behrstock, Hagen, and Sisto, hierarchically hyperbolic spaces axiomatized Masur and Minsky's powerful hierarchy machinery for the mapping class groups. The class of hierarchically hyperbolic spaces encompasses a number of important and seemingly distinct examples in geometric group theory including the mapping class group and Teichmueller space of a surface, virtually compact special groups, and the fundamental groups of 3-manifolds without Nil or Sol components. This generalization allows the geometry of all of these important examples to be studied simultaneously as well as providing a bridge for techniques from one area to be applied to another.

This thesis ...


Symmetric Rigidity For Circle Endomorphisms With Bounded Geometry And Their Dual Maps, John Adamski Jun 2020

Symmetric Rigidity For Circle Endomorphisms With Bounded Geometry And Their Dual Maps, John Adamski

All Dissertations, Theses, and Capstone Projects

Let $f$ be a circle endomorphism of degree $d\geq2$ that generates a sequence of Markov partitions that either has bounded nearby geometry and bounded geometry, or else just has bounded geometry, with respect to normalized Lebesgue measure. We define the dual symbolic space $\S^*$ and the dual circle endomorphism $f^*=\tilde{h}\circ f\circ{h}^{-1}$, which is topologically conjugate to $f$. We describe some properties of the topological conjugacy $\tilde{h}$. We also describe an algorithm for generating arbitrary circle endomorphisms $f$ with bounded geometry that preserve Lebesgue measure and their corresponding dual circle endomorphisms $f^*$ as ...


Averages And Nonvanishing Of Central Values Of Triple Product L-Functions Via The Relative Trace Formula, Bin Guan Jun 2020

Averages And Nonvanishing Of Central Values Of Triple Product L-Functions Via The Relative Trace Formula, Bin Guan

All Dissertations, Theses, and Capstone Projects

Harris and Kudla (2004) proved a conjecture of Jacquet, that the central value of a triple product L-function does not vanish if and only if there exists a quaternion algebra over which a period integral of three corresponding automorphic forms does not vanish. Moreover, Gross and Kudla (1992) established an explicit identity relating central L-values and period integrals (which are finite sums in their case), when the cusp forms are of prime levels and weight 2. Böcherer, Schulze-Pillot (1996) and Watson (2002) generalized this identity to more general levels and weights, and Ichino (2008) proved an adelic period formula which ...


Model Theory Of Groups And Monoids, Laura M. Lopez Cruz Jun 2020

Model Theory Of Groups And Monoids, Laura M. Lopez Cruz

All Dissertations, Theses, and Capstone Projects

We first show that arithmetic is bi-interpretable (with parameters) with the free monoid and with partially commutative monoids with trivial center. This bi-interpretability implies that these monoids have the QFA property and that finitely generated submonoids of these monoids are definable. Moreover, we show that any recursively enumerable language in a finite alphabet X with two or more generators is definable in the free monoid. We also show that for metabelian Baumslag-Solitar groups and for a family of metabelian restricted wreath products, the Diophantine Problem is decidable. That is, we provide an algorithm that decides whether or not a given ...


Dna Complexes Of One Bond-Edge Type, Andrew Tyler Lavengood-Ryan Jun 2020

Dna Complexes Of One Bond-Edge Type, Andrew Tyler Lavengood-Ryan

Electronic Theses, Projects, and Dissertations

DNA self-assembly is an important tool used in the building of nanostructures and targeted virotherapies. We use tools from graph theory and number theory to encode the biological process of DNA self-assembly. The principal component of this process is to examine collections of branched junction molecules, called pots, and study the types of structures that such pots can realize. In this thesis, we restrict our attention to pots which contain identical cohesive-ends, or a single bond-edge type, and we demonstrate the types and sizes of structures that can be built based on a single characteristic of the pot that is ...


Radio Graceful Labelling Of Graphs, Laxman Saha, Alamgir Rahaman Basunia May 2020

Radio Graceful Labelling Of Graphs, Laxman Saha, Alamgir Rahaman Basunia

Theory and Applications of Graphs

Radio labelling problem of graphs have their roots in communication problem known as \emph{Channel Assignment Problem}. For a simple connected graph $G=(V(G), E(G))$, a radio labeling is a mapping $f \colon V(G)\rightarrow \{0,1,2,\ldots\}$ such that $|f(u)-f(v)|\geq {\rm diam}(G)+1-d(u,v)$ for each pair of distinct vertices $u,v\in V(G)$, where $\rm{diam}(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$. A radio labeling $f$ of a graph $G$ is a \emph{radio graceful labeling ...


The Distribution Of The Greatest Common Divisor Of Elements In Quadratic Integer Rings, Asimina S. Hamakiotes May 2020

The Distribution Of The Greatest Common Divisor Of Elements In Quadratic Integer Rings, Asimina S. Hamakiotes

Student Theses

For a pair of quadratic integers n and m chosen randomly, uniformly, and independently from the set of quadratic integers of norm x or less, we calculate the probability that the greatest common divisor of (n,m) is k. We also calculate the expected norm of the greatest common divisor (n,m) as x tends to infinity, with explicit error terms. We determine the probability and expected norm of the greatest common divisor for quadratic integer rings that are unique factorization domains. We also outline a method to determine the probability and expected norm of the greatest common divisor of ...


Oscillations Via Excitable Cells, Derek Orr, Bard Ermentrout May 2020

Oscillations Via Excitable Cells, Derek Orr, Bard Ermentrout

Biology and Medicine Through Mathematics Conference

No abstract provided.


Modeling Vaccination Strategies To Control White-Nose Syndrome In Little Brown Bat Colonies, Eva Cornwell, David Elzinga, Shelby R. Stowe, Alex Capaldi May 2020

Modeling Vaccination Strategies To Control White-Nose Syndrome In Little Brown Bat Colonies, Eva Cornwell, David Elzinga, Shelby R. Stowe, Alex Capaldi

Biology and Medicine Through Mathematics Conference

No abstract provided.


Emergence, Mechanics, And Development: How Behavior And Geometry Underlie Cowrie Seashell Form, Michael G. Levy, Michael R. Deweese May 2020

Emergence, Mechanics, And Development: How Behavior And Geometry Underlie Cowrie Seashell Form, Michael G. Levy, Michael R. Deweese

Biology and Medicine Through Mathematics Conference

No abstract provided.


Domination Number Of Annulus Triangulations, Toshiki Abe, Junki Higa, Shin-Ichi Tokunaga May 2020

Domination Number Of Annulus Triangulations, Toshiki Abe, Junki Higa, Shin-Ichi Tokunaga

Theory and Applications of Graphs

An {\em annulus triangulation} $G$ is a 2-connected plane graph with two disjoint faces $f_1$ and $f_2$ such that every face other than $f_1$ and $f_2$ are triangular, and that every vertex of $G$ is contained in the boundary cycle of $f_1$ or $f_2$. In this paper, we prove that every annulus triangulation $G$ with $t$ vertices of degree 2 has a dominating set with cardinality at most $\lfloor \frac{|V(G)|+t+1}{4} \rfloor$ if $G$ is not isomorphic to the octahedron. In particular, this bound is best possible.


An In-Depth Look At P-Adic Numbers, Xiaona Zhou May 2020

An In-Depth Look At P-Adic Numbers, Xiaona Zhou

Publications and Research

In this study, we consider $p$-adic numbers. We will also study the $p$-adic norm representation of real number, which is defined as $\mathbb{Q}_p = \{\sum_{j=m}^{\infty }a_j p^j: a_j \in \mathbb{D}_p, m\in\mathbb{Z}, a_m\neq 0\} \cup \{0\}$, where $p$ is a prime number. We explore properties of the $p$-adics by using examples. In particular, we will show that $\sqrt{6},i \in \mathbb{Q}_5$ and $\sqrt{2} \in \mathbb{Q}_7 $. $p$-adic numbers have a wide range of applicationsnin fields such as string theory, quantum mechanics ...


Nonlocal Helmholtz Decompositions And Connections To Classical Counterparts, Andrew Haar, Petronela Radu May 2020

Nonlocal Helmholtz Decompositions And Connections To Classical Counterparts, Andrew Haar, Petronela Radu

UCARE Research Products

In recent years nonlocal models have been successfully introduced in a variety of applications, such as dynamic fracture, nonlocal diffusion, flocking, and image processing. Thus, the development of a nonlocal calculus theory, together with the study of nonlocal operators has become the focus of many theoretical investigations. Our work focuses on a Helmholtz decomposition in the nonlocal (integral) framework. In the classical (differential) setting the Helmholtz decomposition states that we can decompose a three dimensional vector field as a sum of an irrotational function and a solenoidal function. We will define new nonlocal gradient and curl operators that allow us ...


Introductory Calculus: Through The Lenses Of Covariation And Approximation, Caleb Huber May 2020

Introductory Calculus: Through The Lenses Of Covariation And Approximation, Caleb Huber

Graduate Student Portfolios, Papers, and Capstone Projects

Over the course of a year, I investigated reformative approaches to the teaching of calculus. My research revealed the substantial findings of two educators, Michael Oehrtman and Pat Thompson, and inspired me to design a course based upon two key ideas, covariation and approximation metaphors. Over a period of six weeks, I taught a course tailored around these ideas and documented student responses to both classroom activities and quizzes. Responses were organized intonarratives, covariation, rates of change, limits, and delta notation. Covariation with respect to rates of change was found to be incredibly complex, and students would often see it ...


Conjugation By Circulant Matrices In Non-Commutative Cryptography, Hannah B. Frederick May 2020

Conjugation By Circulant Matrices In Non-Commutative Cryptography, Hannah B. Frederick

Student Research Submissions

We introduce a procedure in which two trusted individuals, Alice and Bob, may share a secret matrix K from the non-abelian general linear group of matrices. In this procedure, the matrix K is concealed from an eavesdropper, Eve, by a sequence of conjugations by elements from a pre-determined abelian subgroup of the general linear group. We demonstrate that the group of invertible circulant matrices is one abelian subgroup that may be able to withstand a brute force attack. To analyze this we need a technique to determine the order of this group, and to do this we make use of ...


Remote Learning Assignment, Ryan Schneider May 2020

Remote Learning Assignment, Ryan Schneider

Lesson Plans

No abstract provided.


Lattice Paths In Diagonals And Dimensions, Freya Bennett May 2020

Lattice Paths In Diagonals And Dimensions, Freya Bennett

Honors Theses

The Lattice Paths of Combinatorics have been used in many applications, normally under the guise of a different name, due to its versatility in surface variety and specificity of answer. The Lattice Path’s of game development, in finding paths around barriers in mazes, is called Path Finder with the A∗ algorithms as its method of solving.


Combinatorial And Asymptotic Statistical Properties Of Partitions And Unimodal Sequences, Walter Mcfarland Bridges May 2020

Combinatorial And Asymptotic Statistical Properties Of Partitions And Unimodal Sequences, Walter Mcfarland Bridges

LSU Doctoral Dissertations

Our main results are asymptotic zero-one laws satisfied by the diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane; the upper boundary is called the shape. For various types of unimodal sequences, we show that, as the number of squares tends to infinity, 100% of shapes are near a certain curve---that is, there is a single limit shape. Similar phenomena have been well-studied for integer partitions, but several technical difficulties arise in the extension of such asymptotic statistical laws to unimodal sequences. We develop a widely applicable method for obtaining these limit ...


Singular Value Decomposition, Krystal Bonaccorso, Andrew Incognito May 2020

Singular Value Decomposition, Krystal Bonaccorso, Andrew Incognito

Honors Theses

A well-known theorem is Diagonalization, where one of the factors is a diagonal matrix. In this paper we will be describing a similar way to factor/decompose a non-square matrix. The key to both of these ways to factor is eigenvalues and eigenvectors.


Exploration Of Solvable Quintic Polynomials, Stephen Tivenan May 2020

Exploration Of Solvable Quintic Polynomials, Stephen Tivenan

Student Research Submissions

A polynomial f(x) with rational coefficients is solvable by radicals if its roots (in the field of complex numbers C) can be expressed in terms of its coefficients using the basic operations and radicals. It is known that for quintic polynomials there is no generic formula for the roots. That is, some quintic polynomials are solvable and some are not. In this paper, we address the mathematical theory that makes the formula for the roots of a polynomial. Primarily we will focus on our methodology of generating and examining quintic polynomials. In one case study, we will examine quintic ...


Anticommutative Associative Algebras And The Binomial Theorem, Ashley Scurlock May 2020

Anticommutative Associative Algebras And The Binomial Theorem, Ashley Scurlock

Student Research Submissions

We examine the binomial theorem and its components in a noncommutative associative algebra. Specifically, we examine the relationship between the 1-binomial and -1-binomial coefficient, as well as exploring alternatives for the exponential identity for non-commutative and anticommutative elements. Through this investigation we found that the 1-binomial can be mapped to the -1-binomial and that the relationship could be used to prove a defined alternative for the exponential identity for anticommutative elements.


The T, T*, Vi, Vni Model For Human Immunodeficiency Virus Type 1 (Hiv-1) Dynamics, Amy Creel May 2020

The T, T*, Vi, Vni Model For Human Immunodeficiency Virus Type 1 (Hiv-1) Dynamics, Amy Creel

Student Research Submissions

In this research project, I investigated deterministic and stochastic versions of a model for Human Immunodeficiency Virus Type 1 (HIV-1) dynamics. First, an analytical solution to a simplified version of the deterministic model is found. Then, numerical techniques are used to obtain an approximate solution to the deterministic model. Finally, a stochastic version of the model is discussed, and numerical methods are used to find an approximate solution to the stochastic system. These results demonstrate the behavior of HIV-1 in an infected patient under the effects of reverse transcriptase and protease inhibitors, and illustrate how the addition of randomness to ...


Nasa L'Space Mission Concept Academy, Anaily Lorenzo May 2020

Nasa L'Space Mission Concept Academy, Anaily Lorenzo

Research Day

NASA L'Space Mission Concept Academy is a virtual academy in which we had to complete a 12-week team mission concept. Our mission objective was to design a small mission concept that characterized the polar water ice on Earth's moon. Our mission had to fit within the mission concept constraints (mass, volume, and budget). Our spacecraft design and scientific payload had to reflect what can be afforded within these constraints. To complete this mission concept, our team relied heavily on math, engineering, geology, and chemistry.