The 10th Annual UNCG Regional Mathematics and Statistics Conference Saturday, November 1, 2014

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Abstracts of the talks

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

Modeling with Advanced Engineering Applications: The overview of CI MATH 4990, the propagation of heat along the human arm in electric arc phenomenon

Jameel Abbess, Clemson University, Clemson, SC
mentored by Dr. Irina Victorova

Abstract: This creative inquiry seeks to educate students of all STEM majors in the development of mathematical models of physical phenomenon using a multi-disciplinary approach to examining the physical phenomenon behind the propagation of heat through the human arm by means of the electric arc phenomenon. With an end goal of effectively modeling this phenomenon, the creative inquiry involves students of all majors while increasing research, presentation, and mathematical skills in areas directly related to their course of study.

The effect of nature noise in classification of handwritten digits using Deep Learning techniques

Mokhaled Abd Allah, University of North Carolina at Greensboro, Greensboro, NC
mentored by Dr. Shan Suthaharan

Abstract: Handwritten digits classification are affected by noise that caused by nature. These problems decrease the performance of the Machine learning techniques. In order to show this problem, we select Gaussian and Salt and Pepper noise in this paper to reduce the performance for handwritten MNIST dataset. Recent studies showed that Deep Learning method can reach the best result of the handwritten digits classification under noisy environment. Therefore, in this project we applied Deep Learning method to classify the noisy handwritten characters and evaluated the performance of Deep Learning. Then we compare our result with the different Machine Learning methods such as SVM (Support Vector Machine) and Random Forest to measure the success rate of our method and the error rate in the same time.

The effect of assuming a constant population size in models for the spread of Wolbachia

Tim Antonelli, North Carolina State University, Raleigh, NC
mentored by Dr. Alun Lloyd

Abstract: We illustrate the effects of population dynamics on the spread of the bacterium Wolbachia in a well-mixed population of insects. Previous models have almost exclusively assumed that the population size is constant and have modeled frequency of Wolbachia infection only. By incorporating a simple density-dependent per capita emergence rate of new adult insects, we show that models that consider population dynamics can differ drastically in their predictions from frequency-only models. In particular, the invasion threshold--typically thought to be a constant frequency that Wolbachia infection must exceed in order to spread–-may not be constant but rather depend on population size. We find that frequency-only models underestimate the invasion threshold when releasing into a population at carrying capacity, and overestimate the invasion threshold when releasing into a suppressed population. These results apply to a number of different forms of density dependence as well as other gene-drive systems, such as underdominance.

Heat generation under mechanical vibrations for materials with high dissipation: The connected problem of thermo viscoelasticity

Michael Bates, Clemson University, Clemson, SC
mentored by Dr. Irina Victorova

Abstract: In engineering mechanics, the properties of many materials are very dependent on temperature thus emphasizing the importance of the consideration of heat generation due to the influence of mechanical vibrations. Such considerations will lead to the development of a model predicting the point of catastrophic thermal failure, known as heat explosion, through the development of a model considering the loss of energy by means of internally generated heat and the transfer of such heat through implementation of the Maxwell-Cattaneo heat transfer equation.

The Dynamics of Offensive Messages in the World of Social Media

Aida Briceno, University of North Carolina at Greensboro, Greensboro, NC
mentored by Dr. Sherry Towers and Dr. Kamuela Yong

Abstract: The 21st century has redefined the way we communicate, our concept of individual and group privacy, and the dynamics of acceptable behavioral norms. The messaging dynamics on Twitter, an internet social network, has opened new ways/modes of spreading information. As a result cyberbullying or in general, the spread of offensive messages, is a prevalent problem. The aim of this report is to identify and evaluate conditions that would dampen the role of cyberbullying dynamics on Twitter. We present a discrete-time non-linear compartmental model to explore how the introduction of a Quarantine class may help to hinder the spread of offensive messages. We based the parameters of this model on recent Twitter data related to a topic that communities would deem most offensive, and found that for Twitter a level of quarantine can always be achieved that will immediately suppress the spread of offensive messages, and that this level of quarantine is independent of the number of offenders spreading the message. We hope that the analysis of this dynamic model will shed some insights into the viability of new models of methods for reducing cyberbullying in public social networks.

Trade-off of Glial Cell Actions in Response to Ischemic Hypoxia in the Brain

Matthew Buhr, University of South Dakota, Vermillion, SD
mentored by Dr. Carlos Castillo-Chavez

Abstract: A model that explores the dual role of glial cells in the formation of scar tissue and in neural repair, following hypoxia ischemia in the brain, is introduced. We consider multiple pieces of evidence and assumptions that include the fact that scar tissue helps protect the brain during the acute phase of insult by limiting the spread of secondary damage caused by the ischemia; that scar tissue limits recovery by inhibiting both the formation of connections between cells and the repair of damaged neurons; and that repaired neurons offer better functional recovery, but do nothing to halt the spread of ischemic injury. A stochastic, spatially explicit cellular automaton model is used as our insilicate version of the brain's biology. Mathematical analysis is carried out via mean field (MFA) and pair approximation (OPA) models. We show that the MFA, which neglects all spatial autocorrelation, leads to brain death or a brain composed of only damaged and scarred neurons. The OPA demonstrates that explicitly modeling spatial adjacency gives outcomes that include the possibility of scar tissue surrounding damaged cells (preventing further secondary death). Our results inform how the trade-off between scar tissue formation and neural repair impacts future brain health. This supports the use of mixed-treatment regimes.

Solution of Lane-Emden type equations using rational Bernoulli functions

Velinda Calvert, Mississippi State University, Mississippi State, MS
mentored by Dr. Moshen Razzaghi

Abstract: In this talk, we present a method to solve Lane-Emden type equations using rational Bernoulli functions by applying the collocation method. This method reduces solving the nonlinear differential equations to solving a system of algebraic equations. Some examples are given to demonstrate the efficiency and accuracy of the proposed method.

Classification of Intellectual Merit using Random Forests and MapReduce Programming Model

Tejo Sindhu Chennupati, University of North Carolina at Greensboro, Greensboro, NC
mentored by Dr. Shan Suthaharan

Abstract: Modeling of User Knowledge of student participants is important in academic environment, and it is determined by five major objectives (or features): degree of study time of goal objectives, degree of study time of related objectives, degree of repetition, exam performance of goal objectives, and exam performance of related objectives. The goal of the proposed work is to model 'Student's Intellectual Merit' based on the above five features. To achieve this goal we have selected the User Knowledge Modeling dataset available at the University of California at Irvine's (UCI) Machine Learning repository. This dataset consists of 403 observations out of which 258 are used as training dataset and 145 are used as testing dataset. It also provides 4 levels of user knowledge.
In this paper, a Random Forest based approach is proposed and the user knowledge is modeled for classification. The proposed approach first manipulates the dataset using the concept of (key,value) pairs of the modern MapReduce programming model, and then applies the Random Forest algorithm for modeling the user knowledge. The simulations are conducted on a Virtual Machine environment. The results show that the integration of the concept of (key, value) pairs can improve the performance of Random Forest technique significantly. It can also help enhance the proposed random forest model when the dataset grows to big data and adopts Hadoop Distributed File Systems.

Field Work Validation of an Optional Unrelated Question RRT Model

Anu Chhabra, Delhi University, New Delhi, India
mentored by Dr. Sat Gupta

Abstract: The main focus of the paper is to validate an Optional Unrelated Question RRT Model for simultaneously estimating the mean of a sensitive variable and the sensitivity level of the underlying sensitive question without using the traditional split sample approach. The data comes from a survey of undergraduate female students in age group of 17-21 years at a college of University of Delhi, India. The research question in the survey was “have you ever been sexually abused by friend or family member”. The field work shows that there is a response bias in face-to-face survey condition. The survey condition where an optional unrelated question RRT model is used did not produce reasonable results but we suspect it because of the inadequate scrambling device we used. We plan to repeat this part with a better scrambling devise in December 2014/January 2015.

On a convergence/divergence problem from integral calculus

Aziz Contractor, Clayton State University, Morrow, GA
mentored by Dr. Elliot Krop

Abstract: This summer, in my second semester calculus class, we studied the convergence and divergence of infinite series. Motivated by results on the divergence of the sum of 1/n, the sum of 1/nlog(n), and the sum of 1/nlog(n)log(log(n)), I considered the “ultimate” case in which we continue the above pattern until the number of logarithms in the denominator is as large as possible, before the value becomes zero. This led to an exploration of more advanced tests. In this talk I will show my solution to the problem by means of Ermakov’s test, which I will prove. I will also discuss future directions for exploration inspired by this problem.

Mathematical Model of African Sleeping Sickness

Kira Crawford and Aaleah Lancaster, Bennett College, Greensboro, NC
mentored by Dr. Hyunju Oh Dr. Jan Rychtář

Abstract: We will present a mathematical SIR compartment model pertaining to African sleeping sickness using Tsetse vectors to a multi-host population. The proportion of cattle being treated with insecticide-treated cattle (ITC) per day is taken into effect to control Tsetse flies and Trypanosoma brucei rhodesiense. There is an analytical expression used for the basic reproduction number in the presence of ITC treatment using the hosts (cattle, humans, and pigs). The sensivity was observed based on the basic reproduction number in regards to the three hosts in the presence of ITC treatment. A result has shown that the basic reproduction number is more sensitive to changes in the Tsetse morality and the proportion of cattle treated with insecticides per day. It will induce a minimum cost for controlling T.b rhodesiense.

The Growing Heart Changes Courses: An Analysis of Trabeculation's Effect on Hemodynamics in Math and Physical Models of a Developing Vertebrate Heart

John Cruickshank, The University of North Carolina, Chapel Hill, NC
mentored by Dr. Laura Miller

Abstract: As the vertebrate heart develops, changes in the shape and texture of the endocardium can result in distinct and non-intuitive alterations to patterns of blood flow through the heart, which may in turn via epigenetic signaling from shear- and strain-sensitive endocardial cells trigger further endocardial development. By this self-reading mechanism, the heart builds itself. We are interested in analyzing the process; in particular, an early stage of vertebrate heart development when little bumps, called trabeculae, form along the inner ventricular endocardium. Trabeculae are well-conserved within vertebrate hearts, but their effect on the hemodynamics of developing hearts is not well understood. We present a mathematical model of the developing heart and use computational fluid dynamic techniques to quantify the effect of trabeculation on flow through our heart model. We examine the effect of trabeculae height relative to the depth of the heart chambers. Further, we present a physical fluid dynamic model, analogous to our simulated model, and use particle image velocimetry (PIV) analysis to interrogate our simulated results.

The Kansa's Method with Staggered Technique

Thir Dangal, The University of Southern Mississippi, Hattiesburg, MS
mentored by Dr. C.S. Chen

Abstract: The Kansa's method is a simple meshless numerical method of solving partial differential equations with high accuracy. However, the resultant matrix of the Kansa's method is normally dense and highly ill-conditioned. In this study, we introduce a staggered technique to the collocation process to overcome these difficulties. By utilizing the staggered technique, we can handle much larger computations than the traditional Kansa's method. The capability of the proposed method is examined by solving two-dimensional Poisson equations with regular and irregular shape of computational domains by uniform and non-uniform interpolation points. The results show that remarkable improvement of accuracy and stability can be achieved by adding the staggered technique to the Kansa's method.

Generalizing Functions to Functions of Matrices

Jared Dangar, Kennesaw State University, Kennesaw, GA
mentored by Dr. Joshua Du

Abstract: Finding the properties of matrices and their functions is critical for the advancement of engineering and mathematics as a whole. While methods exist, most are complicated and only approximate the series of a function of matrices to an nth degree. As a different approach, we will look at diagonalizable matrices and their properties to find quick and efficient results to matrix functions. We will begin with elementary functions of a single matrix (trigonometric, exponential, etc.) to find basic properties to act as building blocks for more complex functions of two or more matrices under special cases (exponential properties). Once these functions are discovered, calculus of matrices functions will be explored.

Multiple Solutions of a Resonant Boundary Value Problem

Hunter Denham, Wake Forest University, Winston Salem, NC
mentored by Dr. Stephen B. Robinson

Abstract: In our research, we examined the second order differential equation y''+ y+ c sin⁡y+tan^(-1)⁡y =0 with boundary conditions y(0)=0=y(π). This arose from the two papers of Landesman and Lazer (1970) with y''+y+tan^(-1)⁡y+k=0 and Schaaf and Schmitt (1988) with y^''+y+c sin⁡y=0. Our goal is to see how many solutions our differential equation actually has, either finitely many or infinitely many solutions, and how this number depends on the constant c. Our methodology was based mostly on the shooting method. So far, we have found that there are two correlations, one between the value of c and the maximum number of bumps that the solution has, and another between the value of c and the occurrence of change in the concavity of the solution. Moreover, we can show that as c gets larger, the number of solutions also gets larger. However, these are not necessarily the positive solutions described by Schaaf and Schmitt.

Looking at Kurtosis: Is my data normal?

Erin Donahue, Elon University, Elon, NC
mentored by Dr. Laura Taylor

Abstract: The validity of statistical conclusions is paramount in everyday lives. In order to statistically analyze data, basic criteria must be met in order to ensure the legitimacy of our conclusions. For many common inferential procedures, one of these assumptions states that the data come from a population that follows the normal distribution. This criterion is often referred to as the “normality assumption.” It is vital to certain statistical analysis procedures to confirm that the normality assumption has been met otherwise bias is introduced into the results and can render the conclusions inaccurate. A common method for checking this criterion includes observing a histogram of the data for a bell-shape. This method can be subjective, and there is no clear delineation for researchers to reject normality. This presentation investigates the use of kurtosis, a measure of peakedness and the weight of the tails of a distribution, in order to assess the shape of the data. Kurto sis is a commonly reported statistic for basic procedures in several software packages, but it is not widely used by researchers. A normal distribution has a kurtosis of three, and our research aims to understand if this fact can be used to less subjectively evaluate the shape of data. Using simulation methods, with a software program that can randomly produce data, the software code will randomly simulate data from the normal distribution with enough runs in order to generate data with various kurtosis values and compare the validity of analysis procedures for these varying kurtosis values. Then by calculating confidence intervals, the proportion of these data sets that capture the true population mean is determined. The capture rate is expected to drop as kurtosis values get further away from the normal distribution kurtosis value of three. Can kurtosis be used as a less subjective way to validate the normality assumption? Current simulation results indicate that kurtosis is not the most useful method to assess normality. The results showed a surprising pattern in the capture rate of 95% confidence intervals as kurtosis increased.

Dynamic Model of Dengue Fever

Chasity Dorsett and Marie Paulemond, Bennett College, Greensboro, NC
mentored by Dr. Hyunju Oh Dr. Jan Rychtář

Abstract: Dengue Fever is one of the most important vector-borne infections that have emerged as a worldwide problem beginning in the 1950’s. This infection is transmitted by its principal vector Aedes Stegomyia aegypti. In this paper, we will present a SIR compartment model that includes Human as the host, Mosquitoes as the vector, and Eggs as the vertical Transmission. From these basic models, we are trying to eliminate the spread of Dengue Fever by killing all eggs and mosquitoes. From these SIR compartment models we deduced the basic reproduction number R0 and force of infected, λ for Dengue Fever. For elimination of the disease we find that the Basic Reproduction number, R0, is sensitive to the proportion of infected eggs, g, natural death rate for mosquitoes, µM, and daily rate of biting from the mosquitoes, a. If infection of eggs is more than 83%, epidemic starts. The optimal solution is when the proportion of infected eggs, g, is at a maximum of 50% and minimum natural death rate for mosquitoes is at 0.3. The optimal result provides the minimize cost using Dichlorodiphenyltrichloroethane (DDT) and the reduce risk of using DDT for populations at risk.

Identification of Transcription Factor Programs under Methylation Control during Brain Development

Madeline Edwards, Elon University, Elon, NC
mentored by Dr. Hehuang Xie

Abstract: Transcription factors (TFs) often bind to specific DNA sequences to promote or block gene expression. The interactions between TFs and target DNA sequences may be regulated by DNA methylation. It has been well recognized that DNA methylation plays an important role in neural differentiation, which is determined by a cascade of TFs. However, the epigenetic regulated TF programs critical to brain development remain largely unexplored. To fill in such a knowledge gap, we first analyzed mammalian brain methylomes to identify genomic loci differentially methylated during development. We compiled a set of experimentally validated TF binding sites from TRANSFAC 7.0, JASPAR 2014, and UniPROBE databases, and applied HOMER to identify TFs of which binding sites enriched in genomic regions hypomethylated in neurons and glial cells, respectively. Using MEME Suite and ClusterZ, we then determined pairs of TFs with binding sites frequently overlapped. With the Cytoscape program, we created a network of possible TF interactions, which can either be complex formation or regulatory. This predicted network provides novel insight to the epigenetic regulation controlling brain development.

Derivative Sign Patterns for Infinitely Differentiable Function in Three-Dimensions

Madeline Edwards, Elon University, Elon, NC
mentored by Dr. Jeffery Clark

Abstract: A derivative sign pattern is a sequence of positive and negative signs that represent the signs of a function and its derivatives over its domain. Special cases of the function's domain in one-dimension analysis include R, where there are four possible sign patterns, and (0,1), with infinitely many possible sign patterns. In the case of R, a function that is infinitely differentiable will have a sign pattern that can be determined from the original function and the first derivative. Prof. Kenneth Schilling expanded from the one-dimensional case to the two-dimensional case for the entire plane. His specific cases included R^+, R^+×(0,1) and R^+×R^+. Building on Schilling's Derivative Sign Pattern Theorem, the expansion to the three-dimensional case is analyzed. Specific cases of interest include the domain R and the unit cube. In the three-dimensional case there is interesting geometry among the derivative sign patterns.

Modeling Traffic At An Intersection

Saniita FaSenntao and Kaleigh Mulkey, Kennesaw State University, Kennesaw, GA

Abstract: The main purpose of this project is to build a mathematical model for traffic at a busy intersection. We use elements of Queueing Theory to build our model: the vehicles driving into the intersection are the “arrival process” and the stop light in the intersection is the “server.” We collected traffic data on the number of vehicles arriving to the intersection, the duration of green and red lights, and the number of vehicles going through the intersection during a green light. We built a SAS macro code to simulate traffic based on parameters derived from the data. In our program we compute the number of vehicles in the queue every time a vehicle arrives and leaves the intersection, the service time, and the total time the vehicle spends in the queue, or the sojourn time. We describe the probability distribution of the queue length in the long run and analyze its dependence on λ and the durations of the green and red light. Using regression we build a model for the dependence of the average queue length and the average service time on λ and the durations of the green and red light. Based on the regression results we propose traffic models that achieve optimal queue lengths and sojourn times.

A Geometric Approach to Voting Theory

Lee Fisher, Appalachian State University, Boone, NC
mentored by Dr. Vicky Klima

Abstract: Arrow's theorem informs us that while some voting methods may be more fair than others, a certain set of reasonable fairness conditions cannot simultaneously be satisfied by any one voting system. Well-known voting systems include the plurality method, where voters are told to pick just one candidate and the candidate who receives the largest number of votes wins; and the Borda count, where candidates are ranked and given points in equally-spaced descending order. This presentation proposes a new voting system which functions as a generalization of the Borda count. In our new system, we assign a value of one to first place, zero to last place, and intermediate rankings are assigned values in arbitrary intervals of non-increasing order. We then use geometric methods to consider the winning results space associated with each candidate. We will compare this voting system against the fairness criteria stated in Arrow's theorem. We will show this voting procedure is monotone, equivalent to the Borda count only in the three-candidate case, and like both the plurality and Borda count methods does not necessarily satisfy the Condorcet winner criterion; which selects the candidate who wins each pairwise comparison to win the election.

Blow Up Failure in the Destabilization of the White Hen Egg Lysozyme Protein

William Frazier, East Tennessee State University, Johnson City, TN
mentored by Dr. Jeff Knisley

Abstract: Molecular Dynamics (MD) is a computational tool that is used to simulate the folding dynamics of a protein. The process begins with a Protein Data Bank coordinates file for a crystallized protein. After a sequence of pre-processing steps, the simulation predicted the coordinates for each atom in femtosecond time steps for up to nanoseconds of simulated dynamics. If a MD simulation fails it is possible the crystal structure is unstable or flawed, and typically, this results in failure during the pre-processing phase. However, it is possible that a stable crystal structure can destabilize during the simulation itself without a corresponding breakdown in the mathematical model itself. In the process of simulating a destabilization of the protein causing it to unfold, we identified a blow up failure across several versions of the simulation. Our goal is to reduce to reduce the occurrence of the blow failure via alterations to the MD simulation that subsequently allows us to achieve scientific objects with the simulation of such proteins.

A new algorithm for Galois groups of quintic polynomials

Robin French, Elon University, Elon, NC

Abstract: Finding solutions of polynomial equations is a central problem in mathematics. Of particular importance is the ability to solve a polynomial by radicals''; i.e., using only the coefficients of the polynomial, the four basic arithmetic operations (addition, subtraction, multiplication, division), and roots (square roots, cube roots, etc.). For example, the existence of the quadratic formula shows that all quadratic polynomials are solvable by radicals. In addition, degree three polynomials and degree four polynomials are also solvable by radicals, which was shown in the 16th century. However, the same is not true for all degree five polynomials. Therefore, we are left with the following question: how do we determine which degree five polynomials are solvable by radicals? To answer this question, we study an important object that is associated to every polynomial. This object, named after 19th century mathematician Evariste Galois, is known as the polynomial's Galois group. The characteristics of the Galois group encode arithmetic information regarding its corresponding polynomial, including whether or not the polynomial is solvable by radicals. In this talk, we will discuss a new algorithm for determining the Galois group of a degree five polynomial.

Comparison of Nonlinear and Linear Stabilization Schemes for Advection-Diffusion Equations

Ryan Grove, Clemson University, Clemson, SC
mentored by Dr. Timo Heister

Abstract: Standard finite element discretizations of advection-diffusion equations introduce unphysical oscillations around steep gradients. Therefore, stabilization must be added to the discrete formulation to obtain correct solutions. The SUPG, dCG91, and Entopy Viscosity schemes are compared using stationary and non-stationary test equations. Differences in maximum overshoot and undershoot, smear, and convergence orders are compared using code written using deal.ii.

Numerical Method for Random Differential Equations

Joseph Hart, North Carolina State University, Raleigh, NC
mentored by Dr. Pierre Gremaud

Abstract: Differential equations are used to model many physical and biological systems. However, when parameters or initial conditions are uncertain, modeling the system because far more complex. Traditional the Monte Carlo method is used to repeatedly sample values for the uncertain parameters, solve the deterministic system, and then draw statistical interpretation of the results. Some have proposed PDF Methods, which seek to find the PDF for the solution of the system with the uncertain parameters considered as random variables. To find this PDF, one must solve a deterministic partial differential equation. This solution is subject to algebraic constrains of being nonnegative and having constant mass of 1 since it is a PDF. Our work purposes a numerical algorithm for computing this PDF in the case of a system of two ordinary differential equations with random initial conditions. The results of our algorithm provide more information than the Monte Carlo approach and it is more computationally efficient in many cases.

Evaluating the Effects of Treatment Terms on White-Nose Syndrome

Leigh Ann Herhold and Dan Thomas Oliver, North Carolina State University, Raleigh, NC
mentored by Drs. Eric Labor, Brian Reich, Krishna Pacifici

Abstract: Bats are the primary predator of agricultural pests and insect vectors of human disease in the United States. The recent emergence of White-nose syndrome in bats is causing widespread fatalities in several bat species in the U.S. This highly fatal fungus poses a threat to agriculture and human health, and demands attention. As a result, wildlife managers at the U.S. Fish and Wildlife Service are posing intervention strategies to contain and slow the spread of white-nose syndrome. While disease dynamic models for white-nose syndrome exist, models that explore the impact of intervention treatment effects and disease progression are lacking. In this project, we explore through statistical simulation the impact of different generic treatment policies to find the optimal intervention strategy.

The hereditary approach to time dependent modeling with fractional exponential function

Lauren Holden, Clemson University, Clemson, SC
mentored by Dr. Sofeya Alekseeva

Abstract: The hereditary approach to time dependent mechanical behavior is being implemented. The methods of optimization were to get the best parameter estimates for the fractional exponential function in the case of creep as well as relaxation for polymer based composites.

Reverse Mathematics and Marriage Problems

Noah Hughes, Appalachian State University, Boone, NC
mentored by Dr. Jeffry L. Hirst

Abstract: The program of reverse mathematics deals with calibrating the logical strength of mathematical theorems. We analyze several theorems regarding marriage problems via the techniques of reverse mathematics. A marriage problem M consists of a set of boys B, a set of girls G and a relation R, a subset of B x G, where R contains (b,g)'' means boy b knows girl g.'' A solution of the marriage problem is an injection f mapping B into G, such that for every b in B, (b,f(b)) is in R. Using the standard anthropocentric terminology, we see that f assigns a unique spouse to each boy from among his acquaintances. In this talk, we will discuss the program of reverse mathematics, review past results regarding marriage problems, share recently completed work concerning the necessary and sufficient conditions for a marriage problem to have a unique solution, as well as show how each of these theorems fit within the framework of reverse mathematics.

Series Representations of Pi

Christina Jones, Kennesaw State University, Kennesaw, GA
mentored by Dr. Joshua Du

Abstract: Using the Taylor Series of arctangent and trigonometric formulas, it is possible to write different series representations of π. By using trigonometric formulas such as the half angle formulas and triple angle formulas the sine and cosine of smaller angles can be found, and therefore the tangent of these angles can be found. After the tangent of an angle is found, a formula to represent π at that angle can be found. After a series representation is found, the series can be used to approximate values of π and to prove that π is an irrational number.

An Investigation of the Efficiency of Slime Mold as a Maze-Solving Algorithm

Marschall Furman, Nicholas Kapur, Wanlin Zheng, North Carolina State University, Raleigh, NC
mentored by Krishna Pacifici, Eric Laber, Brian Reich

Abstract: Physarum polycephalum, commonly known as slime mold, has recently been noted for its ability to find the shortest path between food sources. Unlike numerical approaches, the slime mold is a viable alternative because it will not break down across large-scale networks. We designed an experiment using a template of the streets of Manhattan and strategic food placement to further test the slime’s ability to efficiently find an optimal path. Our study used image tracking analysis to quantify and record the slime’s movement patterns for subsequent analyses. These analyses were compared to a variety of computational optimization algorithms as a means to assess the efficiency and success of the slime mold's actions. Our current results suggest that the slime mold has the ability to find a nearly perfect route over the course of 6 days. Thus far, it has performed in a relatively efficient manner, comparable to automated search algorithms. The ultimate goal is to develop a dynamic model to simulate the slime's search routine, as well as to determine situations in which slime mold optimization can be of use in real world planning.

Lagrange Interpolation Polynomial Scheme for the Numerical Solution of the time dependent Partial Differential Equations using Localized MPS

Balaram Khatri Ghimire, The University of Southern Mississippi, Hattiesburg, MS
mentored by Dr. C.S. Chen

Abstract: We propose Lagrange Interpolation Polynomial Scheme(LIPS) for the numerical solution of time dependent Partial Di fferential Equations(PDEs) combined with Localized Method of Particular solution(LMPS) and Kansa's method. It is a challenging task for using high order accuracy and variable time interval of temporal discretization while solving time dependent problems. Most of the existing time discretization methods such as Laplace transform scheme, time marching scheme, method of lines etc.use the fixed time interval, which in fact, is not suitable to some steady state and transient state problems. To overcome this diffi culty, we employ multi-order Lagrange Interpolation Polynomial Scheme to use variable time interval. Numerical examples governed respectively by two- dimensional di ffusion equation and two-dimensional Advection-Diffusion equation are performed to demonstrate the accuracy and the stability of this new scheme. The numerical results are performed on both fixed time interval and variable time interval. The results are compared with the traditional time marching scheme. In both cases, the numerical results show that new scheme is more accurate and e fficient than the conventional ones.

Estimating Critical Stock Price for an American Option with Dividends Near Expiration by Method of Sub- and Super-solution

Myung Jun Kim, Swarthmore College, Swarthmore, PA
mentored by Dr. Nsoki Mavinga

Abstract: We are concerned with the study of the critical stock price near expiration for an American option with dividend in the Black-Scholes model. We use the method of sub- and super-solutions to find an upper and lower bounds for the critical stock price near maturity and then prove that these upper and lower bounds approach to each other as expiration approaches, leading to conclude the asymptotic behavior of this critical stock price near the expiration.

Statistical Image Filtering and Denoising Techniques for Synthetic Aperture Radar Data

Troy Kling, University of North Carolina Wilmington, Wilmington, NC
mentored by Dr. Maxim Neumann

Abstract: Images of Earth's surface gathered by Uninhabited Aerial Vehicle Synthetic Aperture Radar (UAVSAR) contain a plethora of information about the nature of the scattering media covering the ground. Depending on the type of instrument used to scan the landscape, a combination of radiometric, polarimetric, and interferometric data can be gathered. This data can be used for a wide range of purposes, from conducting biomass studies to building digital elevation maps.
Despite its usefulness, synthetic aperture radar confers a speckled appearance to the images it takes of Earth's surface. This speckle noise hinders efforts to analyze image contents, and therefore it is necessary to develop new, efficient techniques for removing this noise. Non-local approaches are particularly well-suited for this task, since they are able to reduce speckle noise while preserving important structures in the image.

Particular Solution of Matern RBFs and its implementation in Method of Particular Solution

Anup Lamichhane, The University of Southern Mississippi, Hattiesburg, MS
mentored by Dr. C.S. Chen

Abstract: We derive the particular solution of Matern radial basis function for the Laplace differential operator in 2D and 3D. The derived particular solution is essential for the implementation of the method of particular solutions (MPS) for solving various types of partial differential equations. To find a good shape parameter of Mate function, we apply the technique of Leave-One-Out-Cross- Validation (LOOCV). Numerical examples are given to demonstrate the effectiveness of the derived particular solutions using Matern function.

Bumps and Ridges: Effects of Trabeculae in Heart Morphogenesis

Andrea Lane, The University of North Carolina, Chapel Hill, NC
mentored by Dr. Laura Miller

Abstract: Heart morphogenesis is governed by hemodynamic forces, epigenetic signaling, and other biological regulatory networks. We present a fluid dynamic model of the stage in heart development when little bumps, called trabeculae, begin to form along the inner ventricular endocardium. Trabeculae are well-conserved constructs within all vertebrate hearts. The effect of trabeculae on the hemodynamics of developing hearts is not well understood. In this study, computational fluid dynamics is used to quantify the effects of Reynolds number and occlusion of the atrioventricular valve on the resulting flows. Dynamic processes such as vortex formation are dependent on Re and thus are important to the generation of shears and strains, which will aid in proper development and functionality.

Evaluating the Strength of Evidence in DUI Cases Presented in North Carolina

Rebecca Law and Brady Melton, North Carolina State University, Raleigh, NC
mentored by Dr. Eric Laber

Abstract: Criminal penalties for driving while intoxicated (DWI) in North Carolina are based on hard thresholds; for example, having a blood alcohol content (BAC) at or above 0.08 is considered legally impaired. However, BAC measurements are typically taken using breathalyzers, which are subject to measurement error. Additionally, breathalyzer readings in North Carolina are truncated, i.e. a person blowing a 0.079 would have a breathalyzer reading of 0.07. The purpose of our research is to explore the impact of this error on the strength of evidence in North Carolina DWI cases and to construct recommendations for both law enforcement and courtroom decisions. Using data collected from breathalyzer tickets in Orange County, we have estimated the measurement error using a truncated random effects model and have calculated a prediction interval to determine any individual’s true BAC given the individual’s breathalyzer results. We also ran a parallelized simulation study to determine the effects of the distribution parameters on our model, and plan on exploring factors such as temperature, humidity, and machine calibration. We have created two lookup tables to determine an individuals true BAC, one based on prediction intervals, and the other on the probability that an individual’s true BAC is above 0.08. Using these lookup tables, the courts can determine the strength of evidence in DWI cases in North Carolina.

Pattern Formation in Continuous Neural Networks

Stephen Meier, Joint School of Nanoscience and Nanoengineering, Greensboro, NC
mentored by Dr. Joseph M. Starobin and Jarrett L. Lancaster

Abstract: Information processing in the brain remains an open topic in current Neuroscience research. Literature suggests pattern formation as a possible mechanism for information processing. Here we identify and model several factors which contribute to pattern formation in propagating action potentials.

Parameter Sensitivity Analysis for CO-mediated Sickle Cell De-polymerization

Yao Messan, North Carolina A&T State University, Greensboro, NC
mentored by Dr. Liping Liu, Dr. Mufeed Basti

Abstract: Sickle cell anemia is an abnormality that causes a deformation in the shape of the red blood cell that hinders the circulation of the red blood cell through the blood vessel. The deformation is caused by the association of monomers to each other to form polymers (polymerization). The oxygenation of the sickle cell may lead to the melting of polymers (de-polymerization). This study employed is a global sensitivity analysis that enumerates the overall effect of the model input parameters on the output by perturbing the model input parameters within large ranges. The MPSA is implemented by employing a Monte Carlo method over a broad range of parameters values and comparing the cumulative distribution functions of the acceptance and unacceptance groups of the parameters. All four concentrations as model outputs and four binding/melting rates as input parameters are considered in this study. As a result, for the model output de-oxygenated monomers, the most sensitive parameter is the CO-binding rate of monomers, while the most insensitive parameter is the CO-binding rate of polymers being that it does not affect the de-oxygenated monomers at all. For the model output CO-bound monomers, the most sensitive parameter is the melting rate of the de-oxygenated polymers, while both CO-binding rate of polymers and melting rate of polymers are insensitive. For the model output de-oxygenated polymers, the most sensitive parameter is the melting rate of de-oxygenated polymers, while the other three parameters are insensitive. For the model output CO-bound polymers, the most sensitive parameter is the CO-binding rate on polymers, while the most insensitive parameter is the melting rate of CO-bound polymers.

A Nonlinear Model of Cancer Tumor Treatment With Cancer Stem Cells

Alexander Middleton, Winthrop University, Rock Hill, SC
mentored by Dr. Kristen Abernathy

Abstract: According to the American Cancer Society, cancer is one of the leading causes of death, second only to heart disease. We present a system of nonlinear, ﬁrst-order, ordinary differential equations that describes tumor growth based on healthy cell, tumor cell, and cancer stem cell populations. We include terms within our model which reﬂect the differing effects of chemotherapy and anti-angiogenic therapy to respective cell populations. We perform stability analysis on the equilibrium solutions to predict the long-term behavior of the cell populations. With analysis, it is shown that chemotherapy, with the co-administration of anti-angiogenic treatment, can produce three states: recurrence or persistence of cancer, and a cure state. Results are supported numerically and bifurcation diagrams are included to illustrate the different behavior of cell populations depending on the amount of treatment administered.

Groups as Metric Spaces

Dani Moran, UNCG, Greensboro, NC
mentored by Dr. Greg Bell

Abstract: A group G can be given a natural metric structure based on its generating set. By considering a concept of coarse equivalence, we can get rid of the dependence on the generating set in order to talk about the group itself as a metric space. In this talk we will introduce these concepts in detail, and discuss the construction of a graph of groups, and what metric properties such a construction preserves. This is an extension of work by Antolin and Dreesin.

Item Response Analysis of PTSD Questionnaires for Veterans

Eli Mullis, University of North Carolina Wilmington, Wilmington, NC
mentored by Dr. Cuixian Chen

Abstract: The goal of this project is to explore the possibility of shortening the Minnesota Multiphasic Personality Inventory-2 (MMPI-2) exam for veterans and military personnel using Multidimensional Item Response Theory (MIRT) models. The MMPI is a popular, general-purpose standardized test for measuring adult personality, and has existed in various forms since 1943. The MMPI-2 specifically consists of 567 items, with an alternate abridged version consisting of 370. Item Response Theory is a mathematical model used to design and assess tests based on item difficulty and respondent’s skill level. This project will be carried out by processing scanned PDFs of MMPI-2 responses by personnel into a large Excel spreadsheet. This step should ideally be easy to automate using Python scripts, as we have access to a very large number of respondents. After the data is organized into a usable format, analysis can be performed using the R mirt package.

Investigation of Two Dimensional and Three Dimensional Flow Over Mountains With Shallow-Water Numerical Models

Guy Oldaker IV, North Carolina A&T State University, Greensboro, NC
mentored by Dr. Liping Liu, Dr. Yuh-Lang Lin

Abstract: This project studies how 2-D and 3-D shallow-water models can be used to describe weather phenomenon. Specifically, we investigated how certain non-dimensional parameters, namely the Froude number (F) and non-dimensional mountain height (M), are able to dictate various flow regimes. First, we implemented a shallow-water model that was assumed to be hydrostatic, inviscid and incompressible. We began by solving a 1-D advection equation (x-velocity) and observed linear and nonlinear effects. In the linear case, the wave is well-behaved with constant amplitude, while the nonlinear wave exhibits distortion and wave steepening. Then, we moved to a 2-D shallow-water model by including the continuity equation and performed the same simulations. A stability analysis was performed on the leap-frog in time and 4th order centered in space scheme used throughout the project. It was found that the system was stable for values of Courant number less than 0.75. Next, we added bott om topography to the 2-D model equations and explored how different values of the Froude number F=U/√gH and non-dimensional mountain height M=h_m/H produced varied flow regimes. In particular, we simulated the following: 1) Supercritical flow, 2) Subcritical flow, 3) Flow with upstream and downstream propagating jumps, 4) Flow with upstream propagating and downstream stationary jumps. We examined linear and nonlinear cases for each. Lastly, we repeated these simulations using a 3-D shallow-water model.

Bayesian Model Calibration Techniques for Physical Applications

Thomas Oseroff, North Carolina State University, Raleigh, NC
mentored by Dr. Ralph Smith

Abstract: In this presentation, we discuss the implementation of Bayesian model calibration techniques to estimate parameters in physical models. Bayesian methods provide a natural framework to quantify uncertainties inherent to parameters, models and data since they treat inputs as random variables having associated probability density functions. We focus on Markov Chain Monte Carlo (MCMC) and Delayed Rejection Adaptive Metropolis (DRAM) algorithms due to their robustness with regard to nonlinear parameterizations and potentially correlated parameter sets. These algorithms incorporate the geometry of the parameter space by employing a proposal distribution in which the covariance matrix is updated as proposed chain elements are accepted in the posterior distribution. The performance of MATLAB and Python implementations of the algorithms is illustrated for spring and heat equation examples.

Evolutionary Games on Graphs

Christopher Paoletti and James Withers, Emmanuel College, Boston, MA
mentored by Drs. Benjamin Allen, Christine Sample, Yulia Dementieva

Abstract: Evolutionary game theory is a mathematical approach to studying the evolution of social behavior. Interactions that affect reproductive fitness are conceptualized as games and spatial population structures are represented as graphs. Using this framework, we investigate the effects of self-interaction on the evolution of cooperative strategies. In our model, the population is represented as a cycle with self-loops and individuals can employ one of two competing strategies in a matrix game. Interaction and replacement can be governed by the same graph or by two different graphs, which can be weighted or un-weighted. We calculate fixation probabilities for these strategies and derive exact conditions for natural selection to favor one strategy over the other. We determine the behavior of these conditions in the limits of weak selection and large population size, and show that the limiting conditions do not depend on the order in which limits are taken. We also consider evolutionary success when mutation is present. All calculations are performed for two different update rules (birth-death and death-birth). We conclude that self-interaction can promote the evolution of cooperative strategies in spatially structured populations.

The discretization of ordinary differential equations

Sarah Parsons, Wake Forest University, Winston Salem, NC
mentored by Dr. Stephen Robinson
With a focus on the ordinary differential equation boundary value problem, -u^"=u+sin⁡(u), where u(0)=0=u(π), research was conducted to characterize and prove the existence of solutions to its discrete analogue, Ax=λ_n x+v, where A is a Laplacian matrix of size n, v is an oscillating nonlinear vector consisting of trigonometric functions, and λ_n is the principal eigenvalue. The goal for this research was to apply the results of the discrete version to the problem from ordinary differential equations and to compare the solutions to the results of previous research completed in 1988 by Renate Schaaf and Klaus Schmitt. Using mostly two-dimensional calculus, multivariable calculus, and linear algebra, the discretized equation was transformed into a maximum/minimum problem and solutions for lower dimensional cases were proven to exist. Furthermore, a characterization of the graph was determined to be trough-like, parabolic, and to consist of infinitely many minimums and saddles. For the n = 4 case and for higher dimensions, ideas from ODE were adapted and a diagram of lattice points was created to reflect the graph of the function. Solutions for the higher dimensional matrices were proven to exist using a proof involving the Implicit Function theorem, Extreme Value theorem, and an Irrational Steps Around a Circle theorem. The results of this project verified and characterized those of Schaaf and Schmitt and may provide further insight into the nature of ordinary differential equations that other methods do not.

Reasonance and Margin Flexibility upon Oblate Jellyfish

Antonio Porras, The University of North Carolina, Chapel Hill, NC
mentored by Dr. Laura Miller

Abstract: To further explore how jellyfish optimize their swimming, we analyzed the role of flexibility in jellyfish jet propulsion using computational fluid dynamics. Flexibility has been shown to play a role in propulsion throughout the animal kingdom when examining it in context with resonance phenomenon. In our study, we explored Reynolds number effects on the resonance properties of the bell, and we focused on how flexible oblate bells optimize jellyfish propulsion. We used the Immerse Boundary Method to simulate jellyfish flow movement to measure its vertical velocity, resonant frequencies, bell elasticity and viscosity. Jellyfish have been shown to tune their swimming movement to the resonant frequency of their bell (DeMont and Gosline, 1987). More recently, work has been done to show how flexibility at the bell margin influences propulsion in oblate jellyfish (Colin et al, 2013). Our study concluded that for high viscous environments, the resonant frequency of the bell increases and as the stiffness of the bell increases, so does its natural frequency.

Ecological models with U-shaped density dependent dispersal

Jordan Price, Auburn University Montgomery, Montgomery, AL
mentored by Dr. Jerome Goddard II

Abstract: Dispersal of an animal population is considered to be density dependent when dispersal decisions are made based on the presence of conspecifics. Recently, several ecologists have noted density dependent dispersal in multiple species of animals from insects to birds to bears. In fact, the relationship between population density and dispersal has been shown (empirically) to be U-shaped. In this talk, we will model the effects of U-shaped density dependent dispersal on the patch-level dynamics of a population using one of the most versatile theoretical population frameworks, the reaction diffusion population model. In particular, we will explore the dynamics of a diffusive logistic population model on a one-dimensional domain with nonlinear boundary conditions modeling U-shaped density dependent dispersal via study of the model’s positive steady state solutions. We obtain results through use of the quadrature method and Mathematica computations and will briefly explore their biological implications.

Pattern Formation in Continuous Neural Networks

Aaron Rapp, Western Carolina University, Cullowhee, NC
mentored by Dr. Mark Budden

Abstract: Ramsey theory has posed many interesting questions for graph theorists that have yet to be solved. Many different methods have been used to find Ramsey numbers, though very few are actually known. Because of this, more mathematical tools are needed to prove exact values of Ramsey numbers and their generalizations. Budden, Hiller, Lambert, and Sanford have created a lifting from graphs to 3-uniform hypergraphs that has shown promise in extending known Ramsey results to hypergraphs. This talk will build upon their work by considering other important properties of their lifting and analogous liftings for higher-uniform hypergraphs.

Modeling Traffic At An Intersection

Dennys Rosales, Kennesaw State University, Kennesaw, GA

Abstract: Vehicles arrive according to a Poisson process and "service" time (time it takes a vehicle to go through the green light) is considered constant. With green light and red light returning at fixed intervals of time, the service time has a general distribution whose average we derive through simulations. Other variables that are derived from simulations are average amount of vehicles in the system, average sojourn time, and how many vehicles were serviced. All these simulations are a product of Maple. We will collect data from traffic and compare the simulation results to the actual traffic measurements. Eventually we will attempt to see if there are any ways in which we can optimize the traffic at s busy intersection without any major changes, such as adding another lane.

Exploring the Role of Yeast Nutrient in Hard Cider Brewing

Jamie Rowell, Western Carolina University, Cullowhee, NC
mentored by Dr. John Wagaman

Abstract: In beer brewing, it is widely known that trace elements, particularly zinc, are required by yeast in order to grow and ferment. The general consensus among large-scale brewers and home brewers alike is that the most sensitive and time-consuming step of beer production is the fermentation of wort. Problems encountered during fermentation can often be contributed to a lack of necessary trace elements needed for the fermentation of yeast. While the wort provides trace elements for the yeast, zinc is generally not available in the required amount in the wort. To prevent such problems, brewers may supplement wort with additional zinc salts during malt boiling or yeast storage. In this talk, we explore the use of statistical models in explaining the role of yeast-nutrient in hard cider brewing.

Pilesize Dynamic One-Pile Nim

James Rudzinski, UNCG, Greensboro, NC
mentored by Dr. Clifford Smith

Abstract: A combinatorial game is a two-player sequential game where players take turns changing the position of the game to reach a winning condition. This talk will give an overview of impartial games and the Nim value analysis of a particular combinatorial game. The game is a one-pile counter pickup game for which the maximum number of counters that can be removed on each successive move changes during the play of the game.

Finite Copula Mixture Models and Estimation

Sumen Sen, Old Dominion University, Norfolk, VA
mentored by Dr. Norou Diawara

Abstract: Combining copulas as a finite mixture model help us to not only fully understand the different dependence patterns between observed random variables, but also add more flexibility into the model. Finite mixture model is a probabilistic model represented as a weighted sum of a few parametric densities. Many authors have shown that this type of mixture model is very useful for uncovering hidden structures in the data, and used the mixture model of copulas for clustering in data mining. Mixture of three copulas (Gaussian, Gumbel and Gumbel-Survival) is also used to model dependence of monthly returns between a pair of stock indexes. Due to complexity of these mixture models, most of the cases, MLE method of estimation is not convergent and regular EM algorithm is very slow. We propose a new estimation algorithm to estimate parameters for such models. This estimation method is inspired by the inference function for margins(IFM) method. The algorithm is tested on simulation data.

Questionnaire Refinement in Neuropsychological Assessments

William Smith, University of North Carolina Wilmington, Wilmington, NC
mentored by Dr. Cuixian Chen

Abstract: The 100 item Trauma Symptom Inventory (TSI) diagnoses and assesses the activity of post-traumatic stress disorder (PTSD) and other psychological disorders. Over the past few years the TSI has been administered to thousands of veterans and active marines stationed in Camp Lejeune, a base in Jacksonville, North Carolina, who were referred by military neurologists. In utilizing Item Response Theory (IRT) on these test results, multiple statistical analyses provided insight into the underlying characteristics of the TSI: item difficulty, item discrimination, and local dependency. The TSI is time consuming, therefore a shorter test with tailored assessments is needed. From the IRT model, the selected subset of items will provide an equally as informative test as the longer version, while still maximizing precision along all segments. According to psychologists, “poor effort” constitutes up to 50% of the difference in cognitive testing performance, which may stem from the lengt h of the TSI.

The Madness of March: Predicting the NCAA Division I Men's College Basketball Tournament Using Markov Chains

Alex Smyth, East Tennessee State University, Johnson City, TN
mentored by Dr. Jeff Knisley

Abstract: Every year, millions of people fill out brackets trying to predict the outcome of the NCAA Division I men's basketball post-season tournament. The tournament consists of the winner of each of the NCAA's 32 Division I conferences along with 38 other of the most proven teams in the nation. However, there are 9.2 quintillion different possible combinations for completing the NCAA bracket. Although there have been many different methods created for efficiently predicting the outcome of the tournament, this paper will explore the use of Markov Chains to predict the outcome of the NCAA tournament.

Counting Roots and Galois Groups

Nicole Soltz, Elon University, Elon, NC

Abstract: Let $f(x)$ be an irreducible polynomial over a field $F$ with roots $a$, $b$, and $c$ (in some algebraic closure), and let $K=F(a)$ be the extension obtained by adjoining one root of $f$ to $F$. We present two methods for determining the Galois group of $f(x)$. One involves answering the question: is $(a-b)(a-c)(b-c)\in F$? The other involves answering the question: how many roots of $f$ are in $K$? We end by discussing an application to computing Galois groups of cubic polynomials defined over an extension of the $p$-adic numbers.

Modeling the Population Dynamics of Daphnia Magna

Sarah Stokely, North Carolina State University, Raleigh, NC
mentored by Dr. H.T. Banks

Abstract: Daphnia magna serve as model organisms that can provide valuable insight into their surrounding environments. These organisms have complex life cycles with individual life parameters that are incredibly variable. This variability increases due to density dependent effects. In our work, we used a Leslie matrix model that incorporated density dependent mortality and time-delayed fecundity effects in order to examine the behavior of Daphnia populations. The model was moderately successful at capturing the initial peak as well as the semi-oscillatory behavior of older populations. In future work, we would like to use this approach to predict the effects of climate change, environmental toxins, and to develop a model that will work with aggregate data.

Nonnegative Matrix Factorization Methods

Ryan Story, University of North Carolina at Wilmington, Wilmington, NC
mentored by Dr. Cuixian Chen

Abstract: Nonnegative Matrix Factorization (NMF) is a low-dimensional approximate parts based representation of a data matrix. We will discuss different constraints on such a factorization. Namely, constraints to an orthogonal subspace. Sparseness constraints better construct parts-based representations of data, and orthogonal constraints ameliorate clustering performance.

Isomorphy Classes of Trivolutions of $SL_2(k)$

Ontario Stotts, Wake Forest University, Winston Salem, NC
mentored by Dr. Loek Helminck

Abstract: Trivolutions are group automorphisms of order 3. $SL(2,k)$ is the group of 2-by-2 matrices with determinant 1. In this paper we will give a characterization for the isomorphy classes of trivolutions of $SL(2,k)$ with $k$ any field of characteristic not 2 nor 3. This work is analogous to work by Helminck and Wu on automorphisms of order 2.

Influence of Individual's Mobility and Neighborhood Size on the Evolution of Cooperation

David Suarez, University of North Carolina at Greensboro, Greensboro, NC
coauthored by Praveen Suthaharan (NC State) and Elizabeth Bergen (Cornell University)
mentored by Dr. Jan Rychtář and Dr. Jonathan Rowell

Abstract: Researchers have long sought to identify which mechanisms can overcome basic selection pressures and support the establishment of cooperation within a population. In a slight variation of their traditionally conception, cooperators are individuals who pay a cost for another individual to receive a benefit, whereas wildtype individuals are free-riders or defectors who receive but do not provide benefits, thus paying no cost. In models of well- mixed populations, free-riders are favored; however, cooperation can emerge and evolve in spatially structured populations. In this paper we extend previous results on the evolution of cooperation to a system of a finite, fixed population of mobile competitors. We conduct a stochastic simulation study examining how parameters controlling an individual's mobility and neighborhood size influences the likelihood that cooperation can evolve within the population. We find that both greater mobility and larger neighborhood size inhibit the evolution of cooperation because it allows the free-riders to find the cooperators faster and exploit them more.

Toxoplasmosis Vaccination Strategies

David Sykes, University of North Carolina at Greensboro, Greensboro, NC
mentored by Dr. Jan Rychtář

Abstract: The protozoan Toxoplasma Gondii is a parasite often found in wild and domestic cats, and it is the cause of the disease Toxoplasmosis. More than 60 million people in the United States carry the parasite, and the Centers for Disease Control have placed toxoplasmosis in their disease classification group Neglected Parasitic Infections in which there are five parasitic diseases targeted as priorities for public health action. In recent years, there has been significant progress toward the development of a practical vaccine, so vaccination programs may soon be a viable approach to controlling the disease. Anticipating the availability of a toxoplasmosis vaccine, we are interested in determining when cat owners should vaccinate their own pets. To investigate this, we have created a mathematical model describing the conditions under which vaccination is advantageous for a person living with cats. Analysis of our model shows that under any fixed set of parameters the population vaccination level will stabilize around a value that can be directly computed. We find that populations may achieve herd immunity if the cost of vaccine is zero, there is a critical cost threshold above which no one will use the vaccine, and a vaccine cost slightly below this critical threshold results in high usage of the vaccine, conferring a significant reduction in population seroprevalence.

Interaction Neighborhoods and Memory Effects in a Spatial Prisoner's Dilemma Game

Eli Thompson, Miami University, Oxford, OH, and Jasmine Everett, Bennett College, Greensboro, NC
mentored by Dr. Jan Rychtář and Dr. Jonathan Rowell

Abstract: Evolutionary Game Theory and the Prisoners Dilemma Game (PD) have been used to study the evolution of cooperation. We consider a population of asexually reproducing, age-structured individuals in a two- dimensional square lattice structure. The individuals, either cooperators or defectors, play the PD with their neighbors to accumulate reproductive fitness. We focus on the effects of memory of past interactions and neighborhood size on the evolution of cooperation. We show that larger neighborhood sizes are detrimental to cooperation. Further, we show that larger memories actually hurt the spread of cooperation in small neighborhood sizes. For larger neighborhood sizes, however, longer memories are more favorable to the spread of cooperation than shorter memories.

Quadratic Splines Maximum Entropy Method to Approximate the Invariant Density of the Frobenius-Perron Operators

Tulsi Upadhyay, The University of Southern Mississippi, Hattiesburg, MS
mentored by Dr. Jiu Ding

Abstract: The quadratic B-splines are introduced in the maximum entropy method for the numerical approximation of unique invariant densities of the Frobenius-Perron (FP) operators, P S, related to nonsingular transformations S : [0,1] → [0, 1]. Since the proposed scheme is based on a modified maximum entropy method, which overcomes the ill-conditioning caused by the traditional maximum entropy method, more accurate results are obtained by implementing large number of moment functions. The method has a third order convergence rate which is justified by the numerical results. We compare the new results with the results from three existing methods.

The Role of the Pericardium in the Circulatory System of the Juvenile Tunicate – A Computational Modeling Approach

Maria Williams, The University of North Carolina, Chapel Hill, NC
mentored by Drs. Laura Miller and Lindsay Waldrop

Abstract: Shortly after settlement and metamorphosis, juvenile tunicates of Ciona savignyi develop a beating heart, where a contracting myocardium is enclosed by a stiff pericardium. The myocardium contracts peristaltically down the length of the heart tube and reverses direction every few beats. We present a computational model of a circulatory system based on the juvenile tunicate in the Immersed Boundary with Adaptive Mesh Refinement (IBAMR) software package. Morphological features, including the number of branching vessels, the diameter of vessels and the diameter of the pericardium and myocardium, of juvenile tunicates were measured with light microscopy and used to create the geometry of the model circulatory system. Fluid flow results from the model were validated using measurements of blood cell movements in the animal. This model can be used to explore the effects of scaling, fluid forces, circulatory geometry, and pumping mechanism in developing tunicates. Since the size, pumping frequency, and circulatory geometry of the juvenile tunicate are similar to that of the early stages of the vertebrate heart, this model could be used to explore hypotheses relating fluid flow to major developmental events in the hearts of vertebrates.

Numerical Solutions of American Options with Dividends

Chi Zhang, Swarthmore College, Swarthmore, PA
mentored by Dr. Nsoki Mavinga

Abstract: We study the Black-Scholes model for the American options with dividends. We analyze the problem as a free-boundary problem for the heat equations and use finite difference methods to get numerical solutions. We apply an explicit scheme to the option pricing and find that this scheme is numerically unstable. To overcome this instability issue, we consider the fully-implicit method and the Crank-Nicolson method. We use Matlab to implement and compare results of these three methods.

3D Mathematical Modeling and Simulations of Cell Mitosis by a Phase Field Approach

Jia Zhao, University of South Carolina, Columbia, SC
mentored by Dr. Qi Wang

Abstract: During a cell cycle, mitosis is a process, in which a mother cell duplicates into two generically similar daughter cells. In the initial stage of mitosis, the mother cell, attached on a substrate, would undergo a dramatical shape change by detaching from the substance and forming a round surface. At the late stage of mitosis, a contractile ring would form in cell orbit and the mother cell would split into two daughter cells, which is known as cytokinesis for eukaryotic cells. Recently, we have developed a series of three-dimensional hydrodynamic models by a phase field approach, studying cellular mitosis. Qualitatively patterns of cell rounding, blebbing and division process have been captured, which agrees with experiment observation from our biologist collaborations. In this talk, our preliminary study on the mechanism and controlling factors of cell mitosis would be present. 3D numerical simulations will be shown, as well.

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