An Enticing Study Of Prime Numbers Of The Shape 𝑝 = 𝑥^2 + 𝑦^2, 2020 CUNY New York City College of Technology

#### 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 𝑥^{2} + 𝑦^{2} 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, 2020 Marshall University

#### 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, 2020 Utah State University

#### 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", 2020 University Of New Mexico

#### "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, 2020 The Graduate Center, City University of New York

#### 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 ...

Convexity And Curvature In Hierarchically Hyperbolic Spaces, 2020 The Graduate Center, City University of New York

#### 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 ...

Averages And Nonvanishing Of Central Values Of Triple Product L-Functions Via The Relative Trace Formula, 2020 The Graduate Center, City University of New York

#### 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 ...

Symmetric Rigidity For Circle Endomorphisms With Bounded Geometry And Their Dual Maps, 2020 The Graduate Center, City University of New York

#### 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 ...

Translation Distance And Fibered 3-Manifolds, 2020 The Graduate Center, City University of New York

#### 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 ...

Quadratic Packing Polynomials On Sectors Of R2, 2020 The Graduate Center, City University of New York

#### 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 ...

Model Theory Of Groups And Monoids, 2020 The Graduate Center, City University of New York

#### 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 ...

Radio Graceful Labelling Of Graphs, 2020 Balurghat College

#### 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, 2020 CUNY Bernard M Baruch College

#### 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, 2020 University of Pittsburgh

#### 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, 2020 St. Olaf College

#### 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, 2020 University of California Berkeley

#### 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, 2020 Yokohama National University

#### 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, 2020 CUNY New York City College of Technology

#### 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, 2020 University of Nebraska - Lincoln

#### 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, 2020 University of Montana, Missoula

#### 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 ...