The Annual UNCG Regional Mathematics and Statistics Conference

Abstracts

This page will start to be updated in September and then about once a week, so do not worry if your name will not appear here right after you register.

Dimension Reduction Methods for Highly Correlated Data

Ekram Adam, North Carolina A&T State University, Greensboro, NC
mentored by Dr. Kossi Edoh

Abstract: In the era of big data, prediction modeling encounters high dimensionality and multicollinearity in predictor variables. These two issues are often simultaneously addressed by Principal Component Analysis (PCA), which uses an orthogonal transformation of the sample covariance matrix of predictors. Despite the mathematical clarity, PCA derives principal components independent of the outcome variable. To overcome this issue, Sufficient Dimension Reduction (SDR) was proposed, which utilizes the information of outcome variable for calculating principal components via the inverse mean function, E(X|Y=y). Our objective was to investigate the prediction performance of SDR using both simulated and real data sets. The SDR method was compared to PCA and other regression methods regarding the mean squared prediction error. In simulation studies, we considered various types of multicollinearity structures such as independent, autoregressive, constant, and block-wise covariance structures. Results of the study showed that the prediction model performance relies highly on covariance structures, sample sizes, and sparsity of parameter space. The methods discussed were applied to the predictive model of several types of crime with 125 predictors. This article studies a potential utility of SDR in prediction modeling to achieve a dimension reduction.

Mathematical Modeling of HER2 Signaling Pathway: Implications for Breast Cancer Therapy

Sameed Ahmed, University of South Carolina, Columbia, SC
mentored by Dr. Xifeng Liu

Abstract: The cancer stem cell hypothesis states that there is a small subset of tumor cells, called cancer stem cells (CSCs), that are responsible for the proliferation and resistance to therapy of tumors. CSCs have the ability to self-renew and differentiate to form the nontumorigenic cells found in tumors. Over-expression of human epidermal growth factor receptor 2 (HER2) plays a role in regulation of CSC population in breast cancer. Current cancer therapy includes drugs that block HER2, however, patients can develop anti-HER2 drug resistance. Downstream of HER2 is nuclear factor κB (NFκB). The aberrant regulation of NFκB leads to cancer growth, which makes it a promising target for cancer therapy, especially for those who have developed resistance to anti-HER2 treatment. Our collaborator's lab has discovered that interleukin-1 (IL1), which is downstream of HER2, is responsible for NFκB activation, thus making it a potential target for cancer treatment. We have developed a mathematical model to represent the dynamics of this signaling pathway. Simulations of the model match experimental results, confirming the new pathway. We will use the mathematical model to make predictions for different scenarios, and it will be updated and expanded based upon new experiments.

Completing Partial Latin Squares Arising from Latin Arrays

Kevin Akers, Concord University, Athens, WV
mentored by Dr. Michael Schroeder

Abstract: Completing partial Latin squares has been studied since the 1940s. Recently, Kuhl and Schroeder looked at a specific problem where an r x r Latin array A is copied n times down the diagonal of a blank array. Call this partial Latin square nA. In 2015, they proved that if n > r, then nA is completable for any r x r Latin array A, and if n < r, there exists an r x r Latin array A such that nA is not completable. They failed to resolve the case when n = r. At the Summer 2016 Marshall University REU, we improved upon their techniques. In this work, we show that rA is completable for every r x r Latin array A.

Using Curve-Shortening Flow to Solve Dido's Problem

Davis Atkinson, NC State University, Raleigh, NC
mentored by Dr. Andrew Cooper

Abstract: Dido�s problem is to maximize the area enclosed by a specified length of curve, when part of the curve is fixed. We approach Dido�s problem using a partial differential equation known as curve-shortening flow, which moves each point of the curve in the direction of curvature. Gage and Hamilton have shown that curve-shortening flow solves the related isoperimetric problem. We conjecture that curve-shortening flow will provide us with the optimal curve to solve Dido�s problem as well.

Proper Connection Number of the Direct Product of Cycles

Hailey Belleperche and Jillian Allen, Virginia Commonwealth University, Richmond, VA
mentored by Dr. Ghidewon Abay-Asmerom, Dr. Moa Apagodu and Dr. Dewey Taylor

Abstract: The proper connection number of a graph is the least integer k for which the graph has an edge coloring with k colors, with the property that any two vertices are joined by a properly colored path. We investigate the proper connection number of the direct product of cycles. In particular, we show that the proper connection number of a direct product of cycles is 2.

Stereoscopic Skin Segmentation for Musical Conducting Recognition

Love Bennett, Concord University, Athens, WV
mentored by Lonnie Bowe

Abstract: A variety of skin segmentation techniques were surveyed, with one being implemented using a ZED Stereolabs stereoscopic camera for hand tracking and musical gesture recognition.

Using Gompertzian Growth to Model Chronic Myeloid Leukemia and Its Treatment

Lindsay Bradley, Winthrop University, Rock Hill, SC
mentored by Dr. Kristen Abernathy and Dr. Zach Abernathy

Abstract: Chronic Myeloid Leukemia (CML) is a prevalent type of cancer where the presence of cancer stem cells is well studied. In this talk, we modify existing Gompertzian growth models to study the dynamics of CML and the effects of treatment on CML. In the absence of treatment, we demonstrate that the cure state is always unstable. We then present conditions on treatment parameters to guarantee a locally stable cure state. We conclude with numerical simulations and remaining open questions.

Spider Monkeys in Fragmented Landscapes: A Discrete Mathematical Model

Matthew Buhr, University of South Dakota, Vermillion, SD
mentored by Dr. Jose Flores

Abstract: We implement a discrete model to study the population dynamics of Ateles hybridus in a single patch. Since data suggest a population level of under one thousand inhabitants, a discrete model is the most suitable. The different patches resemble a landscape which has been fragmented over the past few years particularly in Colombia. Given the population, the population is divided into categories by sex: male and female. Furthermore, the population is broken down so that the female population is broken into subgroups: adult females and young females, to account for an age of reproductive ability. Additionally, females are the dispersing sex in spider monkeys. In our population, a young female acquires its reproductive ability around their seventh year, at which point they disperse from their group or family in search of another group where they will spend their reproductive life. This activity will require the adult females to select a target patch other than their original one, and successfully cover the distance between their current patch and their selected one. We also consider the possibility that their new patch has an unfit operational sex ratio in which some females who make a poor decision on a new patch may never reproduce. An additional factor includes a target patch that is close to its carrying capacity in which the female could have a considerable amount of trouble staying alive, hence having to make a second decision. Because of the given variables in female dispersal throughout the patches in question, we consider three ecological process. These are the natural per-capita birth and death rate, the average time for females to reach reproductive ability, and eventually, a forced migration process at time of female adulthood. We analyze equilibria, and modify parameters to simulate different initial conditions in a real-life model to conclude how to best handle spider monkey populations.

(q,t) Symmetry in Macdonald Polynomials

Jacob Coleman, West Virginia Wesleyan College, Buckhannon, WV
mentored by Dr. Elizabeth Niese

Abstract: We examine (q,t) symmetry in the Macdonald polynomial Fl(q,t) using combinatorial methods. For hooks, we show full (q,t) symmetry using existing bijections attributable to Carlitz. We then present two maps between subsets of the standard fillings of a Ferrers diagram of an integer partition l and the set of l-sub ballot words to obtain (q,t) symmetry for some shapes. Our first bijection maps fillings with zero comajor index into the l-sub ballot words, encoding information about the inversion number of the filling. The second function maps between certain integer partitions with zero inversions and the l-sub ballot words. Finally, we present some conjectures to guide future work on this topic.

Heart Attack Patients: Predicting Life or Death After One Year Using Data Mining Classification Techniques

Jessica Cook, Morgan Dozier, and Ryan Bostic, UNC Wilmington, Wilmington, NC
mentored by Dr. Cuixian Chen

Abstract: In the United States, a heart attack occurs every 20 seconds. Every minute a fatality from a heart attack occurs. A group of patients from Miami, Florida who had suffered from a heart attack were analyzed using classification data mining techniques to identify whether or not a patient was going to be dead or alive one year after suffering from the heart attack. We had nine prediction variables to provide a deeper understanding of the way the patient�s heart is/was functioning. Classification methods of Logistic Regression, Linear Discriminant Analysis, Quadratic Discriminant Analysis, and K-Nearest Neighbors. Methods of cross validation were applied to improve model prediction accuracy and there are plans to move forward using methods of bagging and random forest. Our preliminary studies show that Quadratic Discriminant Analysis will produce the highest accuracy of 96% with 5-fold cross validation. From our data analysis, we can predict for future heart attack patients, with sufficient accuracy, whether or not said patient will be living or passed one year after suffering from the heart attack.

A Mathematical Analysis on the Transmission Dynamics of Neisseria Gonorrhoeae

Christine Craib, UNC Wilmington, Wilmington, NC
mentored by Dr. Wei Feng

Abstract: In this project, we analyze an epidemiological model describing the transmission of gonorrhea, with a core sexual activity class and a noncore sexual activity class. We discuss the behavior of the model around the two equilibrium points, a disease-free equilibrium and a coexistence equilibrium. The focus of the project is to identify equilibrium points, analyze the stability of these points, and discuss the results in terms of the epidemiological model. Ultimately, the goal of the project is to find conditions of an endemic state, and the conditions that ensure the eradication of gonorrhea.

Yellow Fever Vaccination and The Theory of Games

Chasity Dorsett and Hakimah Smith, Bennett College, NC
mentored by Dr. Hyunju Oh and Dr. Jan Rychtar

Abstract: Yellow Fever is a hemorrhagic disease transmitted to humans by infected mosquitoes. Yellow Fever infects about 200,000 humans per year, and the Center for Disease Control and Prevention recommends the vaccine for humans aged 9 months and older who are traveling to or living areas at risk for yellow fever virus transmission in South America and Africa. We constructed a schematic diagram of the yellow fever model with the presence of a vaccine, whose protection may decrease over time. We derived a threshold vaccination rate. We showed that endemic exists when the mortality rate of the mosquitoes is less than the given threshold; the vaccination rate is greater that the given threshold and recovery rate is greater than given threshold. In contrast, when the reproductive number is greater than one and the vaccination rate greater than a given threshold, the disease will die off. We will use this analysis to plan vaccination strategies. Our goal is to determine when humans should receive the yellow fever vaccination and when being vaccinated is the best choice.

k-proper Connection Number of the Direct Product of Cycles

Idalmy Escobar and Hermella Tessema, Virginia Commonwealth University, Richmond, VA
mentored by Dr. Ghidewon Abay-Asmerom, Dr. Moa Apagodu and Dr. Dewey Taylor

Abstract: An edge-colored graph is k-proper connected if every pair of vertices is connected by k internally pairwise vertex-disjoint properly colored paths. The k-proper connection number of a connected graph is the smallest number of colors that are needed to color the edges of the graph in order to make it k-proper connected. We determine the k-proper connection number for a direct product of cycles.

On Congruence Subgroups of Hilbert Modular Groups of Real Quadratic number Fields

Lance Everhart, University of North Carolina at Greensboro, Greensboro, NC
mentored by Dr. Sebastian Pauli

Abstract: In this report the beginnings of the computations and tabulations, using MAGMA, of the congruence subgroups of $\PSL_2(\OO_d)$ where $\OO_d$ is the maximal order of $\QQ(\sqrt{d})$. We will discuss methods of obtaining generators and values of invariants of the congruence subgroups. For our tables, we focus primarily on the genus, euler number and signature of these subgroups. In this talk we will focus mainly on the creation of generators of the general quadratic Hilbert modular groups.

Propensity Score Matching Analysis for Measuring the Influence of Mothers� Education Levels on Children�s Academic Performance

Christian Felton, North Carolina A&T State University, Greensboro, NC
mentored by Dr. Seong-Tae Kim

Abstract: One in three children faces a father absence in the United States, which causes serious familial and societal issues. In the presence of father absence, mother�s attitude toward children�s education is critical for their academic performance. This parental attitude is often highly associated with her education level. The objective of this study is to investigate if mother�s education level affects children�s academic performance using a nationally representative sample data. Propensity score matching methods are applied to a subset of the database of the Early Childhood Longitudinal Program, Kindergarten Class of 2010-11 (ECLS-K:2011) to overcome the selection bias issue in the observational study. We compare the influence of mothers� high and low education levels on their first-grade students� test scores in reading and mathematics. The propensity score matching results such as average treatment effect and significance test are compared to those from ordinary least square regression. This study would help educators properly support students whose mothers have a low education level.

Elucidation of Cysteine Richness in Arabidopsis thaliana Genome Using Exact Distribution of Clump Statistics

Dustin Ford, Philander Smith College, Little Rock, AK
mentored by Dr. Jocelyn A. Moore and Dr. Deidra A. Coleman

Abstract: Arabidopsis thaliana is a model plant with a fully sequenced genome, small size and rapid growth rate. A. thaliana and other plants are immobile leaving them vulnerable to a multitude of predators, such as fungi, insects, bacteria, and animals. Plants have to rely on innate defense mechanisms in the forms of peptides and proteins, for example defensins, proteins rich in cysteine and disulfide bridges. A potential mathematical measure of cysteine richness emerged from recent methodology used to establish the over-abundance of a motif in a sequence. This measure specifically is the number of clumps of the motif. In this work, the observed number of clumps of cysteine within A. thaliana is obtained, and then the probability of outcomes or more extreme outcomes are obtain using the algorithm to compute the distribution of the number of clumps. With this, we conclude the statistical significance of the observed number of clumps through performing appropriate hypothesis testing and consequently establishing cysteine richness. In future work, cysteine richness will be confirmed using an alternative measure, namely coverage of the number of clumps, which has also proven to be a useful measure of over-abundance

A Study of Identifying Socioeconomic and Health-Related Factors and Their Relative Importance with HRQOL among U.S. Adults

Ashley Fowler, North Carolina A&T University, Greensboro, NC
mentored by Dr. Seong-Tae Kim

Abstract: Personal perception of health status is important because it is not only a predictor of morbidity and mortality but it is also a component of the professed need for health care services. Health-Related Quality of Life (HRQOL) is a multi-dimensional concept which is related to the physical, mental, emotional, and social functioning domains. HRQOL critically depends on socioeconomic status and health conditions. The objective of this study aims to identify socioeconomic factors and health-related factors associated with HRQOL and their relative importance using a population-based sample of adults in the United States. We considered four self-reported HRQOL variables � overall health status, physical health status, mental health status, and activity limitations � with socioeconomic and health-related factors for approximately 460,000 noninstitutionalized adults from the 2014 Behavioral Risk Factor Surveillance System (BRFSS) data. We used logistic regression along with subset and relative weight analysis. After adjusting for sociodemographic variables such as age, gender, and race, the HRQOL variables of overall and physical health status and activity limitations are most significantly affected by employment status, income, and internet accessibility, as well as arthritis, depression, and exercise. We also observed that mental health status is highly associated with smoking, alcohol, and marital status in addition to depressive disorder. This study identified both socioeconomic and health-related factors which are statistically significantly associated with HRQOL variables. Not only would our study contribute to public health strategies for health promotion activities, but also provide a potential guideline for healthier lifestyles.

Simple Solutions for Missing Data Problem Big Data Domain

Nitin Gaikwad, University of North Carolina at Greensboro, Greensboto, NC
mentored by Dr. Shan Suthaharan

Abstract: In big data analytics, missing data in datasets severely affect the performance of classification algorithms. The classification techniques applied to big data environment require fast and efficient statistical approaches to alleviate the problem of missing data. Although the simple approaches based on mean, median, and mode statistics of data have been extensively studied in statistical theory, the big data requirements encourage researchers revisit and study them further for big data problem domain. In our research, we have used mammographic-masses dataset to study missing-data problem, understand its impact on machine learning techniques, and replace the missing data using the three simple statistical approaches, mean, median and mode to make the dataset complete. We have selected and studied the performance of random forest technique using the complete data set to understand how these simple solutions have supported the classification algorithm perform better. We will report our simulation results, and discuss the findings and the procedures used.

Analysis of Individual Greensboro Police Officers' Stopping Patterns Using Propensity Scores

Ivanti Galloway, Wake Forrest University, Winston Salem, NC
mentored by Dr. Jan Rychtar

Abstract: In October 2015, a New York Times article highlighted a disparity between the proportion of black versus non-black drivers pulled over in traffic stops in Greensboro, NC. In response to these allegations, we examined 563 individual officers in the Greensboro Police Department (GPD) to determine if the driver's race played a role in their traffic stops. We used propensity score weighting, which compared an officer's particular stops to similar stops made by peers. This method was based on RAND Corporation's study for the Cincinnati Police Department. For our purposes, two stops were similar if they occurred for the same reason at a similar time of day and at a similar location in town. After applying our propensity score weights, we conducted a false discovery rate analysis. In this analysis, 10 out of the 563 officers had z-statistics that indicated racial bias against black drivers. These results are based off of 295,228 stops that occurred between January 1, 2009 and September 30, 2015.

Positive Impact of the STS Model in the Operating Room for Cesarean Cases

Monika Goel, University of North Carolina at Greensboro, Greensboro, NC
coauthored by Wynn Fussell, CNRA, MSN, ANP; Jennifer L. Fencl, DNP, RN, CNS, CNOR; Jenny Clapp, MSN, RNC-OB, CNS Cone Health System
mentored by Dr. Sat Gupta

Abstract: Many organizations have implemented Skin to Skin (STS) in their labor and delivery units as a successful model of care, demonstrating positive outcomes for the mother and newborn. Positive outcomes discussed in the literature include early neonatal temperature and blood sugar regulation, enhanced mother/baby bonding etc. Although STS is well documented for the labor and delivery setting, there is limited literature discussing STS in the operating room for cesarean birth. The purpose of this study is to examine STS in the operating room with a focus on intravenous (IV) anti-anxiety medication or narcotic in response to and treatment of maternal anxiety. A retrospective chart review assessing two data points in time; pre STS implementation (n=100) and post STS implementation (n=100) was conducted. This study is just the first step in understanding the impacts of the STS model in the operating room. Preliminary results support the STS model of care on the operating room for C-sections, yielding several statistically significant positive outcomes for both the mother and the newborn infant. Further research needs to be conducted to explore and define the impact of STS in the operating room.

Patrol Zone Realignment for the Huntington, WV Police Department

Elizabeth Hance, Marshall University, Huntington, WV
mentored by Dr. Michael Schroeder

Abstract: The Huntington Police Department patrol zones have, due to changes in crime distribution and the makeup of the department, become ineffective over the last 14 years. We first analyzed data and looked for trends from 2004 to 2014. To create better zones, we began by creating naive maps, or maps drawn by hand using human intuition and data analysis. These maps were then optimized using a mathematical technique called gradient descent. The naive maps were tested, and optimized maps were found by measuring them using a fitness function. We mimic the overall patrol patterns of an officer using a discrete event simulation, which measures the workload distribution and response time. The effectiveness of each plan were further evaluated, and in this talk we present our findings.

Machine Learning Methods for Syncope Data Analysis

Joey Hart, NC State University, Raleigh, NC
mentored by Dr. Pierre Gremaud

Abstract: Syncope is a sudden loss of consciousness with loss of postural tone and spontaneous recovery; it is a common condition, albeit one that is challenging to accurately diagnose. Uncertainties about the triggering mechanisms and their underlying pathophysiology have led to various classifications of patients exhibiting this symptom. In this talk an analysis of syncope types using machine learning is presented. We hypothesize that syncope types can be characterized by analyzing blood pressure and heart rate time series data obtained from the head-up tilt test procedure. Our proposed method is applied to clinical data from 157 subjects; each subject was identified by an expert as being either healthy or suffering from one of three conditions: cardioinhibitory syncope, vasodepressor syncope and postural orthostatic tachycardia. Clustering confirms the three disease groups and identifies two distinct subgroups within the healthy controls. These two distinct healthy subgroups raise new medical questions. Additionally, the successful clustering of the three disease groups validates the merit of machine learning as a tool for future syncope research and provides direction for this ongoing work.

Recreation of Axelrod: An Evolution of Cooperation

Victoria Hayes, UNCG, Greensboro, NC
mentored by Dr. Jan Rychtar

Abstract: The Iterated Prisoner�s Dilemma is a commonly studied game in Game Theory. Many real life situations can be modeled by such a game. Robert Axelrod implemented a project that analyzed the Prisoner�s Dilemma through a computer tournament. Axelrod wanted to determine the best strategy to implement in such interactions. Various strategies competed in his tournament, and as a result, Axelrod described certain characterisitcs that must be present for a strategy to be successful. We examine the results of Axelrod�s first computer tournament and discuss the qualities needed to make a successful strategy.

College Football: Predictors Winning Seasons in the Power 5 Conferences

William Hayes, Brittany Palmer, and Sean Vanhille, UNC Wilmington, Wilmington, NC
mentored by Dr. Tracy Chen

Abstract: Data mining techniques apply statistical methods, like classification, to assist in the analysis and interpretation of large data sets. The computer program R was specifically designed to assist in statistical computations and is a powerful tool to apply to large data sets. Utilizing R and applying principles of data mining, can results from previous seasons predict the next season�s winning percentage in athletics? Specifically, classification techniques like logistic regression, linear discriminant analysis, quadratic discriminant analysis, and K-Nearest Neighbors were applied to a data set containing the win percentages of the NCAA football Power 5 Conferences from years 2005-2013 to explore which method best predicted a winning or losing season in 2013. Cross validation techniques were utilized to improve prediction model accuracy with plans of applying further techniques to improve accuracy such as bagging or random forests. Preliminary analyses in R suggest logistic regression as providing the most accurate predictive model. Since athletic events bring in millions of dollars to organizations, the ability to create accurate predictive models of success by identifying the most important variables for that organization to focus upon will assist in efforts to maximize both performance and profits.

Minimizing Energy, the Geometry of Tension

Lizzy Huang, Duke University, Durham, NC
mentored by Dr. Mark Stern

Abstract: Many questions in topology and physics can be expressed in terms of finding a function $f$ between a curved space $M$ (the domain) and another curved space $N$ (the target) which minimizes a natural energy functional: $\int_M |df|^2 dx$. Functions that minimize this energy are called harmonic maps. One method to obtain a harmonic map is to consider a family of maps $f_t$ which follow a path of 'steepest descent' - akin to rolling down a hill when the energy is given by the height, then discuss the relationship between the limiting map of $f_t$ and the harmonic map. In this talk, I will discuss a modification of this approach in which an unbounded potential energy is added to the total energy. I will discuss the limiting maps in cases of special significance to topology.

Modeling Nucleosomal DNA in Living Yeast

Caitlin Hult, UNC, Chapel Hill, NC
mentored by Dr. Greg Forest

Abstract: The genome in living yeast cells is a highly dynamic system where entropic interactions and nuclear confinement drive the formation of domains of high chromosomal interaction, known as topologically associating domains. In this talk, we further investigate the dynamic organization of all 16 chromosomes in living yeast cells during interphase using coarse-grained, entropic polymer chain models. In particular we study the role that the polymeric nature of the chromosomes plays in nucleosome dynamics. We start by investigating loop configuration and how it regulates cell function in the yeast genome as related to nucleolus size, segregation, and level of compaction. We have developed a microscope simulator computational program to translate simulated data from our models into equivalent microscope images, as well as a pipeline to view and analyze experimental images obtained by the Bloom lab using live cell microscopy. Through such visualization tools and comparison with experimental data, we aim to shed insights into nucleolus dynamics and structure that are beyond current experimental resolution.

Degree Six Polynomials and Their Solvability by Radicals

Peter Jakes, Elon university, Elon, NC

Abstract: For about 500 years, formulas have existed to find exact solutions to quadratic, cubic and quartic polynomials. However, it was proven later that not all solutions to quintic polynomials can be found exactly, or solved by radicals. As a result, a method was created in the 20th century using a property of each function called its Galois group in order to determine which degree five polynomials could be solved exactly and which could not. This project expands upon this discovery by exploring degree six polynomials. By using computer software, the Galois group of a degree six polynomial can be determined by only using two resolvent polynomials, improving upon prior methods. From this information, it can then be determined whether or not the polynomial is solvable by radicals. Further research can explore higher degree polynomials as well as reducible polynomials, as the current method is only viable for irreducible polynomials.

Signaling Models in Open-Market, Cooperative Settings: an Application of Game Theory to Islamic Finance

Ausha Khan, University of North Carolina Wilmington, Wilmington, NC
mentored by Dr. Yaw Chang

Abstract: In 2010, the International Monetary Fund conducted a study investigating the performance of Islamic banks during the Great Recession. It was concluded that Islamic banks were more resilient than conventional banks. The I.M.F. attributed this phenomenon to the strong risk management methods that ameliorated conventional exploitative, high-risk practices. Islamic finance prohibits riba and gharar, or "unjust increases" and "exploitative usury", respectively. For over a millennium, scholars have debated which financial practices and instruments can be categorized as riba or gharar, coming to the conclusion that Islamic finance is characterized by cooperation between all parties involved in a transaction, bound by religious principles that will circumvent exploitative practices, allowing all parties to gain. Dr. Mahmoud El-Gamal utilizes a game theoretical approach, the prisoner�s dilemma, to illustrate that both parties in any transaction benefit greater should there be mutual cooperation, as set up by the Trading in Risky Assets Model. El-Gamal proves that trading in risk is �at worst efficiency neutral and at best efficiency enhancing.� In this presentation, we present a signaling model that employs cooperative risk-and-return sharing between competitive open-market entities. We conduct a case study, utilizing this model, on the manufacturing of the 2014 World Cup Brazuca.

k-kings in Strong Products of Digraphs

Peter LaBarr and Morgan Norge, Virginia Commonwealth University, Richmond, VA
mentored by Dr. Dewey Taylor

Abstract: A k-king in a digraph D is a vertex that can reach every other vertex in D by a directed path of length at most k. We consider k-kings in the strong product of digraphs. In particular, we determine the relationship between k-kings in the strong product of digraphs and k-kings in the factors of the product.

Optimization of Border Patrol Strategies

Aaleah Lancaster, Bennett College, NC
mentored by Dr. Hyunju Oh and Dr. Jan Rychtar

Global Dynamics of a Cancer Stem Cell Treatment Model

Heidi Whiteside, Winston-Salem State University, Winston-Salem, NC
mentored by Dr. Kristen Abernathy

Abstract: We provide global stability arguments for a cancer treatment model with chemotherapy and radiotherapy that accounts for the cancer stem cell hypothesis. Employing the method of localization of compact invariant sets, we resolve the global dynamics of the no-treatment, constant radiation, and combination chemotherapy and radiotherapy cases. In our analysis of the combination treatment model, we show that the presence of a chemotherapy agent lowers the required radiation strength for a globally asymptotically stable cure state.

The Direct Sum of Two Cyclic Groups of Order Three in the Rubik�s Cube

Trevor Williams, West Virginia Wesleyan College, Buckhannon, WV
mentored by Dr. Scott Zinzer

Abstract: The Rubik's Cube is a vastly complicated structure. With only a few twist and turns we can turn a solved Rubik's Cube into one of 43,252,003,274,489,856,000 states. Certain moves in the Rubik�s Cube have order three. By putting these into a set they form a group that is isomorphic to the direct sum of two cyclic groups of order three.

Creating Art with Mathematical Symmetries

Kelsey Windham, Georgia College and State University, Milledgeville, GA
mentored by Dr. Marcela Chiorescu

Abstract: The connection between math and art has been known for thousands of years. Using a photograph, a software and mathematical tools such as: domain coloring, complex wave functions and wallpaper groups, we show how we can create mathematical art with interesting patterns.