Mathematics Commons^™

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 …

Identifying Important Parameters In The Inflammatory Process With A Mathematical Model Of Immune Cell Influx And Macrophage Polarization, Marcella Torres, Jing Wang, Paul J. Yannie, Shobha Ghosh, Rebecca A. Segal, Angela M. Reynolds

Department of Math & Statistics Faculty Publications

In an inflammatory setting, macrophages can be polarized to an inflammatory M1 phenotype or to an anti-inflammatory M2 phenotype, as well as existing on a spectrum between these two extremes. Dysfunction of this phenotypic switch can result in a population imbalance that leads to chronic wounds or disease due to unresolved inflammation. Therapeutic interventions that target macrophages have therefore been proposed and implemented in diseases that feature chronic inflammation such as diabetes mellitus and atherosclerosis. We have developed a model for the sequential influx of immune cells in the peritoneal cavity in response to a bacterial stimulus that includes macrophage …

Critical Fault-Detecting Time Evaluation In Software With Discrete Compound Poisson Models, Min-Hsiung Hsieh, Shuen-Lin Jeng, Paul Kvam

Department of Math & Statistics Faculty Publications

Software developers predict their product’s failure rate using reliability growth models that are typically based on nonhomogeneous Poisson (NHP) processes. In this article, we extend that practice to a nonhomogeneous discrete-compound Poisson process that allows for multiple faults of a system at the same time point. Along with traditional reliability metrics such as average number of failures in a time interval, we propose an alternative reliability index called critical fault-detecting time in order to provide more information for software managers making software quality evaluation and critical market policy decisions. We illustrate the significant potential for improved analysis using wireless failure …

Mean Value Theorems For Riemannian Manifolds Via The Obstacle Problem, Brian Benson, Ivan Blank, Jeremy Lecrone

Department of Math & Statistics Faculty Publications

We develop some of the basic theory for the obstacle problem on Riemannian manifolds, and we use it to establish a mean value theorem. Our mean value theorem works for a very wide class of Riemannian manifolds and has no weights at all within the integral.

Perturbed Obstacle Problems In Lipschitz Domains: Linear Stability And Nondegeneracy In Measure, Ivan Blank, Jeremy Lecrone

Department of Math & Statistics Faculty Publications

We consider the classical obstacle problem on bounded, connected Lipschitz domains D⊂Rⁿ. We derive quantitative bounds on the changes to contact sets under general perturbations to both the right-hand side and the boundary data for obstacle problems. In particular, we show that the Lebesgue measure of the symmetric difference between two contact sets is linearly comparable to the L¹-norm of perturbations in the data.

Finite Blaschke Products: A Survey, Stephan Ramon Garcia, Javad Mashreghi, William T. Ross

Department of Math & Statistics Faculty Publications

A finite Blaschke product is a product of finitely many automorphisms of the unit disk. This brief survey covers some of the main topics in the area, including characterizations of Blaschke products, approximation theorems, derivatives and residues of Blaschke products, geometric localization of zeros, and selected other topics.

A Probability Model For Strategic Bidding On The Price Is Right, Paul H. Kvam

Department of Math & Statistics Faculty Publications

The TV game show “The Price is Right” features a bidding auction called “Contestants’ Row” that rewards the player (out of 4) who bids closest to an item’s value, without overbidding. This paper considers ways in which players can maximize a winning probability based on the player's bidding order. We consider marginal strategies in which players assume opponents are bidding individually perceived values of the merchandise. Based on preceding bids of others, players have information available to create strategies. We consider conditional strategies in which players adjust bids knowing other players are using strategies. The last bidder has a large …

The Range And Valence Of A Real Smirnov Function, Timothy Ferguson, William T. Ross

Department of Math & Statistics Faculty Publications

We give a complete description of the possible ranges of real Smirnov functions (quotients of two bounded analytic functions on the open unit disk where the denominator is outer and such that the radial boundary values are real almost everywhere on the unit circle). Our techniques use the theory of unbounded symmetric Toeplitz operators, some general theory of unbounded symmetric operators, classical Hardy spaces, and an application of the uniformization theorem. In addition, we completely characterize the possible valences for these real Smirnov functions when the valence is finite. To do so we construct Riemann surfaces we call disk trees …

Optimal Weak Parallelogram Constants For L-P Spaces, Raymond Cheng, Javad Mashreghi, William T. Ross

Department of Math & Statistics Faculty Publications

Inspired by Clarkson's inequalities for L-p and continuing work from [5], this paper computes the optimal constant C in the weak parallelogram laws parallel to f + g parallel to(r )+ C parallel to f - g parallel to(r )= 2(r-1 )(parallel to f parallel to(r) + parallel to g parallel to(r)) for the L-p spaces, 1 < p < infinity.

Multipliers Between Model Spaces, Emmanuel Fricain, Andreas Hartmann, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper we examine the multipliers from one model space to another.

A Unified Inter-Host And In-Host Model Of Antibiotic Resistance And Infection Spread In A Hospital Ward, Lester Caudill, Barry Lawson

Department of Math & Statistics Faculty Publications

As the battle continues against hospital-acquired infections and the concurrent rise in antibiotic resistance among many of the major causative pathogens, there is a dire need to conduct controlled experiments, in order to compare proposed control strategies. However, cost, time, and ethical considerations make this evaluation strategy either impractical or impossible to implement with living patients. This paper presents a multi-scale model that offers promise as the basis for a tool to simulate these (and other) controlled experiments. This is a “unified” model in two important ways: (i) It combines inter-host and in-host dynamics into a single model, and (ii) …

Quantum Groups And Knot Invariants, Greg A. Hamilton

Honors Theses

Knot theory arguably holds claim to the title of the mathematical discipline with the most unusually diverse applications. A knot can be defined topologically as an embedding of S1 in R3. Naturally, two knots are topologically equivalent if one cannot be smoothly deformed into the other. The question of whether two knots are equivalent is highly non-trivial, and so the question of knot invariants used to distinguish knots has occupied knot theorists for over a century. Knot theory has found application in statistical mechanics [1], symbolic logic and set theory [2], quantum fi theory [3], quantum computing [4], etc. …

A Comprehensive Analysis Of Team Streakiness In Major League Baseball: 1962-2016, Paul H. Kvam, Zezhong Chen

Department of Math & Statistics Faculty Publications

A baseball team would be considered “streaky” if its record exhibits an unusually high number of consecutive wins or losses, compared to what might be expected if the team’s performance does not really depend on whether or not they won their previous game. If an average team in Major League Baseball (i.e., with a record of 81-81) is not streaky, we assume its win probability would be stable at around 50% for most games, outside of peculiar details of day-to-day outcomes, such as whether the game is home or away, who is the starting pitcher, and so on.

In this …

Multipliers Of Sequence Spaces, Raymond Cheng, Javad Mashreghi, William T. Ross

Department of Math & Statistics Faculty Publications

This paper is selective survey on the space l_A^p and its multipliers. It also includes some connections of multipliers to Birkhoff-James orthogonality.

Birkhoff–James Orthogonality And The Zeros Of An Analytic Function, Raymond Cheng, Javad Mashreghi, William T. Ross

Department of Math & Statistics Faculty Publications

Bounds are obtained for the zeros of an analytic function on a disk in terms of the Taylor coefficients of the function. These results are derived using the notion of Birkhoff–James orthogonality in the sequence space ℓ^p with p ∈ (1,∞), along with an associated Pythagorean theorem. It is shown that these methods are able to reproduce, and in some cases sharpen, some classical bounds for the roots of a polynomial.

Approaching Cauchy’S Theorem, Stephan Ramon Garcia, William T. Ross

Department of Math & Statistics Faculty Publications

We hope to initiate a discussion about various methods for introducing Cauchy’s Theorem. Although Cauchy’s Theorem is the fundamental theorem upon which complex analysis is based, there is no “standard approach.” The appropriate choice depends upon the prerequisites for the course and the level of rigor intended. Common methods include Green’s Theorem, Goursat’s Lemma, Leibniz’ Rule, and homotopy theory, each of which has its positives and negatives.

A New Almost Difference Set Construction, David Clayton

Honors Theses

This paper considers the appearance of almost difference sets in non-abelian groups. While numerous construction methods for these structures are known in abelian groups, little is known about ADSs in the case where the group elements do not commute. This paper presents a construction method for combining abelian difference sets into nonabelian almost difference sets, while also showing that at least one known almost difference set construction can be generalized to the nonabelian case.

Differential Equations Models Of Pathogen-Induced Single- And Multi-Organ Tissue Damage, Fiona Lynch

Honors Theses

The rise of antibiotic resistance has created a significant burden on healthcare systems around the world. Antibiotic resistance arises from the increased use of antibiotic drugs and antimicrobial agents, which kill susceptible bacterial strains, but have little effect on strains that have a mutation allowing them to survive antibiotic treatment, defined as “resistant” strains. With no non-resistant bacteria to compete for resources, the resistant bacteria thrives in this environment, continuing to reproduce and infect the host with an infection that does not respond to traditional antibiotic treatment.

A number of strategies have been proposed to tackle the problem of antibiotic …

Differential Privacy For Growing Databases, Gi Heung (Robin) Kim

Honors Theses

Differential privacy [DMNS06] is a strong definition of database privacy that provides indi- viduals in a database with the guarantee that any particular person’s information has very little effect on the output of any analysis of the overall database. In order for this type of analysis to be practical, it must simultaneously preserve privacy and utility, where utility refers to how well the analysis describes the contents of the database.

An analyst may additionally wish to evaluate how a database’s composition changes over time. Consider a company, for example, that accumulates data from a growing base of customers. This company …

Toward A Scientific Investigation Of Convolutional Neural Networks, Anh Tran

Honors Theses

This thesis does not assume the reader is familiar with artificial neural networks. However, to keep the thesis concise, it assumes the reader is familiar with the standard Machine Learning concepts of training set, validation set, and test set [1]. Their usage is intended to help ensure that the Machine Learning system can generalize its training from input examples used during its training to “similar” kinds of examples never used during its training.

The concept of a Convolutional Neural Network (CNN) is one of the most successful computational concepts today for solving image classification problems. However, CNNs are difficult and …

Evaluating Infection Prevention Strategies In Out-Patient Dialysis Units Using Agent-Based Modeling, Joanna R. Wares, Barry Lawson, Douglas Shemin, Erika D'Agata

Department of Math & Statistics Faculty Publications

Patients receiving chronic hemodialysis (CHD) are among the most vulnerable to infections caused by multidrug-resistant organisms (MDRO), which are associated with high rates of morbidity and mortality. Current guidelines to reduce transmission of MDRO in the out-patient dialysis unit are targeted at patients considered to be high-risk for transmitting these organisms: those with infected skin wounds not contained by a dressing, or those with fecal incontinence or uncontrolled diarrhea. Here, we hypothesize that targeting patients receiving antimicrobial treatment would more effectively reduce transmission and acquisition of MDRO. We also hypothesize that environmental contamination plays a role in the dissemination of …

Evaluating Infection Prevention Strategies In Out-Patient Dialysis Units Using Agent-Based Modeling, Joanna R. Wares, Barry Lawson, Douglas Shemin, Erika M. C. D'Agata

Department of Math & Statistics Faculty Publications

Patients receiving chronic hemodialysis (CHD) are among the most vulnerable to infections caused by multidrug-resistant organisms (MDRO), which are associated with high rates of morbidity and mortality. Current guidelines to reduce transmission of MDRO in the out-patient dialysis unit are targeted at patients considered to be high-risk for transmitting these organisms: those with infected skin wounds not contained by a dressing, or those with fecal incontinence or uncontrolled diarrhea. Here, we hypothesize that targeting patients receiving antimicrobial treatment would more effectively reduce transmission and acquisition of MDRO. We also hypothesize that environmental contamination plays a role in the dissemination of …

The Role Of Mathematical Modeling In Designing And Evaluating Antimicrobial Stewardship Programs, Lester Caudill, Joanna R. Wares

Department of Math & Statistics Faculty Publications

Antimicrobial agent effectiveness continues to be threatened by the rise and spread of pathogen strains that exhibit drug resistance. This challenge is most acute in healthcare facilities where the well-established connection between resistance and suboptimal antimicrobial use has prompted the creation of antimicrobial stewardship programs (ASPs). Mathematical models offer tremendous potential for serving as an alternative to controlled human experimentation for assessing the effectiveness of ASPs. Models can simulate controlled randomized experiments between groups of virtual patients, some treated with the ASP measure under investigation, and some without. By removing the limitations inherent in human experimentation, including health risks, study …

Partitioning Groups With Difference Sets, Rebecca Funke

Honors Theses

This thesis explores the use of difference sets to partition algebraic groups. Difference sets are a tool belonging to both group theory and combinatorics that provide symmetric properties that can be map into over mathematical fields such as design theory or coding theory. In my work, I will be taking algebraic groups and partitioning them into a subgroup and multiple McFarland difference sets. This partitioning can then be mapped to an association scheme. This bridge between difference sets and association schemes have important contributions to coding theory.

Real-Time Translation Of American Sign Language Using Wearable Technology, Jackson Taylor

Honors Theses

The goal of this work is to implement a real-time system using wearable technology for translating American Sign Language (ASL) gestures into audible form. This system could be used to facilitate conversations between individuals who do and do not communicate using ASL. We use as our source of input the Myo armband, an affordable commercially-available wearable technology equipped with on-board accelerometer, gyroscope, and electromyography sensors. We investigate the performance of two different classification algorithms in this context: linear discriminant analysis and k-Nearest Neighbors (k-NN) using various distance metrics. Using the k-NN classifier and windowed dynamic time …

An Inner-Outer Factorization In ℓp With Applications To Arma Processes, Raymond Cheng, William T. Ross

Department of Math & Statistics Faculty Publications

The following inner-outer type factorization is obtained for the sequence space ℓ^p: if the complex sequence F = (F₀, F₁,F₂,...) decays geometrically, then for an p sufficiently close to 2 there exists J and G in ℓ^p such that F = J * G; J is orthogonal in the Birkhoff-James sense to all of its forward shifts SJ, S²J, S³J, ...; J and F generate the same S-invariant subspace of ℓ^p; and G is a cyclic vector for S on ℓ …

Classifying Coloring Graphs, Julie Beier, Janet Fierson, Ruth Haas, Heather M. Russell, Kara Shavo

Department of Math & Statistics Faculty Publications

Given a graph G, its k-coloring graph is the graph whose vertex set is the proper k-colorings of the vertices of G with two k-colorings adjacent if they differ at exactly one vertex. In this paper, we consider the question: Which graphs can be coloring graphs? In other words, given a graph H, do there exist G and k such that H is the k-coloring graph of G? We will answer this question for several classes of graphs and discuss important obstructions to being a coloring graph involving order, girth, and induced subgraphs.

Concrete Examples Of H(B) Spaces, Emmanuel Fricain, Andreas Hartmann, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper we give an explicit description of de Branges-Rovnyak spaces H(b) when b is of the form q^r, where q is a rational outer function in the closed unit ball of H^∞ and r is a positive number.