Mathematics Commons^™

An Introduction To Obstacle Problems, Calvin Reedy

Honors Theses

The obstacle problem can be used to predict the shape of an elastic membrane lying over an obstacle in a domain Ω. In this paper we introduce and motivate a mathematical formulation for this problem, and give an example to demonstrate the need to search for solutions in non-classical settings. We then introduce Sobolev spaces as the proper setting for solutions, and prove that unique solutions exist in W1,2(Ω).

Overdose Prevention Sites Placement Informed By Simulation, Jing Dong

Honors Theses

In Philadelphia, people are experiencing the greatest opioid crisis in a century. Plac- ing the Overdose Prevention Site (OPS) can alleviate this crisis. However, the journey to the successful launch of the first OPS in the USA is rough. It was first accused of having a collision with federal drug laws. While Safehouse won the lawsuit and the OPS was judged to be legal in 2020, other pressure rose afterward such as the against from the public and the COVID19, which delayed the plan to open the OPS. Without solid research on the effectiveness of OPS, we thought it is …

Computational Difficulty And Invariants Of The Snake Cube Puzzle, Adrian Negrea

Honors Theses

The snake cube is a popular puzzle that has been analyzed for its computational difficulty and shown to be NP-complete. Conceiving of the puzzle as a Hamiltonian path in an n x n x n graph, we offer a novel mathematical analysis by considering invariants of the puzzle. This allows us to determine necessary conditions for a particular snake cube to be solvable and eliminate a large class of possible puzzles as unsolvable. In particular, we establish upper and lower bounds on the possible number of straight components in solvable snake cube puzzles.

A Generalized Polar-Coordinate Integration Formula, Oscillatory Integral Techniques, And Applications To Convolution Powers Of Complex-Valued Functions On $\Mathbb{Z}^D$, Huan Q. Bui

Honors Theses

In this thesis, we consider a class of function on $\mathbb{R}^d$, called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on $\mathbb{Z}^d$. As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function $P$, we construct a Radon measure $\sigma_P$ on $S=\{\eta \in \mathbb{R}^d:P(\eta)=1\}$ which is invariant under the symmetry group of $P$. With this measure, we prove …

Counting Conjugacy Classes Of Elements Of Finite Order In Compact Exceptional Groups, Qidong He

Honors Theses

Given a compact exceptional group $G$ and $m,s\in\mathbb{N}$, let $N(G,m)$ be the number of conjugacy classes of elements of order $m$ in $G$, and $N(G,m,s)$ the number of such classes whose elements have $s$ distinct eigenvalues. In string theory, the problem of enumerating certain classes of vacua in the string landscape can be rephrased in terms of the study of these quantities. We develop unified combinatorial algorithms based on Burnside's Lemma that can be used to compute both quantities for each of the five compact exceptional groups. Thus, we provide a combinatorial, alternative method to that of Djoković and extend …

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.

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.

Boundary, Costs And Trade-Offs In Reserve Design Systems, Justus Hurd

Honors Theses

Due to limitations in funding and natural resources, it is infeasible to construct perfect reserve systems for large populations of critical species. For this project, our objective is to formulate a reserve design model that minimizes the distance between reserve sites meeting a threshold of biodiversity features subject to a species coverage constraints. Coupled with other spatial characteristics including reserve size and configuration, the boundary of a reserve system is of key importance. While positive area effects are gained when selecting additional sites, negative boundary length effects are also experienced. For example, it is costly to implement and maintain boundary …

Predictive Modeling Of Iphone 7 Charge Rates Using Least Squares Curve Fitting, Grace Cahill

Honors Theses

In a time where individuals depend on their cell phones, the need for a long lasting and quick charging battery life is imperative. As information regarding how long a battery can remained charged is highly advertised, there is no information regarding how long it would take for a dead phone battery to completely charge. This study determined the amount of time it will take an iPhone 7 to charge from 0% to 100% using the standard charging cable under four different charging conditions. The charge percentage was recorded every two minutes until it was fully charged with this process being …

Almost Difference Sets In 2-Groups, Xin Yutong

Honors Theses

Difference sets have been studied for decades due to their applications in digital communication, cryptography, algebra, and number theory. More recently, mathematicians have expanded their focus to the field of almost difference sets. Almost difference sets have similar functionalities with difference sets, yet with more potential of finding new constructions. In this paper I will introduce the definitions, properties, and applications of difference sets and almost difference sets, and discuss our effort and results in the exploration of almost difference sets in cyclic and non-cyclic groups.

Estimating Value-At-Risk Of An Unconventional Portfolio, Elizabeth N. Mejía-Ricart

Honors Theses

Since the 2008 financial crisis, interest rates and bond yields have been low all through the recovery and expansion that followed, and they are still low. As a result, more investors have been attracted to US equities, a space of possibly higher returns. However, these returns come with a potential downside: risk of loss. One of the methods to assess this potential downside is value-at-risk (VaR), which gained momentum in the late 1990s. At the time, the market risk amendment to the 1988 Basle Capital Accord required commercial banks with significant trading activities to put aside capital to cover market …

Computer-Assisted Coloring-Graph Generation And Structural Analysis, Wesley Su

Honors Theses

Graphs are a well studied construction in discrete math, with one of the most common areas of study being graph coloring. The graph coloring problem asks for a color to be assigned to each vertex in a graph such that no two adjacent vertices share a color. An assignment of k colors that meets these criteria is called a k-coloring. The coloring graph Ck(G) is defined as the graph where every vertex represents a valid k-coloring of graph G and edges exist between colorings that di↵er by one vertex. We call graph G the base graph of the k-coloring graph …

Internal Migration Of Foreign-Born In Us: Impacts Of Population Concentration And Risk Aversion, Thin Yee Mon Su

Honors Theses

Internal migration in the US has been declining since the 1990s and research has mostly focused on labor market dynamics and aging population to explain the migration trends. This paper analyzes migration patterns of foreign-born groups in the US from 2000 to 2019. Along with the migration determinants such as education and employment, the paper focuses on population concentration as a factor that shapes foreign-born decisions to relocate in the US. Population concertation is defined to be a measure of how geographically concentrated each foreign-born group is across the US. I find that the likelihood of migrating to another state …

Biasing Medial Axis Rapidly-Exploring Random Trees With Safe Hyperspheres, David Qin

Honors Theses

Motion planning is a challenging and widely researched problem in robotics. Motion planning algorithms aim to not only nd unobstructed paths, but also to construct paths with certain qualities, such as maximally avoiding obstacles to improve path safety. One such solution is a Rapidly-Exploring Random Tree (RRT) variant called Medial Axis RRT that generates the safest possible paths, but does so slowly. This paper introduces a RRT variant called Medial Axis Ball RRT (MABallRRT) that uses the concept of clearance -- a robot's distance from its nearest obstacle -- to efficiently construct a roadmap with safe paths. The safety of …

Fast Medial Axis Sampling For Use In Motion Planning, Hanglin Zhou

Honors Theses

Motion planning is a difficult but important problem in robotics. Research has tended toward approximations and randomized algorithms, like sampling-based planning. Probabilistic RoadMaps (PRMs) are one common sampling-based planning approach, but they lack safety guarantees. One main approach, Medial Axis PRM (MAPRM) addressed this deficiency by generating robot configurations as far away from the obstacles as possible, but it introduced an extensive computational burden. We present two techniques, Medial Axis Bridge and Medial Axis Spherical Step, to reduce the computational cost of sampling in MAPRM and additionally propose recycling previously computed clearance information to reduce the cost of connection in …

United States Suicide Analysis: 1999-2016, Malynn Clark

Honors Theses

The purpose of this thesis is to create information visualizations surrounding suicide trends from 1999-2016 in the United States. The original data was obtained from the Centers for Disease Control and Prevention’s Compressed Mortality Database. This database permits users to download several fields of information regarding deaths for the years given. Using this information, many graphs below show trends and patterns for suicide. One notable trend includes the higher proportion of male to female suicides for all categories explored including: age group, race, and metro/nonmetro status. The goal is to bring awareness and understanding surrounding the suicide epidemic in the …

Portfolio Optimization Methods: The Mean-Variance Approach And The Bayesian Approach, Hoang Nguyen

Honors Theses

This thesis is a discussion on the mean-variance approach to portfolio optimization and an introduction of the Bayesian approach, which is designed to solve certain limitations of the classical mean-variance analysis. The primary goal of portfolio optimization is to achieve the maximum return from investment given a certain level of risk. The mean-variance approach, introduced by Harry Markowitz, sought to solve this optimization problem by analyzing the means and variances of a certain collection of stocks. However, due to its simplicity, the mean-variance approach is subject to various limitations. In this paper, we seek to solve some of these limitations …

#Whyididntreport: Using Social Media As A Tool To Understand Why Sexual Assault Victims Do Not Report, Abby Garrett

Honors Theses

Sexual assault has gone largely under-reported, and social media movements, like #WhyIDidntReport, have brought great awareness to this issue. In order to take advantage of the large amounts of data the #WhyIDidntReport movement has generated, the study uses tweets to explore reasons why victims do not report their assault. The thesis cites current research on the topic of assault to generate a list of explanations victims use to describe their lack of reporting and compares the distributions with existing studies. We use a supervised learning technique to automatically categorize tweets into one of eight categories. This approach uses social sensing …

The Number Of Fixed Points Of And-Or Networks With Chain Topology, Lauren Geiser

Honors Theses

Boolean networks are sets of Boolean functions, which are functions that contain Boolean variables and the logical operators AND, OR, and NOT. In the simple case, the variables can be in one of two states—either 1 or 0, which can be interpreted in different ways such as ON or OFF, or TRUE or FALSE, depending on the application. Arranging model systems into Boolean functions, we can study steady states of these networks. This refers to the overall state of the dynamical system given an initial condition and another theoretical condition such as a subsequent point in time. Boolean networks have …

Topology Of Fractals, Amelia Pompilio

Honors Theses

No abstract provided.

Basis Reduction In Lattice Cryptography, Raj Kane

Honors Theses

We develop an understanding of lattices and their use in cryptography. We examine how reducing lattice bases can yield solutions to the Shortest Vector Problem and the Closest Vector Problem.

Positivity Among P-Partition Generating Functions Of Partially Ordered Sets, Nate Lesnevich

Honors Theses

We find necessary and separate sufficient conditions for the difference between two labeled partially ordered set's (poset) partition generating functions to be positive in the fundamental basis. We define the notion of a jump sequence for a poset and show how different conditions on the jump sequences of two posets are necessary for those posets to have an order relation in the fundamental basis. Our sufficient conditions are of two types. First, we show how manipulating a poset's Hasse diagram produces a poset that is greater according to the fundamental basis. Secondly, we also provide tools to explain posets that …

On A Generalization Of Lucas Numbers, Skylyn Olyvia Irby

Honors Theses

In this paper, we consider a generalization of Lucas numbers. Recall that Lucas numbers are the sequence of integers defined by the recurrence relation: L_n = L_{n−1} + L_{n−2} with the initial conditions L_1 = 1 and L_2 = 3(or L_0 = 1 and L_1 = 3 if the first subscript is zero). That is, the classical Lucas number sequence is 1, 3, 4, 7, 11, 18, .... The goal of the present paper is to study properties of certain generalizations of the Lucas sequence. In particular, we consider the following generalizations of the sequence: l_n = al_{n−1} + l_{n−2} …

6th-12th Grade Math Teachers And Their Experiences With The Mississippi College- And Career-Readiness Standards, Dorothy Reid

Honors Theses

This thesis identifies and describes 6th-12th grade math teachers and their experiences with the Mississippi College- and Career- Readiness Standards. There are two parts to this thesis: 1) a survey distributed to public school math teachers across the state and 2) the written thesis. In my thesis, I craft teacher narratives from the quantitative and qualitative results of the survey. Listening to the teachers’ narratives provides beneficial insights to the implementation of the MCCRS at the classroom level. Teachers have many different experiences. My thesis offers policy recommendations, based on the teacher narratives, to three levels of education: teachers, schools …

Primes In Arithmetical Progression, Edward C. Wessel

Honors Theses

This thesis will tackle Dirichlet’s Theorem on Primes in Arithmetical Progressions. The majority of information that follows below will stem from Tom M. Apostol’s Introduction to Analytical Number Theory. This is the main source of all definitions, theorems, and method. However, I would like to assure the reader that prior knowledge of neither the text nor analytical number theory in general is needed to understand the result. A rough background in Abstract Algebra and a moderate grasp on Complex and Real Analysis are more than sufficient. In fact, my project’s intent is to introduce Dirichlet’s ideas to the mathematics student …

Scheduling Problems, Aamir Kudai

Honors Theses

Manufacturing industry is growing exponentially. The need of using algorithms and computational techniques to enhance processes is increasing every day. Algorithms help us solve almost all kind of computational problems. Not only choosing the right algorithm for a problem is important but also optimizing its time and space efficiency is crucial. BorgWarner Transmission Systems located in Water Valley, Mississippi is one among the leading manufacturing companies. This paper will demonstrate a real-world audit scheduling problem happened at BorgWarner and the techniques used to solve it. A gentle introduction to some of the heuristic algorithms such as Genetic algorithm, Randomized algorithm, …

A Logistic Regression Analysis Of First-Time College Students’ Completion Rates At The University Of Southern Mississippi, Jesse Homer Robinson

Honors Theses

The demand for employees with a college degree is steadily on the rise in a plethora of competitive job markets throughout the United States. This increase in demand has aided in the increasing college enrollment rates throughout the country. However, unlike enrollment trends, the rate of college completion has not had the same fortunate rise.

The goal of this study is to research and compare differences among those first-time college students who completed college within four years, six years, or did not complete. The primary source for data in this study was the Office of Institutional Research at USM. Both …

An Explicit Formula For Dirichlet's L-Function, Shannon Michele Hyder

Honors Theses

The Riemann zeta function has a deep connection to the distribution of primes. In 1911 Landau proved that, an explicit formula where ρ = β + iγ denotes a complex zero of the zeta function and Λ(x) is an extension of the usual von Mangoldt function, so that Λ(x) = log p if x is a positive integral power of a prime p and Λ(x) = 0 for all other real values of x. Landau’s remarkable explicit formula lacks uniformity in x and therefore has limited applications to the theory of the zeta function. In 1993 Gonek proved a version …

Probabilistic Modeling Of Student Interactions During A Passing Period At The University Of Dayton, Allyson Pacifico

Honors Theses

The University of Dayton is composed of five colleges and schools: College of Arts and Sciences, School of Law, School of Business Administration, School of Education and Health Sciences, and School of Engineering. The University of Dayton is composed of about 11,000 students on campus who all have distinct class schedules and paths they take between their classes. In this study, I wanted to know the probability of meeting my friends with a different class schedule as I walk between classes. The data consisted of one to two students from each college, except for the School of Law, who documented …

Automating The Calculation Of Hilbert-Kunz Multiplicities And F-Signatures, Gabriel Johnson

Honors Theses

We describe an application written to automate a calculation for the mathematical research of Dr. Spiroff of University of Mississippi & Dr. Enescu of Georgia State University. This work represents a way to overcome the barriers of the mathematical calculations in obtaining theoretical results.