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.
- Davis Atkinson and Jonathan Dunay: Using Curve-Shortening Flow to Solve Dido's Problem
- Hailey Belleperche and Jillian Allen: Proper Connection Number of the Direct Product of Cycles
- Love Bennett: Stereoscopic Skin Segmentation for Musical Conducting Recognition
- Jacob Coleman: (q,t) Symmetry in Macdonald Polynomials
- Jessica Cook, Morgan Dozier, and Ryan Bostic: Heart Attack Patients: Predicting Life or Death After One Year Using Data Mining Classification Techniques
- Christine Craib: A Mathematical Analysis on the Transmission Dynamics of Neisseria Gonorrhoeae
- Chasity Dorsett and Hakimah Smith: Yellow Fever Vaccination and The Theory of Games
- Idalmy Escobar and Hermella Tessema: k-proper Connection Number of the Direct Product of Cycles
- Peter Jakes: Degree Six Polynomials and Their Solvability by Radicals
- Aysha Khan: Signaling Models in Open-Market, Cooperative Settings: an Application of Game Theory to Islamic Finance
- Peter LaBarr and Morgan Norge: k-kings in Strong Products of Digraphs
- Mark Leadingham II: Chaos: Pulsars and their Quirks
- Nicholas Lindell: Cops and Robbers: The Infinite Chase
- Andrew Linzie: Extreme Learning Machines for Stock Market Prediction
- Sydney Meeks and Dana West: Using A Game Theoretic Approach to Analyze the Rift Valley Fever Virus Vaccine
- Kristen Melton: A Lower Bound on the Hadwiger Number of a Random Subgraph of the Kneser Graph
- Jessica Nash: Analysis of Steady States for Classes of Reaction-Diffusion Equations with Hump-Shaped Density Dependent Dispersal on the Boundary
- Grace Ohanian: Pythagorean Triples
- Emily Pazdera: Conway's Cats
- David Perez-Suarez: Greenberg Unrelated Question Binary RRT Model under Inverse Sampling
- Daniel Plaugher: Modeling the Spread of Invasive Species Using PDE's
- Nathan Pool: The Automatic Construction of Fractals of Arbitrary Dimension
- Joshua Postel: Statistical Exploration of Leukemia Mortality Data
- Michelle Rave: Prevention Methods: Mathematical Modeling of Dengue Fever
- Kelly Reagan: A Mathematical Model of Dengue Fever Incorporating Human Travel
- William Reese: Numerical Simulation of Liquid Crystal Interactions with Colloids and Electric Fields
- Joshua Safley: Resource Allocation Trade-Offs Between Survival and Reproduction
- Stephen Steward: Incorporating the Cancer Stem Cell Hypothesis into a Treatment Model of Glioblastoma Multiforme
- Cuyler Warnock: Investigating Proofs of the Quadratic Reciprocity Law
- Heidi Whiteside: Global Dynamics of a Cancer Stem Cell Treatment Model
- Trevor Williams: The Direct Sum of Two Cyclic Groups of Order Three in the Rubik�s Cube
- Kevin Akers: Completing Partial Latin Squares Arising from Latin Arrays
- Lindsay Bradley: Using Gompertzian Growth to Model Chronic Myeloid Leukemia and Its Treatment
- Dustin Ford: Elucidation of Cysteine Richness in Arabidopsis thaliana Genome Using Exact Distribution of Clump Statistics
- Elizabeth Hance: Patrol Zone Realignment for the Huntington, WV Police Department
- Aaleah Lancaster: Optimization of Border Patrol Strategies
- Laura Layton: Modeling Gun Ownership As a Social Disease
- Megan McGurl: A Comparative Study of the Mathematical Curriculum of France and the State of Georgia
- Adam O'Neal: On Sets of Cardinality 2 of Nondecreasing Diameter
- Jay Saini: Racial Bias in Individual Officers in the Greensboro Police Department
- Jade Schrader: Galois 2-adic Fields of Degree 16
- Alison Tighe: Various Methods for Assessing the Ability of Baseball Managers
- Kelsey Windham: Creating Art with Mathematical Symmetries
- Ekram Adam: Dimension Reduction Methods for Highly Correlated Data
- Sameed Ahmed: Mathematical Modeling of HER2 Signaling Pathway: Implications for Breast Cancer Therapy
- Matthew Buhr: Spider Monkeys in Fragmented Landscapes: A Discrete Mathematical Model
- Lance Everhart: On Congruence Subgroups of Hilbert Modular Groups of Real Quadratic number Fields
- Christian Felton: Propensity Score Matching Analysis for Measuring the Influence of Mothers� Education Levels on Children�s Academic Performance
- Ashley Fowler: A Study of Identifying Socioeconomic and Health-Related Factors and Their Relative Importance with HRQOL among U.S. Adults
- Nitin Gaikwad: Simple Solutions for Missing Data Problem Big Data Domain
- Ivanti Galloway: Analysis of Individual Greensboro Police Officers' Stopping Patterns Using Propensity Scores
- Monika Goel: Positive Impact of the STS Model in the Operating Room for Cesarean Cases
- Joey Hart: Machine Learning Methods for Syncope Data Analysis
- Victoria Hayes: Recreation of Axelrod: An Evolution of Cooperation
- William Hayes, Brittany Palmer, and Sean Vanhille: College Football: Predictors Winning Seasons in the Power 5 Conferences
- Lizzy Huang: Minimizing Energy, the Geometry of Tension
- Caitlin Hult: Modeling Nucleosomal DNA in Living Yeast
- Austin Lawson: Straight Finite Coarse Decomposition Complexity
- Michael Leshowitz : Development of Honesty in Repeated Signalling Games
- Jonathan Milstead: Computing Galois Groups of Eisenstein Polynomials over p-adic Fields
- Erik Palmer: An Approach Towards Modelling the Mixture of Stimuli-Responsive Hydrogels
- Jasmin Petterson-Hassan: Clustering Heterogeneous Human Activities with Multi-Dimensional Dynamic Time Warping
- James Rudzinski: Monotone Organization of Symmetric Chains
- Sandi Rudzinski: Symbolic Computation of Resolvents
- Ghasemali Salmani Jajaei: Snug Circumscribing Simplexes for Convex Hulls
- Isabella Sanders: K-Balanced Tree Partitioning
- Aswini Sen: Study of the Sensitivity of Supervised Classification Models towards the Increased Complexity of Data
- Pooja Sen and Anirudh Reddy Karra: A Study of Diabetic Conditions in the U.S. Population Using Random Forest Classification
Undergraduate Students Talks
Undergraduate Students Posters
MORE TO COME
Graduate Students
MORE TO COME
Dimension Reduction Methods for Highly Correlated Data
Ekram Adam, North Carolina A&T State University, Greensboro, NCmentored 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, SCmentored 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, WVmentored 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, NCmentored 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, VAmentored 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, WVmentored 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, SCmentored 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, SDmentored 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, WVmentored 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, NCmentored 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, NCmentored 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, NCmentored 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, VAmentored 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, NCmentored 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, NCmentored 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, AKmentored 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, NCmentored 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, NCmentored 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, NCmentored 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, NCcoauthored 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, WVmentored 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, NCmentored 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, NCmentored 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, NCmentored 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, NCmentored 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, NCmentored 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, NCmentored by Dr. Chad Awtrey
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, NCmentored 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, VAmentored 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, NCmentored by Dr. Hyunju Oh and Dr. Jan Rychtar
Abstract: U.S. Customs and Border Protection (CBP), is responsible for securing the border between U.S. ports of entry and has divided the 2,000-mile U.S. border with Mexico among nine Border Patrol sectors. CBP reported spending about $3 billion to support Border Patrol's efforts on the southwest border in fiscal year 2010 alone, and apprehending over 445,000 illegal entries and seizing over 2.4 million pounds of marijuana. The number of border patrol officers has been increased but because of the limitation of patrolling personnel and budget, it is critical to allocate resources appropriately. The goal of this project is to optimize border patrol routes for the border with Mexico. We used several �games� to find general equations for the best patrol strategy. In our �games� the infiltrators' goal is to enter US successfully while patrols intend to capture infiltrators to prevent illegal cross-border activities. Using the games we were able to find the optimal strategy, and general equations for any sized �game� with a grid. Our findings in the future should be used in real life to test the theory. If our findings are found to be correct, they could change the way the government sees border patrol.
Straight Finite Coarse Decomposition Complexity
Austin Lawson, UNCG, Greensboro, NCmentored by Dr. Greg Bell
Abstract: Coarse geometry is the study of large-scale properties of metric spaces. One can define a coarse structure on any set without reference to a metric and study the coarse properties of the resulting space. In this talk we are concerned with general coarse spaces with infinite asymptotic dimension. We describe how Dydak's countable asymptotic dimension can be applied to such spaces and generalize the theorem of Dydak and Virk to show that countable asymptotic dimension coincides with straight finite coarse decomposition complexity. We also show that these notions imply coarse property A, which generalizes a result of Dranishnikov and Zarichnyi.
Modeling Gun Ownership As a Social Disease
Laura Layton, CW Stanford Middle School, Hillsborough, NCmentored by Dr. Anita Layton
Abstract: Gun violence is a leading cause of premature deaths in the US. The goal of this study is to better understand how different social conditions may impact gun-ownership rate. We use a SIR-type model that treats gun ownership like a contagious social disease. The model divides the population into three classes: non-susceptible, susceptible, and gun owners. The model is used to assess the effectiveness of different approaches in lowering gun-ownership rate.
Chaos: Pulsars and their Quirks
Mark Leadingham II, West Virginia Wesleyan College, Buckhannon, WVmentored by Dr. Scott Zinzer
Abstract: Chaos Theory is the study of systems that are highly dependent on their initial conditions. Any seemingly insignificant change will create completely different results than those expected. Deterministic Chaos focuses on strange results occurring from seemingly predictable systems. The applications are apparent in computer science, physics, and astronomy, which is the interest of this talk. Astronomical objects called pulsars exhibit chaotic behavior in the emission of their radio pulses. We will discuss famous attractors (Lorenz & Rossler) as well as the methods of describing these phenomena, such as building models, the fourth-order Runge-Kutta Method, and the concept of embedding dimensions.
Development of Honesty in Repeated Signalling Games
Michael Leshowitz, The University of North Carolina at Greensboro, Greensboro, NCmentored by Dr. Jan Rychtar
Abstract: A symbiotic relationship inherently depends on the development of a reliable information channel. This network provides an unintended opportunity for rouge agents to reap benefits by counterfeiting honest signals. In a plant-pollinator system, plants have a multitude of potential signals, primarily visual and olfactory. Some signals are a byproduct of rewards, while other can be positively correlated through pollinator conditioning. Successful pollinators cross reference signals with prior experiences in order to determine the integrity of their sender. We examine the parameters of honest signaling as high-yield and low-yield plants compete for visitation rates in a repeated Sir Philips Sidney Game.
Cops and Robbers: The Infinite Chase
Nicholas Lindell, University of Georgia, Athens, GAmentored by Dr. Robert W. Bell
Abstract: Cops and Robbers, or the game of pursuit, is a two-player, turn-based game played on the vertices of a finite graph. To start, the cop player C places her cop pawn(s) on the graph, then the robber R his single pawn. Each turn consists of a player moving each of his/her pawns to an adjacent vertex or staying still. C wins if she can "capture" the robber by moving a cop to the vertex R occupies; if not, then R wins. Of mathematical interest are the minimum number of cop pawns required and strategies to ensure that C can always win on a given graph. We examine, in particular, a variation of the classic game on infinite graphs, where C instead wins by "chasing away" the robber by preventing him from visiting any vertex infinitely often. We develop strategies for this infinite game and give results about several tilings of the plane. Larger questions involve how theorems about the classic game carry over to the infinite case.
Extreme Learning Machines for Stock Market Prediction
Andrew Linzie, Gardner-Webb University, Boiling Springs, NCmentored by Dr. Miroslaw Mysttkowski
Abstract: Machine Learning has been a prolific research subject from the 1990s till now with relevant applications in handwriting classification, speech recognition, self-driving cars, and more. Machine learning techniques can be applied to nearly any quantifiable subject relating to everyday life like stock market prediction. Stocks are an example of a time series, price as a function of time, and predictions can be made with methods derived from artificial neural networks. An extreme learning machine is a type of artificial neural network, but the weights are chosen by a different method than traditional single or multilayer perceptrons trained with error back propagation. Recent research by G. Huang, et. al, has shown that extreme learning machines can give better training times without sacrificing accuracy or generalization of predictions. This presentation focuses on using extreme learning machines as a method of stock market price prediction. Using available historical stock market data we trained few ELM models and evaluated their performance against each other and against other stock market prediction methods.
A Comparative Study of the Mathematical Curriculum of France and the State of Georgia
Megan McGurl, Georgia College and State University, Milledgeville, GAmentored by Dr. Simplice Tchamna-Kouna
Abstract: The purpose of our study is to make a comparison between the high school math content of the State of Georgia and the high school math content of France, whose students perform better at the international level than U.S high school students. A report by the Pew Research Center (2015) found that U.S students are scoring higher on national math assessments than they did two decades ago. However, U.S students� performance is still behind all the other major industrial countries (Desilver, 2015). We investigate the causes of this low performance of U.S students in math. We conduct a statistical analysis of sample high school exit exams given to students in France to better understand the causes of the gap observed between the groups of students. We propose changes in the content of the math curriculum of Georgia to help the close this performance gap.
Using A Game Theoretic Approach to Analyze the Rift Valley Fever Virus Vaccine
Sydney Meeks and Dana West, Bennett College, Greensboro, NCmentored by Dr. Hyunju Oh and Dr. Jan Rychtar
Abstract: Rift Valley Fever (RVF) is a vector borne disease that is prevalent in Sub-Saharan Africa, from Egypt to South Africa. RVF is caused by the Rift Valley Fever Virus (RVFV), whose primary vector is a mosquito with an infected mouthpart. RVFV primarily affects livestock, or ruminants, such as sheep, goats and cattle, so much so that if the disease is eradicated in the ruminant population the disease will ultimately die off. We constructed an SIR flow diagram to examine the flow of the disease and its impact on the ruminant and mosquito populations. There are vaccines for RVFV for ruminants, which grant lifelong immunity to the vaccinated ruminant, therefore helping to decrease the RVF outbreaks. Using the basic reproductive number R0, we found the vaccination rate required to eradicate the disease. Using the Game Theory we will find the vaccination rate necessary to achieve ruminant herd immunity, taking into consideration the disease incidence and risk of infection, combined with the side effects and other costs of the vaccine. These findings will allow us to predict the vaccination level of the ruminant population, while accounting for the costs of the vaccine.
A Lower Bound on the Hadwiger Number of a Random Subgraph of the Kneser Graph
Kristen Melton, Winthrop University, Rock Hill, SCmentored by Dr. Arran Hamm
Abstract: Hadwiger's Conjecture is one of the most famous open problems in graph theory; it states that h(G) is greater than or equal to the chromatic number, Χ(G), for any graph, G. h(G) denotes the Hadwiger number of G. A theorem of Kostochka gives a lower bound on h(G) in terms of the average degree of G. This talk will be focused on giving a lower bound on h(G) where G is a (binomial) random subgraph of a Kneser graph. Recall: A Kneser graph with parameters n and k, denoted KG(n,k), has the set of k-element subsets of {1, � , n} as its vertex set where two k--sets are adjacent if and only if they are disjoint. So G is given by keeping each edge of KG(n,k) independently with probability p. For certain values of n, k, and p we improve upon the bound given in Kostochka's theorem.
Computing Galois Groups of Eisenstein Polynomials over p-adic Fields
Jonathan Milstead, University of North Carolina at Greensboro, Greensboro, NCmentored by Dr. Sebastian Pauli
Abstract: The most efficient algorithms for computing Galois groups of polynomials over global fields are based on Stauduhar's relative resolvent method. These methods are not directly generalizable to the local field case, since they require a field that contains the global fields in which all roots of the polynomial can be approximated. We present splitting field-independent methods for computing the Galois group of an Eisenstein polynomial over a p-adic field. Our approach is to combine information from different disciplines . We primarily, make use of the ramification polygon of the polynomial. The polygon is the newton polygon of a related polynomial and it allows us to quickly calculate several invariants that serve to reduce the number of possible galois groups. Algorithms by Greve and Pauli very efficiently returns the Galois group of polynomials where the ramification polygon consists of one segment as well as information about the subfields of stem field. Second, we look at the factorization of linear absolute resolvents to further narrow the pool of possible groups.
Analysis of Steady States for Classes of Reaction-Diffusion Equations with Hump-Shaped Density Dependent Dispersal on the Boundary
Jessica Nash, University of North Carolina at Greensboro, Greensboro, NCmentored by Dr. Ratnasingham Shivaji
Abstract: We study a two point boundary-value problem describing steady states of a logistic growth population dynamics model with diffusion. In particular, we focus on a model in which the population exhibits hump-shaped density dependent dispersal on the boundary.
Pythagorean Triples
Grace Ohanian, Muhlenberg College, Allentown, PAmentored by Dr. Byungchul Cha
Abstract: A primitive Pythagroean triple is a 3-tuple of integers (x,y,z) that satisfies the Pythagorean Theorem. Every triple can be generated using matrix multiplication represented on a ternary tree structure known as a Berggren Tree. We know that this set of pythagorean triples relates to a dynamical system between the positive quadrant of the unit circle and the unit interval on the real line. Using already known matrices, we can create similar dynamical systems to represent other quadratic forms as well. Since the idea of utilizing parameterizations as a means of simplifying equations is not exclusive to the Pythagorean Theorem, we present four other quadratic equations that were examined in detail, from the starting point of a unique parameterization. Using the parameterizations for various forms, we employ a method analogous to the creation of a dynamical system for Pythagorean triples with the goal of identifying similar mappings related to each of the quadratic forms studied. This allows us to study connections between the set of unique digit-expansions and continued fractions. Additionally, we explore the concept of invariant measure as it applies to the additional quadratic forms that we considered.
On Sets of Cardinality 2 of Nondecreasing Diameter
Adam O'Neal, Marshall University, Huntington, WVmentored by Dr. Michael Schroeder
Abstract: Our problem comes from the field of combinatorics known as Ramsey theory. Ramsey theory, in a general sense, is about identifying the threshold for which some object, associated with a particular parameter, goes from never or sometimes satisfying a certain property to always satisfying that property. Research in Ramsey theory has applications in design theory and coding theory. For integers m,r, and t, we say that a string of n integers colored with r colors is (m,r,t)-permissible if there exist t monochromatic subsets B_1, B_2, ..., B_t such that (a) |B_1| = |B_2| = ��� = |B_t| = m, (b) the largest element in B_i is less than the smallest element in B_(i+1) for 1 ≤ i ≤ t − 1, and (c) the diameters of the subsets are nondecreasing. Define f(m,r,t) to be the smallest integer n such that every string of length n is (m,r,t)-permissible. In this paper, we begin by fixing m = r = 2 and show that f(2,2,t) > 5t − 5 and f(2,2,t) ≤ 5t − 2. We conjecture that f(2,2,t) = 5t − 4 and prove the conjecture in certain cases. We conclude by investigating colorings with more than two colors.
An Approach Towards Modelling the Mixture of Stimuli-Responsive Hydrogels
Erik Palmer, University of South Carolina, Columbia, SCmentored by Dr. Paula Vasquez
Abstract: Stimuli-responsive Hydrogels respond to temperature, pH, UV or ionization making them ideally suited for a variety of applications. Recent advances in the field include the creation of multi-responsive interpenetrated self-assembled polymer networks (IPSANs) by combining two Stimuli-responsive gels. The result is a single hydrogel whose properties can be tuned via two different types of stimuli; making them especially desirable. However, the development of mathematical models, algorithms and numerical simulation tools for the investigation of IPSANs remains largely unexplored. In this talk, we will present an elastic dumbbell chain model that was originally conceived to analyze properties of Lung mucus. This approach leverages the parallel processing power of GPUs to create a unique micro-macro scale driven design capable of efficiently recreating experimentally measured rheological data from single hydrogels. In addition we will discuss the numerical and computational challenges of adapting the lung mucus model to the hydrogel mixing problem using two copolymers, a UV-responsive one based on poly-ethylene oxide (tPEO) and another pH-responsive one based on poly-acrylic acid (tPAA).
Conway's Cats
Emily Pazdera, Lenoir-Rhyne University, hickory NCmentored by Dr. Douglas Burkholder
Abstract: We examine whether the Trap-Neuter-Release (TNR) program can be effectively used to control the populating of free-roaming feral cats. We do this by constructing a Game of Cats, which is similar to but far more complex than Conway�s Game of Life. After construction of the game we are observing mathematical populations of cats to see the effects of sterilization of both randomly selected male and female cats. Preliminary Report.
Greenberg Unrelated Question Binary RRT Model under Inverse Sampling
David Perez-Suarez, University of North Carolina at Greensboro, Greensboro, NCmentored by Dr. Sat Gupta
Abstract: Getting accurate information is vital in all surveys, especially in public health research where respondents are often faced with sensitive and personal questions. Even though there are assurances of anonymity, subjects often lie and give out untruthful responses that automatically lead to problematic response bias. One method that can reduce this bias is the randomized response technique (RRT) that was introduced by Warner (1965). Greenberg et al. (1969) gave an alternative RRT model by introducing an unrelated question element to the model. In the unrelated-question model, a predetermined proportion of subjects are randomly asked to answer an innocuous unrelated question. In turn, the researcher does not know which question any particular respondent answered. In such models, the responses can be unscrambled at the aggregate level but not at individual level. Unrelated-question RRT models have been used in many field surveys and are proven to be quite effective. The main objective of our research project is to examine variations of the unrelated question model by using inverse sampling. We will present both theoretical and simulation results to show that the new variations work better that the original unrelated question model.
Clustering Heterogeneous Human Activities with Multi-Dimensional Dynamic Time Warping
Jasmin Petterson-Hassan, North Carolina A&T State University, Greensboro, NCmentored by Dr. Seong-Tae Kim
Abstract: Measuring similarity among sequential and temporal data has a long history in time series analysis, which is critical to clustering and classification analysis. Dynamic Time Warping (DTW) finding an optimal path between two time series is one of the most frequently used methods for measuring similarity, and has been considerably applied to one-dimensional data. In the era of big data, there is a high demand in similarity analysis for multi-dimensional temporal data such as activity recognition and motion detection. This project aims to study heterogeneous human activities using multi-dimensional dynamic time warping (MD-DTW) algorithm in a three-dimensional temporal data. We utilized the publicly available data of human activities associated with mobile device usage. The MD-DTW algorithm is implemented through multiple steps: normalization of each dimension, Gaussian filtering, calculation of distance matrix, and synchronization with regular DTW. The results of MD-DTW are reported for both simulated and real data, which is also compared to those from one-dimensional DTW. Our study illustrates a novel application of DTW to multi-dimensional data, which deepens insights of high-dimensional temporal data
Modeling the Spread of Invasive Species Using PDE's
Daniel Plaugher, West Virginia Wesleyan College, Buckhannon, WVmentored by Dr. Scott Zinzer
Abstract: Unbeknownst to many, a common issue being fought is the invasion of nonnative species. In my project, I focus on one invasive species in particular, Tree of Heaven [Ailanthus altissima]. Using partial differential equations, I model the spread of these trees with the heat diffusion equation. Namely, through collecting field data, I compare theoretical solutions with those seen in the field data itself.
The Automatic Construction of Fractals of Arbitrary Dimension
Nathan Pool, Elon university, Elon, NCmentored by Dr. Jeffrey Clark
Abstract: Have you ever gazed into a work of art or a coastline on a map and noticed a repetitive pattern the closer that you observe it? This distinct quality is characteristic of fractals. These geometric figures diverge from the structure of traditional polygons with their fractional dimension. Not only do they have dimensions of non-integer value, but they have self-similarity. Kenneth Falconer addressed an equation for dimensions of self-similar sets � the iterated function systems used to construct fractals from initial figures. The research I am conducting uses this equation by taking into account the number of similarity transformations to produce a fractal and the contraction factor used in the transformations. I began this research by writing an algorithm to construct fractals out of a regular polygon corresponding to the number of vertices inputted. The end product of this research will have the ability to construct fractals of any given dimension using just one function. With access to this data, people will be able to more adequately analyze the correlation and relationship between fractals� aesthetic characteristics and their corresponding dimensions. I hope to push this research even further by examining the connection between dimension and sound in the area of fractal music. As many classical composers made use of self-similar themes in their music, perhaps there is a way to portray that self-similar composition technique through the mapping fractals� coordinates to corresponding frequencies. Furthermore, dimension has the potential to be connected to the sound just as it has a connection to self-similarity. Having a successful algorithm for creating fractals of arbitrary dimension will give access to countless fractals varying in dimension in any way that researchers desire.
Statistical Exploration of Leukemia Mortality Data
Joshua Postel,University of Michigan-Dearborn, Dearborn, MImentored by Dr. Keshav Pokhrel
Abstract: We compare the childhood leukemia mortality rates in the United States to that of Great Lake States through the interpretation of statistical models. The desired data is extracted from the Surveillance, Epidemiology, and End Results (SEER) Database using SEER*stat software. The tabular form of the data leads to a trade-off between patient details and unsuppressed mortality rates. To address this challenge we built multiple Generalized Linear Models (GLMs), refined the models and compared them to one another. Leukemia mortality rate has decreased over time; however, the models' coefficients reveal that this has not been uniform across race or age. We discuss a privacy exploitation of the SEER database which was discovered during research.
Prevention Methods: Mathematical Modeling of Dengue Fever
Michelle Rave, Elon university, Elon, NCmentored by Dr. Crista Arangala and Dr. Karen Yokley
Abstract: Dengue fever is a virus that is transmitted by mosquitos. It is prevalent in tropical areas of the world. There is no cure, but there are possible prevention methods such as vector control and vaccines. One method of vector control is a bacteria, Wolbachia. Wolbachia infected mosquitos are unlikely to transmit dengue between human hosts. We use ordinary differential equations (ODEs) and an SEIR (Susceptible, Exposed, Infectious and Recovered) model to model the transmission of dengue fever. The set of ODEs is then used to examine the possible prevention methods. Wolbachia is incorporated into the model by creating a Removed category of mosquitos that cannot transmit the disease. Vaccines are separately incorporated into the model by creating Vaccinated categories for humans. Computer simulations of the models are run yielding graphical results. The presence of Wolbachia infected mosquitos shortens the duration of the dengue fever presence in the human population. The greater the number of infected mosquitos released, the shorter the infection becomes. The use of vaccines also shortens the duration of dengue fever presence in the human population. This is dependent on a sufficient number of people being vaccinated in a timely manner.
A Mathematical Model of Dengue Fever Incorporating Human Travel
Kelly Reagan, Elon University, Elon, NCmentored by Dr. Karen Yokley
Abstract: Dengue fever is a disease transmitted by day-biting mosquitoes in tropical, urban areas. There is currently no vaccine for dengue fever, so preventative measures are the only solution to slow the spread of the disease. In order to develop a model to simulate dengue fever spreading between two populations, an in-depth analysis was conducted on a malaria model with human traveling aspects and on a general dengue fever model. Then, parameter values relevant to dengue fever were investigated and implemented into both of the models. The results from both the malaria model and dengue fever model were regenerated. A thorough investigation of previous research in the field allowed for a combination of the malaria model and the general dengue fever model by adding the human travel aspects to the dengue model. The combination resulted in ten ordinary differential equations, which simulate humans of two different communities visiting the other community. Further simulations have been conducted on how population size differences and the length of time spent in each community affects the spread of dengue. It has been seen that the time factor does heavily impact the rate at which dengue spreads across a community.
Numerical Simulation of Liquid Crystal Interactions with Colloids and Electric Fields
William Reeses, NC State University, Raleigh, NCmentored by Dr. Shawn Walker
Abstract: Liquid Crystals (LCs) find much use in modern technology; some examples of this are LCDs, Optical Imaging devices, and Polymer Dispersed Liquid Crystal (PDLC) sheets. Research into LCs continues to increase because of increasing numbers of applications. In the soft matter community there is a growing interest in creating colloidal particles with a designable valence, which would enable self-assembly of the colloids, and lead to new types of materials. LCs provide a potential medium for enabling and mediating colloidal self-assembly. For this research we attempted to simulate and analyze the effects produced by colloidal particles coupled with Ericksen�s one constant model. To accomplish this, several colloids were modeled using level set functions. Through several trials we found that for a given smooth colloidal shape, the minimizing state of the colloid/LC can be significantly affected by imposing appropriate boundary conditions for the director field. Moreover, simulations were done with a prescribed electric field which yielded more control over the equilibrium state of the colloid, especially the colloid�s orientation.
Monotone Organization of Symmetric Chains
James Rudzinski, University of North Carolina at Greensboro, Greensboro, NCmentored by Dr. Clifford Smyth
Abstract: A symmetric chain decomposition (SCD) of a Boolean lattice is a partition into symmetric chains. In this talk we present and analyze an algorithm which organizes the chains of a SCD into a tree. This allows each element of the decomposition to be constructed in an increasing or "monotone" fashion. One could use this to efficiently check for certain monotone properties such as connectivity in subgraphs of a graph.
Symbolic Computation of Resolvents
Sandi Rudzinski, University of North Carolina at Greensboro, Greensboro, NCmentored by Dr. Sebastian Pauli
Abstract: The Galois group of a polynomial f(x) consists of permutations of the roots of the polynomial. If n is the degree of the polynomial f(x) then the Galois group is isomorphic to a transitive subgroup of the symmetric group on n elements. Determining the structure of the Galois group for a given polynomial f(x) is an important problem in computational algebra. One method for determining the Galois group involves the use of resolvent polynomials. The resolvent polynomials are commonly obtained by evaluating an invariant at approximations to the roots of the polynomial f(x). The approximations of these roots have to be found to a high precision. In his thesis, Leonard Soicher outlines an algorithm for computing linear resolvents that does not rely on approximations of the roots. In this talk, we discuss the application of linear resolvent polynomials in finding Galois groups of polynomials and their symbolic computation as given by Soicher.
Resource Allocation Trade-Offs Between Survival and Reproduction
Joshua Safley, The University of North Carolina at Greensboro, Greensboro, NCmentored by Dr. Jan Rychtar
Abstract: All living things must allocate resources to their own survival and towards the production of offspring. Because both are necessary and resources are finite, there exists a trade-off between survival and reproduction. We explore the dynamics between these two needs. Preliminary results suggest that both positive and negative correlations can occur between amount of resources allocated to the survival and reproduction. Additionally, we adapt a model for the foraging and predation risk to determine an optimal allocation strategy in our scenario.
Racial Bias in Individual Officers in the Greensboro Police Department
Jay Saini, The University of North Carolina at Greensboro, Greensboro, NCmentored by Dr. Jan Rychtar
Abstract: In 2015, the New York Times published an article that brought the Greensboro Police Department (GPD) under scrutiny for conducting more traffic stops on black drivers than non-black drivers. To shed light on these allegations, we analyzed the GPD's traffic stop data. We based our analysis on the RAND Corporation's investigation of the New York Police Department. Propensity score weighting was used to align the joint distribution of the stops of each officer to the stops of his/her peers. This allows us to compare the data, accounting for disparate stop locations, times and dates. We then determined the probability that officers were racially biased using testing and false discovery rate analysis. We found thirteen officers within the GPD who are likely to be disproportionately stopping black drivers compared to other officers patrolling the same neighborhoods at the same time.
Snug Circumscribing Simplexes for Convex Hulls
Ghasemali Salmani Jajaei, Virginia Commonwealth University, Richmond, VAmentored by Dr. Jose Dula
Abstract: We propose procedures for enclosing convex hulls of finite m-dimensional point sets with simplexes. A simplex is called snug if it intersects the convex hull in some way. Our goal is to find a tight simplex, one that coincides with the convex hull exactly. We find that if the convex hull of the point set is a simplex, our procedures will find that simplex exactly. If the convex hull of the point set is not a simplex, our procedures generate a simplex that coincides with m+1 facets of the convex hull.
K-Balanced Tree Partitioning
Isabella Sanders, Virginia Commonwealth University, Richmond, VAmentored by Dr. Paul Brooks
Abstract: The problem of partitioning a graph into balanced sub-graphs can be explored as a mixed integer programming (MIP) problem. The sub-graphs are to consist of no more than k nodes and we want to maximize the number of edges in each partition. For this problem we will only look at trees. Since we are looking at a MIP problem computational time can be very slow. The idea is to add constraints to find the optimal solution at a faster rate. Several methods and results are compared from previous work. Graph partitioning can be used for many different applications. Some examples are, minimizing connections between electrical circuits, convex coloring, computer networks, or information storage. In our research we are looking at graph partitioning to group different optimal solutions together based on some underlying root node that they have in common. The overall application would be to explain why some optimal solutions are grouped together and what different characteristics they might have from the other groups.
Galois 2-adic Fields of Degree 16
Jade Schrader, Elon university, Elon, NCmentored by Dr. Chad Awtrey
Abstract: Let K be a Galois extension of the 2-adic numbers of degree 16, and let G be the Galois group of K. We show there are 251 such extensions, up to isomorphism. We also show G can be determined by the Galois groups of the octic subfields of K. We prove that all 14 groups of order 16 occur as the Galois group of some Galois extension K except for the elementary abelian group of order 16. For the other 13 groups, we give a sample defining polynomial for each isomorphism class of extensions.
Study of the Sensitivity of Supervised Classification Models towards the Increased Complexity of Data
Aswini Sen, The University of North Carolina at Greensboro, Greensboro, NCmentored by Dr. Shan Suthaharan
Abstract: In machine learning, we exclude several useful data sets from the studies because of their simplicity, in spite of some of their contributing statistical characteristics. We define a data set as simple (i.e., simplicity), when it gives 100% classification accuracy for at least two classification techniques. In our current research, we have selected the simple and well-known IRIS dataset, generated multiple versions with varying complexities determined by Gaussian additive noise, and studied the classification performance of support vector machine and random forest classification algorithms. This study contributes to the development of parameterized data complexity and the understanding of the relationship between data complexity and classification accuracy. Another advantage is that - instead of giving a single classifier for a given dataset - it can provide a band of potential classifiers out of which the final classifier can be selected depending on the extent to which the data is corrupted. We will report our theory, results, and the findings at the conference.
A Study of Diabetic Conditions in the U.S. Population Using Random Forest Classification
Pooja Sen and Anirudh Reddy Karra, The University of North Carolina at Greensboro, Greensboro, NCmentored by Dr. Shan Suthaharan
Abstract: The main objective of this study is to understand the features that affect the diabetic conditions in the U.S. population. In this research, we have studied these conditions using the well-known National Health Nutrition Examination Survey (NHANES) data set. In 2008 and 2010, the logistic regression, classification tree, and support vector machine were explored to understand the diabetic conditions. However, these models do not show the features that contribute to different levels of diabetic conditions, rather they predict or classify the individuals who have these conditions. Therefore, in our research, we have adopted the ensemble classification tree, called random forest, and generated an ordered sequence of feature with respect to their strengths of contributions to diabetic conditions in the U.S. population. Our experience with data set and the findings will be presented at the conference.
Incorporating the Cancer Stem Cell Hypothesis into a Treatment Model of Glioblastoma Multiforme
Stephen Steward, Winthrop University, Rock Hill, SCmentored by Dr. Kristen Abernathy and Dr. Zach Abernathy
Abstract: In this talk, we extend the work of Kronik, Kogan, Vainstein, and Agur (2008) by incorporating the cancer stem cell hypothesis into a treatment model for Glioblastoma Multiforme. Cancer Stem Cells (CSCs) are a specialized form of tumor cell with normal adult stem cell properties. CSCs are believed to be one of the primary reasons for cancer recurrence since they are more resilient to current treatment practices and are able to repopulate the tumor once their own population has regenerated. We present a system of nonlinear ordinary differential equations that describes the interaction between cancer stem cells, tumor cells, and alloreactive cytotoxic T-lymphocytes (CTLs). Under the assumption of constant treatment, we present sufficient conditions for a treatment threshold that ensures a cure state that is globally asymptotically stable. We also explore a more biologically accurate treatment schedule in which CTLs are injected periodically. We consider cases where treatment is applied continuously over varying time intervals, as well as treatment injections using the Dirac Delta function. We conclude with a discussion of biological implications.
Various Methods for Assessing the Ability of Baseball Managers
Alison Tighe, Winthrop University, Rock Hill, SCmentored by Dr. Thomas Polaski
Abstract: While much effort has been spent on using and combining various metrics to rate baseball players, very few attempts have been made to use in-game data to rate baseball managers. This project studies three potential methods to rate managers based on skills that �a good manager� should possess � specifically producing a productive line-up and using their relief pitchers effectively -- and that use widely available data. These methods may be aggregated to produce a model for managerial prowess. Data from the 2015 season will be used to rate managers according to the three specific methods and in aggregate, and popular managers over the last 15 years will be rated to see whether these rating methods correlate with popular opinion about their relative merits.
Investigating Proofs of the Quadratic Reciprocity Law
Cuyler Warnock, Georgia College and State University, Milledgeville, GAmentored by Dr. Martha Allen
Abstract: For over 300 years, number theorists have investigated quadratic residues and their properties. A quadratic residue modulo p is an integer $a$ such that $x^2 \equiv a \pmod p$ for some $x\in\mathbb{Z}_p$. Number theorists such as Fermat, Euler, Legendre, and Gauss were interested in finding conditions for distinct primes p and q so that p would be a quadratic residue modulo q and q would be a quadratic residue modulo p. Through the contributions of these mathematicians, a surprisingly eloquent relationship was discovered: the quadratic nature of p modulo q is the same as the quadratic nature of q modulo p if and only if $p\equiv 1 \pmod 4$ or $q\equiv 1 \pmod 4$. This result, first proved by Gauss in the late 1790's, has become known as the Quadratic Reciprocity Law. Since that time, there have been hundreds of different proofs published. These proofs exhibit an astounding variety of methods derived from various branches of mathematics. We will investigate three different proofs, each varying in proof technique and complexity.
Global Dynamics of a Cancer Stem Cell Treatment Model
Heidi Whiteside, Winston-Salem State University, Winston-Salem, NCmentored 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, WVmentored 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, GAmentored 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.