# The Annual UNCG Regional Mathematics and Statistics Conference

## Abstracts of the talks

### Nonexistence of Positive Radial Solutions to a Quasilinear Semipositone Systems

Abraham Abebe, UNCG, Greensboro, NC
mentored by Dr. Maya Chhetri

Abstract: We consider a coupled system of quasilinear boundary value problem in a ball. The nonlinearities are smooth non-decreasing functions with p-superlinear growth at infinity. It is known that there is a positive solution of this system when both nonlinearities satisfy the so called semipositone condition at the origin and the bifurcation parameter is small. We complement this existence result by showing that the system has no positive solution for large bifurcation parameter. We prove this result by analyzing the behavior of the positive solution near the center of the ball as well as near the boundary of the ball to arrive at a contradiction.

### Continued Fractions, Ford Circles, and Descartes� Theorem

Benjamin Ackerman, Elon University, Elon, NC
mentored by Dr. Todd Lee

Abstract: First introduced by Lester R. Ford in 1938, Ford circles can be used to visually represent the relationship between continued fractions and rational approximation of a real number. In 2011, Ian Short used Ford circles to prove that the convergents of the continued fraction expansion for a real number coincide with the rationals that are best approximations of that number. We will analyze the kth convergent of a continued fraction chain of Ford circles using a theorem of Descartes.

### Finding Solutions to the Turning All Lights Out Puzzle in 3D

John Antonelli, Elon University, Elon, NC
mentored by Dr. Crista Arangala

Abstract: The typical lights out game consists of the Lights Out Puzzle. It gained popularity when Tiger Electronics created their hand-held Lights Out Puzzle in 1995. The game consists of 25 lights in a 5 x 5 grid. The goal of the lights out game is to turn all of the lights off (or out) by pressing the buttons in the 5x5 grid. Each press not only changes its own state (on or off), but also the states of all the horizontally and vertically adjacent buttons. Anderson/Feil gave mathematical insight into the 5 x 5 classic lights out game by integrating linear algebra and nullspaces to find initial states that are solvable. In previous research, the Lights Out Puzzle only looked into two dimensions, so we analyze the Lights Out Puzzle by adding a third dimension. We will give numerical data that shows the possibilities of finding solutions to a lights out cube of arbitrary dimensions.

### Estimating Parameters for Deterministic and Stochastic Models of Mosquito Larval Growth

Tim Antonelli, NC State University, Raleigh, NC
mentored by Dr. Alun Lloyd and Dr. Fred Gould

Abstract: Dengue infects nearly 400 million people each year and is currently the world�s fastest growing tropical disease. There is no treatment or vaccine available, so control efforts focus on the primary mosquito vector, Aedes aegypti, which often breeds in artificial containers in and around homes. Mathematical models are useful in assessing the feasibility of such control efforts and in guiding experimental design. Skeeter Buster is a detailed model that was developed to predict the response of an Ae. aegypti population in Iquitos, Peru to various control strategies. However, the model consistently overestimates both the proportion of containers with larvae present and the abundance of larvae found in the field, due to an incomplete understanding of food availability and its impact on larval growth. We develop two simple models to describe larval development and mortality: a linear deterministic model and its stochastic Markov chain counterpart. We then employ nonlinear least-squares and maximum-likelihood techniques to estimate the rates of each based on field data from Iquitos. Our results can inform more complex models, such as Skeeter Buster, that require careful calibration to field data.

### Modeling of Breast Cancer Using Mathematical Game Theory

Ke'Yona Barton and Corbin Smith, Bennett College, Greensboro, NC
mentored by Dr. Tsvetanka Sendova, Dr. Hyunju Oh, and Dr. Jan Rychtář

Abstract: here are over one hundred different types of cancer. Among these types of cancers breast cancer is the third most common. African American women have the highest rate of breast cancer. Being that we are African American and attend a Historically Black Women�s College we sought this as our motivation and drive for our research on breast cancer. Our main goal was to help cancer researchers and doctors explore possible drugs to suppress or defeat cancer through mathematical game theory modeling. Through computer applications we were able to model a typical cancer cell and add factors to see how it would affect the cell growth. In a typical cancer patient there are native cells, which are the healthy cells, macrophages, which rid the body of toxic cells, and there are benign tumor cells or metastatic (motile) tumor cells or there can be a combination of all four in a patient. These cells are considered to be our �players� in our modeling. We sought factors that would at least decrease the metastatic tumor cells, being that this is the worst type of cancer, and increase the native and macrophage cells. In order for this condition to be satisfied we found that the fitness of our MOT (motile) had to be less than the fitness of the Mφ (macrophages) and the NAT (native) cells. Fitness is considered the expected payoff off each player. To conduct our fitness we had to create variables according to each player that would give the payoff for each player. We concluded that if we could find a drug to increase the ability of macrophages to consume tumor cells and to increase the cost that the Motile cells pay for movement that cancer could possibly be defeated. If we could not find a drug to do both, we can at least find a drug to increase the cost of movement. If we make it costly for cancer cells to spread giving them a negative payoff, it is no longer beneficial to be a cancer cell and over time the cells would die off.

### Dynamics of Triatomine Infestation in a Population of Houses

Crystal Bennett, North Carolina A&T University, Greensboro, NC
mentored by Dr. Juan P. Aparicio

Abstract: Trypanosoma cruzi, is the causal agent and parasite of Chagas disease, a neglected tropical disease transmitted mainly by blood-sucking triatomine insects in Latin America. Because of the unavailability of a cure for Chagas disease, disease control relies on the control of the vector population. In this work, we developed deterministic and stochastic mathematical models for the dynamics of bug infestation in a community of houses. We used a Levins metapopulation approach in which houses are considered to be patches that can be in one of three states: empty, infested, or treated. First, we considered spatially implicit models for homogeneous and heterogeneous populations. We studied the effect of differences in housing quality in infestation dynamics and the effect of heterogeneity in the distribution of the houses. Then, we developed more realistic spatially explicit, agent-based, metapopulation models. The models were used to assess the effect of different control strategies on house infestation. The results show that spraying only bad houses is more benefficial than spraying the whole community while using the same treatment rate.

### Analysis of Otoacoustic Emissions using Wavelet Transform to Examine the Physiological Basis of Noise Induced Hearing Loss Associated with A Specific Genetic Polymorphism

Ishan Bhat, UNCG, Greensboro, NC
mentored by Susan L. Phillips and Dr. Shanmugathasan Suthaharan

Abstract: Noise Induced Hearing Loss (NIHL) can lead to permanent hearing loss; hence it is important to understand the physiological basis of this problem. Recent research suggests that genetic variability plays an important role in modifying susceptibility to NIHL. However it is still not clear how individuals with different genetic predisposition to NIHL acquire damage to inner ear physiology. The inner ear physiology was evaluated by comparing the amplitude of click evoked otoacoustic emissions (i.e. the ear echoes generated by inner ear in a response to clicks) between participants with and without a specific genetic polymorphism (i.e. genetic variability between individuals) previously associated with NIHL. Fourier Transform has been the major player in calculating the amplitude of the clicks and analyzing otoacoustic emissions across the frequency range. However the Fourier Transform has limitations in separating frequencies on rapidly changing click evoked otoacoustic emissions. It cannot analyze the spectral aspects of the rapidly changing click evoked otoacoustic signals. In our research we hypothesized that individuals with a specific genotype previously associated with NIHL will show significantly reduced amplitude and poorer latencies at the frequency range from 1 to 4 kHz compared to individuals with the genotype not associated with NIHL. We adopt the wavelet transform technique to evaluate amplitude and latency-related aspects of click evoked otoacoustic emissions across the frequency range from 1 to 4 kHz. The results and findings with wavelet transform will be presented at the conference.

### Basic Linear Algebra behind Google's Search Engine

Colleen Brockmyre, Elon University, Elon, NC
mentored by Dr. James Beuerle

Abstract: Although Google's presence in the technology world is huge today, it began just fifteen years ago as a search engine. It was the first search engine to use a Page Ranking system, putting the more important pages higher than others. The internet can be visualized through a web-graph, and that graph can then be represented by a matrix. Using this matrix and the corresponding eigenvectors, Google is able to give every website an importance score. In this presentation, we will discuss Google's implementation and several of its variations.

### Fractals and their application in medical imaging

Todd Calnan, Elon University, Elon, NC
mentored by Dr. Jeff Clark

Abstract: In the 1800s, Leibniz first developed the idea of fractals but mathematics at that time had not advanced to a stage where they could be accurately described. His ideas would be developed over the next 200 years and with the aid of computers the concept of fractals has come to be common knowledge. We look at the use of fractal properties to create algorithms in order to compress medical images efficiently enough to send them over the web rather than physically transporting them across the country.

### Queueing Theory Applied to Traffic Analysis

Zachary Carter and Narae Song, Kennesaw State University, Kennesaw, GA

Abstract: We apply the mathematical theory of M/G/1 queues (Poisson arrivals, general service time, one server) to a traffic intersection. We build our model based on collected data and run simulations in Maple to determine the mean queue length and the mean waiting time of the �customers� in the system. Results from this experiment will be used to reduce traffic congestion and improve efficiency at the intersection.

### Overview of Clemson Mthsc 499

Geoffrey Clark and Collin Kerrigan, Clemson University, Clemson, SC
mentored by Dr. Irina Victorova

Abstract: This creative inquiry is a class at Clemson University that allows undergraduate students the opportunity to do research in the field of mathematical modeling. The overall goal of the class is to model injury of the human arm due to heat explosion. In order to do this an understanding of the mathematical modeling of several aspects must be understood they include: heat transfer, self-heating, viscoelastic material properties, the physics of electric arcs, and the modeling of mechanical vibrations. Students research each of the topics and create presentations based on their understanding and present to the class. This presentation will be an overview of the class and a brief discussion on the modeling of the topics.

### Hybridization of classical unidimensional search with ABC to improve exploitation capability

Pranav Dass, North Dakota State University, Fargo, ND
mentored by Dr. Jagdish Bansal

Abstract: Artificial Bee Colony(ABC)optimization algorithm is relatively a recent, fast and easy to implement population based meta heuristic for optimization. ABC has been proved a competitive algorithm with some popular Swarm Intelligence based algorithms such as particle swarm optimization, firefly algorithm and Ant colony optimization. However, it is observed that ABC algorithm is better at exploration but poor at exploitation. Due to large step size, the solution search equation of ABC has enough chance to skip the optimum . In order to balance this, ABC is hybridized with a local search called as Classical Unidimensional Search (CUS). The proposed algorithm is named as Hybridized ABC (HABC). In HABC,best solution of each iteration is further exploited in both its positive and negative direction in a predefined range which enhances the exploitation in ABC. The experiments are carried out on 6 test problems of different complexities and dimensions in order to prove the efficiency of proposed algorithm and compared with ABC. The results shows that hybridization of CUS with ABC improves the performance of ABC.

### A Recursive Solution to Enumerating Directed Graphs

Christine Dierk, Elon University, Elon, NC
mentored by Dr. Todd Lee

Abstract: Since the introduction of graph theory in Leonhard Euler's paper on "The Seven Bridges of K�nigsberg" (1736), graphs have been used to represent relationships between a number of different things. In this talk, we will explore a specific type of graph, flocks, which are likened to chicken pecking orders. We will define the conditions that must hold in order for a graph to be a flock. We will also discuss the possibility for the existence of flocks of arbitrary size as well as effective algorithms for enumerating higher order flocks.

### Zeros of a Dirichlet Polynomial Connected with the Riemann Zeta-Function

Jordan Eliseo, UNCG, Greensboro, NC
mentored by Dr. Andrew Ledoan

Abstract: We investigate the distribution of zeros of the partial sums of the square of the Riemann zeta-function, which we will denote H(X,s), where X is the length of truncation, and s is a complex number. In particular, we establish zero-free regions for H(X,s), finding that there exists a real number b, such that H(X,s) is nonzero for all Re(s) greater than or equal to b. Furthermore, we show that there exists another real number a, depending on X only, such that H(X,s) is nonzero for all Re(s) less than or equal to a. Using the previous two results, we construct a rectilinear strip in the complex plane up to height T, and provide a sharp estimate for the number of zeros inside this strip for any given T.

### Positive Solutions For a Class of One Dimensional p-Laplacian Problems

mentored by Dr. Maya Chhetri

Abstract: We study the positive solutions of a class of one dimensional p-Laplacian boundary value problems. By utilizing the so-called quadrature method, we establish the number of positive solutions of the BVP depending on the behavior of the nonlinearity at infinity. In particular, we show that if the nonlinearity is p-superlinear (p-sublinear) at infinity, then the BVP has a unique positive solution for small (large) bifurcation parameter and no positive solution for large (small) bifurcation parameter. We also discuss the case when the nonlinearity is asymptotically linear at infinity and provide sufficient conditions on the nonlinearity to guarantee multiple positive solutions for a range of the parameter.

### Mathematical Modeling of Asian Carp Invasion Using Evolutionary Game Theory

Jasmine Everett, Marwah Jasim, Hakimah Smith, Bennett College, Greensboro, NC
mentored by Dr. Tsvetanka Sendova, Dr. Hyunju Oh, and Dr. Jan Rychtář

Abstract: Asian Carp were imported from China in the 1970s to improve water quality of aquaculture and to control aquatic vegetation ponds. However, they subsequently migrated from ponds into the Mississippi river, where they quickly reached high population density. They are considered invasive species, highly detrimental to the ecological balance, as they threaten the native fish population, eating up the algae and other microscopic organisms essential for the survival of the native fish. The Asian Carp now threatens the Upper Mississippi River and the Great Lakes. The goal of our project was to gain a better understanding of the interaction between native species and Asian Carp using mathematical model of evolutionary game theory. Our model consists of three players, one Asian Carp, Silver Carp and two native fishes Gizzard Shad and Largemouth Bass and various parameters of the game. We found best strategy to slow down or even stop the invasion depends on parameters through the simulations using MAPLE.

### Investigation into a mathematical model of epileptic seizures

Andrew Fischer, Elon University, Elon, NC
mentored by Dr. Karen Yokley

Abstract: An ordinary differential equation model describing the rate of change of membrane potential in groups of neurons in the hippocampal portion of the temporal lobe of the brain is being investigated. The o.d.e. model is intended to give insight into the propagation of epileptic seizures, and current research is being conducted using the computational software Mathematica into how changes to individual fixed parameters affect model output. A subset of parameters that have a significant impact on model output was determined, and the sensitivity of the model to the identified parameters has been characterized. The constants representing the delay time step, the equilibrium potential of leakage channels, and membrane capacitance were examined further due to their strong effect on the behavior of the model. Further investigation will result in the identification of optimal values for the constants of interest based on the biological relevance of the model output.

### Precedent Based Voting Paradox with Ranking Wheel Configurations for Odd Numbered Voting Bodies

Jonathan Fisher, Kennesaw State University, Kennesaw, GA
mentored by Dr. Mari Castle

Abstract: We denote a $(q,n)$-rule to be an voting system with $n$ voters and $q > \frac{n}{2}$, where a winning proposition receives at least $q$ votes. An $N$-alternative Ranking Wheel Configuration," denoted $RWC_N$, is a profile of $N$ voter preference rankings which, for a symmetry group of preference rankings, we call a Condorcet $N$-tuple. Recent results from Donald Saari have subsumed and extended significant results in social choice theory concerning the possibility of undesirable outcomes for $(q,n)$-rule voting systems using $RWC_N$. David Cohen has established a model for the Precedent Based Voting Paradox (PBVP) seen in Supreme Court cases. Cohen has shown the existence of the PBVP in 11 unique Supreme Court cases and has suggested that many other cases likely exhibit the conditions required. Here, we apply Saari's aforementioned Unifying Theorem to describe the PBVP, and extend the results to any odd numbered voting body.

### Exploration into the Harmonic Structure of the Tabla

Manu-Sankara Gargeya, The Early College at Guilford, Greensboro, NC
mentored by Dr. Promod Pratap

Abstract: This research examines the harmonic properties of the Tabla, a tunable percussion instrument from India, which is used in various forms of music from South Asia. Unlike other percussion instruments, the design of the percussive membrane of the Tabla allows the instrument to produce multiple harmonics of its fundamental frequency. Through using harmonic peak modeling, we analyzed the different pitches of the Tabla and gained a better understanding of the pitch variations of the Tabla. Through this research, we created a harmonic model of the Tabla and were able to differentiate it from other instruments that we examined.

### Correlation between Body Measurements of Different Genders and Races

Saumya Goel, Grimsley High School, Greensboro, NC

Abstract: The purpose of this experiment was to determine ratios between different signature measurements such as head width and length, height and arm span, and forehead and lower face. In this experiment, body measurements of height, arm span, and head dimensions were collected from 28 students. The measurements were obtained by analyzing the images of the students in ImageJ and determine patterns and ratios between different signature measurements such as head width and head height, data was separated by gender and racial background to see whether any correlation could be found. It was concluded that different racial groups had unique characteristics and ratios relating to their body measurements.

### Immersed Boundary Modeling of Journal Bearings in a Viscoelastic Fluid

Ryan Grove, Clemson University, Clemson, SC
mentored by Dr. John Chrispell

Abstract: The flow created by two eccentric rotating cylinders immersed in both Newtonian and viscoelastic fluids is simulated with a model created using the immersed boundary method. A model of this form allows for the transient behavior of the flow in the bearing to be studied as it transitions from rest to a steady state. The model allows for the study of lubricants inside a journal bearing. For a fixed outer annulus and a rotating inner annulus, the model simulates flow of various Deborah and Reynold's numbers and results are compared and contrasted using marker particles, velocity profiles, streamlines, the trace of the viscoelastic extra stress, stress ellipses, and mean flux. Study of the transient behavior of this type of system is an area where there are gaps in existing literature, especially for viscoelastic fluids.

### On the Stability of Solutions to a Phase Transition Model

Heather Hardeman, Wake Forest University, Winston Salem, NC
mentored by Dr. Stephen Robinson

Abstract: TBA We will discuss the stability of certain solutions of a phase transition model. This model is typically expressed as a partial differential equation: u_t= -\epsilon^2u_{xx}+F'(u) = 0, u_x(0) = u_x(1) = 0, where F(u) is a so-called �double-well� potential. We consider both classical and nonclassical examples for F. Furthermore, the main method we use in this discussion of stability is that of upper and lower solutions.

### On Some Subgraphs Generated by the Lifting of Graphs to 3-Uniform Hypergraphs

Josh Hiller, West Carolina University, Cullowhee, SC
mentored by Dr. Mark Budden

Abstract: Given a simple undirected graph $\Gamma$ having clique number $m$ and its complement $\overline{\Gamma}$ having clique number $n$, we construct a 3-uniform hypergraph $\Gamma^*$ containing cliques of order at most $2m$, whose complement $\overline{\Gamma^*}$ has clique number at most $2n$. This talk focuses on identifying many useful clique-preserving properties of such a lifting $\phi :\mathcal{G} \longrightarrow \mathcal{G}^*$ from the set of all graphs $\mathcal{G}$ to the set of $3$-uniform hypergraphs $\mathcal{G}^*$.

### Unsupervised Classification Using Sparse Submodule Clustering

Eric Kernsfeld, Tufts University, Medford, MA
mentored by Dr. Misha Kilmer

Abstract: In this project, we develop a method for unsupervised clustering by combining two recent innovations from different fields: the Sparse Subspace Clustering algorithm, which groups points according to membership in low-dimensional subspaces, and the t-product, which allows for a matrix-like multiplication for third order tensors. Our method uses a non-trivial self-representation of the data in order to form an affinity matrix, which can be post-processed using any clustering algorithm that clusters data given pairwise similarities. Like Sparse Subspace Clustering, we use a �sparsifying� norm, but unlike Sparse Subspace Clustering, we use the t-product in the self-representation. We present theoretical guarantees on performance as well as test results grouping images at different lighting conditions from the Weizmann Institute face database.

### Security visualization and simulation of suspicious behavior for intruder detection in public space

Swathi Kota, UNCG, Greensboro, NC
mentored by Dr. Shanmugathasan Suthaharan

Abstract: Exploration and development of intruder detection techniques for public space security is a challenging research problem. The difficult nature of acquiring reliable and labeled dataset, which can help to propose Machine Learning based intruder detection techniques is one of the challenges. In a previous research we studied Rowe's exposure metric which considers suspiciousness, noticeability, and visibility parameters and developed strategies to detect an intruder in public space by calculating a threat level parameter. However then we did not explore the techniques for generating datasets and visualization tools. The synthetic labeled datasets and associated visualization and simulation tools can motivate researchers study, explore and develop Machine Learning based intruder detection techniques. Here, we adopted our previous results and generated training synthetic datasets for intruder detection research in public space. Our main focus was on the suspiciousness parameter of Rowe's exposure metric and development of the labeled datasets. We also developed a visualization tool that is capable of showing the suspicious behavior and help analyze the approaches for detecting intruders.

### Mechanical Vibrations along the arm

Nicholas Larrieu, Clemson University, Clemson, SC
mentored by Dr. Irina Victorova

Abstract: This presentation will go over the mathematical modeling of vibrations along the human arm.

### A Reaction-Diffusion Model on the Effect of Insulin in Colon Cancer

Johnakin Martin, Winthrop University, Rock Hill, SC
mentored by Dr. Zachary Abernathy and Dr. Kristen Abernathy

Abstract: Robert Gatenby and Edward Gawlinski (1996) developed a reaction-diffusion model describing how tumor-induced pH levels influence cancer invasion. Using their approach, we present a new reaction-diffusion system which models both the spatial and temporal effects of insulin on neoplastic and cancerous cell growth in the colon. In particular, we are interested in the competition between these cell groups to uptake available insulin in order to promote growth. A stability analysis is performed to classify the fixed points of the system and predict long-term behavior of each cell type.

### Integrating the Functionality of Several Computer-Algebra Systems in a Unified Environment

Benjamin Manifold, UNCG, Greensboro, NC
mentored by Dr. Sebastian Pauli

Abstract: There are many specialized computer algebra systems, designed in the 1990s, that offer highly sophisticated functionality for mathematical computations in a narrow area of mathematics. Each uses its own custom interface, which makes it difficult for a user to access the functionality of these systems together. This can be overcome by making the functionality of several systems accessible from one shell. We present the tasks that need to be completed to make the functionality of one computer algebra system available through another shell. As an example we show how the functionality of the computer algebra system PARI for number theory can be made available in Python.

### Existence and uniqueness results for Emden-Fowler equations

Bevin Maultsby, UNC, Chapel Hill, NC
mentored by Dr. Christopher K.R.T. Jones

Abstract: A variation on Poisson's equation Δ u + f(u)=0 is to replace the regular Laplacian with the non-uniformly elliptic p-Laplacian, defined by
Δp u = div (|∇ u|p-2∇ u).
When p is 2, this is the regular Laplacian and can be used to model Newtonian fluids; if p is not 2, we can model non-Newtonian fluid motion. We are interested in the case p <2, when the operator is singular. Weak positive solutions of
Δp u + f(u) = 0
on a ball in Rn with Dirichlet boundary conditions can be shown to be radially symmetric for a large class of f. The Emden-Fowler transformation allows us to use blow down or blow up methods to rescale the solutions u and convert the original PDE into an ODE. This ODE is used to construct an invariant manifold; we explore how the existence of a unique solution to Δp u + f(u) = 0 comes from our ability to control the amount of winding of this manifold.

### Music Genomics: Applying Seriation Algorithms to Billboard #1 Hits

Nakhila Mistry, Elon University, Elon, NC
mentored by Dr. Crista Arangala

Abstract: Music plays a prominent role in society and companies have even started studying its aspects for commercial purposes. It is only natural to ask what are the characteristics that make certain songs appealing. While much research has been conducted on the mathematical principles of sound, there has been less focus on analyzing the structure of popular songs from a mathematical perspective. One mathematical tool that researchers have used to study this is seriation, ordering. Seriation algorithms are frequently used for companies with an online presence, including Google, Facebook, Amazon, and Pandora, to understand the traits of what users like in order to attract more consumers. We will use these types of seriation algorithms to conduct a mathematical analysis of the structural qualities of music. We will test whether the same structural traits appear in an artist's songs as the songs of the artists that they cite as musical influences. The artists chosen for this project are linked because of the influences they cite, their musical genre, and the popularity of their music. In order to musically link the chosen artists, we will use applied linear algebra methods. Results show that an artist's songs have a higher quantitatively measured connection with the artists they cite as influences rather than the artists who they never mention as musical influences.

### Tree Decompositions of Partially-Minimal Cayley Graphs

Evan Moore, Kennesaw State University, Kennesaw, GA
mentored by Dr. Erik Westlund

Abstract: A tree T decomposes a graph G if there exists a partition of the edge set of G into isomorphic copies of T. In 1963, Ringel conjectured that any complete graph on 2k+1 vertices can be decomposed by any tree with k edges. In 1989, H�ggkvist conjectured more generally that every 2k-regular graph can be decomposed by any tree with k edges. This conjecture remains unresolved in general, despite generating considerable interest and a significant literature. Expanding upon the work of El-Zanati et al. (2000), we present further results on H�ggkvist's conjecture by establishing a new family of Cayley graphs that are decomposable by any tree satisfying the divisibility conditions. Some open problems and future directions will also be discussed.

### Modeling Illicit Drug Use Patterns

Jacob Norton, NC State University, Raleigh, NC
mentored by Dr. Georgiy Bobashev

Abstract: Though our understanding of the mechanisms that drive drug use at the population level is poor, our understanding of the mechanisms driving individual drug use is even worse. Agent-based modeling provides a unique platform that allows us to tease apart the mechanisms that can lead to switching illicit drug use preferences. In particular, due to the structure of agent-based models, both population-level emergent phenomena and individual-level trajectories can be validated. Also, by performing variable selection, we can begin to determine what variables/ parameters play a critical role in individual adaptive behavior and drug switching in illicit drug users. Here, we consider six possible addiction forming substances and attempt to gain understanding into why individuals switch between drugs and show evidence of poly-drug use.

### Robust Variable Selection for Functional Regression Models

Jasdeep Pannu, Auburn University, Auburn, AL
mentored by Dr. Nedret Billor

Abstract: We consider the problem of selecting functional variables using the L1 regularization in a functional linear regression model with a scalar response and functional predictors in the presence of outliers. Since the LASSO is a special case of the penalized least squares regression with L1-penalty function it suffers from the heavy-tailed errors and/or outliers in data. Recently, the LAD regression and the LASSO methods have been combined (the LAD-LASSO regression method) to carry out robust parameter estimation and variable selection simultaneously for a multiple linear regression model. However variable selection of the functional predictor based on LASSO fails since multiple parameters exist for a functional predictor . Therefore group LASSO is used for selecting grouped variables rather than individual variables. In this study we extend the LAD- group LASSO to a functional linear regression model with a scalar response and functional predictors We illustrate the LAD- group LASSO on both simulated and real data.

### Ecological systems with aggregation, grazing, and sigma-shaped bifurcation curves

Heather Pierce, Zach Burnett, and Lyndee Bobo, Auburn University Montgomery, Montgomery, AL
mentored by Dr. Jerome Goddard II

Abstract: Population models built upon the reaction diffusion framework have provided important biological insight into the patch-level consequences of various assumptions made on individual behavior in ecological systems. Such models have seen enormous success both in their empirical validation with actual spatio-temporal distribution data and their ability to yield general conclusions about an eco-system based on the analytical results of theoretical models. Even with these great successes, the varied dynamics of theoretical reaction diffusion models are still not fully understood. In this talk, we will explore the dynamics of a logistic population model on a one-dimensional domain with hostile boundary, aggregation modeled by modifying the diffusion term, and grazing, i.e. a form of natural predation, via study of the model�s positive steady state solutions. We obtain results through use of the quadrature method and Mathematica computations. Specifically, we computation ally ascertain the existence of Σ-shaped bifurcation curves and will briefly explore their biological implications.

### Combinatorics of Quartet Amalgamation

Emili Price, Winthrop University, Rock Hill, SC
mentored by Dr. Joseph Rusinko

Abstract: One technique for reconstructing phylogenetic trees is inputting a set of quartet trees (containing four taxonomic units) and amalgamating them into a single supertree. We prove the minimal number k(|X|) such that every compatible quartet system Q, with |Q| greater than or equal to k defines a unique tree. Our construction leads to a lower-bound on the probability that an arbitrary collection of quartets define a unique tree.

### Time Periodic Nonlocal Dispersal Operators\Equations, Existence of Their Principal Eigenvalues and Applications

Nar Rawal, Auburn University, Auburn, AL
mentored by Dr. Wenxian Shen

Abstract: The boundary conditions such as Dirichlet and Neumann are ubiquitous in applied sciences and have been used widely in connection with local dispersal equations(equations involving Laplacian) to study many real life problems in diffusive systems. Principal Eigenvalue theory for local dispersal equations with Dirichlet, Neumann boundary conditions have been studied and well understood. The criteria for the existence of principal eigenvalue theory for nonlocal dispersal operators without time dependence have also been studied and well understood now. The talk will be focussed on the criteria for the existence of principal eigenvalue of time periodic nonlocal dispersal operators with Dirichlet type, Neumann type, and peridic boundary conditions. In addition to the principal eigenvalue theory, the talk will also be focussed on the existence, uniqueness, and stability of time periodic positive solution of Fisher type or KPP type equations with nonlocal dispersal.

### An Investigation of Genome Features and Their Effect on Meiotic Recombination Rates in Apis mellifera

Caitlin Ross and Dominick DeFelice, UNCG, Greensboro, NC
mentored by Dr. Olav Rueppell

Abstract: Recombination is important for ensuring chromosomes segregate properly during meiosis, but it also increases genetic variance. The honey bee, Apis mellifera, has been observed to have higher recombination rates than many other species, and the higher recombination rates are found genome-wide in A. mellifera. Proper chromosome segregation does not seem to explain this excess recombination, so recombination rates in the honey bee must be influenced by other factors. In our study, we have collected information on a variety of genomic features in 100KB windows throughout the honey bee genome to investigate their correlation with recombination rates. We used eight genetic maps along with a physical map to determine the average recombination rates for each 100KB window. We collected data about occurrences of various sequence motifs, di-, tri-, and quad-nucleotide sequences, low complexity sequences, and microsatellites comprised of dinucleotide repeats from the latest assembly (Amel_4.5) of the A. mellifera genome sequence. We also used available gene data to determine the number and average sizes of genes, introns, and exons and the average distance between genes for each window. Correlations between these genome features and the average recombination rate and its variance will be used to test alternative hypotheses that may explain the evolution of recombination in general.

### Edge Colorings Avoiding Some Proper Cycles

Andrew Schmidt, Clayton State University, Morrow, GA
mentored by Dr. Elliot Krop

Abstract: We consider the problem of finding the minimum order of any complete graph so that for any coloring of the edges by k colors it is impossible to avoid a properly colored cycle of length five. If we consider the condition of excluding some family of monochromatic graphs H in the above definition, we produce the pattern Ramsey number, pr_k(C_5;H). We determine this function in terms of k when H is a triangle, as well as a triangle and a path on five vertices. In particular, we find that pr_k(C_5;K_3) = k+4 and pr_k(C_5;K_3,P_5) = k + 3.

### Investigating Physical Properties of Mucus from Creep Experiment Using Linear Viscoelastic Models

Yue Shan, UNC Chapel Hill, Chapel Hill, NC
mentored by Dr. Greg Forest

Abstract: In order to understand the inability for a patient to clear mucus from the lung, we investigate the biophysical properties of mucus, and try to build a suite of mathematical and experimental tools that one can apply to a patient�s mucus in a clinical setting that give physical properties that help assess if the patient�s lung is healthy in terms of the mucus status. We did creep experiments on mucus under step stress and tried to fit proper models to the experimental data. Different types of models are investigated, and physical properties of mucus are discussed.

### Scattering theory for Schrödinger operators with sparse potentials

Zhongwei Shen, Auburn University, Auburn, AL
mentored by Dr. Wenxian Shen

Abstract: This paper is devoted to the scattering theory of a class of continuum Schrödinger operators with sparse potentials. We first establish the limiting absorption principle for both modified free resolvents and modified perturbed resolvents. Then, we prove the existence and completeness of wave operators. In particular, the absolutely continuous spectrum of the Schrödinger operator with sparse potential coincides with [0,∞).

### Computing Galois groups of degree 12 2-adic fields with trivial automorphism group

Christopher Shill, Elon University, Elon, NC

Abstract: The p-adic numbers are a foundational tool in 21st century number theory. They also have practical applications in many fields including, cryptography and physics. Several researchers have focused on classifying extensions of the p-adic numbers by computing arithmetic invariants associated to each extension. Current research has classified all extensions up to and including degree 11. In this talk we will focus on degree 12 2-adic fields, considering only those with a trivial automorphism group. In particular we describe our method of computing the Galois groups of these extensions.

### A Modified Optional Unrelated Question RRT Model

Jeong Sep Sihm, UNCG, Greensboro, NC
mentored by Dr. Sat Gupta

Abstract: We propose a modified unrelated question randomized response technique (RRT) model which allows respondents the option of answering a sensitive question directly without using the randomization device if they find the question non-sensitive. This situation has been handled before (Sihm and Gupta (2013a), and Gupta et al. (2013b)) using the split sample approach. In this work we avoid the split sample approach which requires larger total sample size. Instead, we estimate the prevalence of the sensitive characteristic by using an Optional Unrelated Question RRT Model and the corresponding sensitivity level from the same sample by using the traditonal Unrelated Question RRT Model. We compare the simulation results of this new model with those of the split-sample based Optional Unrelated Question RRT Model of Gupta et al. (2013b) and the traditional Unrelated Question RRT Model of Greenberg et al. (1969). Computer simulations show that the new model has the smallest variance among the three models when they have the same sample size.

### Uniqueness of positive solutions for a singular nonlinear eigenvalue problem when a parameter is large

Byungjae Son, UNCG, Greensboro, NC
mentored by Dr. Ratnasingham Shivaji

Abstract: We study positive solutions to singular boundary value problems of the form -u''(t)=Kh(t)g(u(t)) in (0,1) with Dirichlet boundary condition. Here, K is a positive parameter and h(t) and g(u(t)) satisfy certain conditions. Then we establish the uniqueness of positive solutions when a parameter is large.

### Weight Loss Through Bariatric Lap-Band Surgery - Some Issues

Robert Stoesen, UNCG, Greensboro, NC
mentored by Dr. Sat Gupta

Abstract: Obesity is a serious chronic disease. In the United States, approximately one-third of the population meets the criteria to be classified as obese, impacting both the morbidity and mortality for these individuals. Emerging in the treatment of obesity are surgical interventions. In particular, laparoscopic banding (Lap-Band) surgery is a widely implemented treatment for weight loss in obese individuals. The literature supports the necessity of lifestyle changes such as diet, exercise, and medical follow-up following bariatric surgery in order to sustain long-term health. Strong and sustained social support may also play a role in managing weight loss and lifestyle change, though few studies have explored the role of social support in weight loss management. In this retrospective study, we examined data for patients who underwent Lap-Band surgery who were offered the opportunity to attend support group meetings during the year following their surgery. We did not observe a statistically significant relationship between the number of support group meetings attended by patients and weight loss, nor did we find that age, sex, or race were contributing factors in such a relationship. The median number of support group sessions attended by Caucasian patients was found to be significantly larger as compared to African American patients. We did not find that one gender was more likely to attend support group meetings than another.

### Two-Fluid Flow in a Capillary Tube

Melissa Strait, NC State University, Raleigh, NC
mentored by Dr. Michael Shearer

Abstract: Phase field model for two-phase flow in a capillary tube, developed by Cueto-Felgueroso and Juanes, results in a PDE with higher-order terms. We find traveling wave solutions of the PDE and determine a bound on parameters to obtain physically relevant solutions. We observe that the traveling wave height decreases monotonically with the capillary number and that the traveling wave height corresponds to the height of the plateaus seen in PDE simulations. We also compare results against classical experiments of G.I. Taylor. This is joint work with Michael Shearer, Rachel Levy, Ruben Juanes, and Luis Cueto-Felgueroso.

Erin Strosnider, Elon University, Elon, NC

Abstract: A foundational result in algebraic number theory states that there are only finitely many nonisomorphic extensions of the p-adic numbers of degree n. Many researchers have focused on developing methods for computing data about these extensions (such as Galois groups and ramification information). Previous research has completely classified all extensions up to and including degree 13. In this talk, we'll survey some of the past research related to computational p-adic field theory, and we will end by illustrating our techniques for classifying degree 14 extensions of the 2-adic numbers.

### Distributed Concentric Circle Representation Technique for Network Intrusion Traffic Classification

Laxmi Sunkara and Manasa Tumkur Vijayendra , UNCG, Greensboro, NC
mentored by Dr. Shan Suthaharan

Abstract: Most recent research shows that classification of network intrusion traffic can be achieved using the Unit-Circle Algorithm (UCA). UCA represents different network traffic types by centralized unit-circles and classifies them based on the circular boundaries that divides the unit-circle regions. The efficiency of this approach has been demonstrated to classify regular and back attack traffic of publicly available network intrusion NSL-KDD datasets. In this research we propose a technique called, Distributed Concentric Circle Algorithm (DCCA) which adopts the concept of UCA, and is capable of classifying larger set of intrusion traffic. This technique is also capable of handling Big Data classification problem. The efficiency of the proposed method is tested for larger and complex network traffic dataset in a Hadoop Distributed File System (HDFS). We present our results and findings of the experiments conducted to classify the network traffic types for intrusion detection using the proposed method at the conference.

### Kleptoparasitic Interactions and Internal States

David Sykes, UNCG, Greensboro, NC
mentored by Dr. Jan Rychtář

Abstract: A kleptoparasitic interaction occurs when one individual (a kleptoparasite) attempts to take resources from another individual. Some animals exhibit different behavior in similar interactions, and we would like to understand why they may have evolved to do so. Internal states, such as health, age, or hunger, can affect what behavioral strategies yield optimal gains. To study this effect, we have created a mathematical model that describes the outcomes of these interactions in terms of the value of contested resources, the cost of a fight (or a similar conflict), and the internal states of individuals involved. Changing the degree to which internal states affect an individual's appraisal of resources changes optimal behavior, as indicated by our model. When this degree is high, it can happen that individuals should forgo stealing from weaker individuals, and this does not happen when the degree is low. This degree can also be set so that the constant strategy of always stealing is optimal behavior; however, for most parameter settings, optimal behavior is not a constant strategy (i.e. a strategy of always making the same decision). Optimal behavior should, in most cases, be adaptive to changes in resource value, cost of conflict and internal health.

### The Coupon Collector Problem: Analysis of Multiple Collections

Michael Thomas, Kennesaw State University, Kennesaw, GA

Abstract: Suppose that every time you purchase a box of cereal from a certain manufacturer, there is a collectable coupon inside the box. A complete collection of coupons has �m� different coupons, each being found with different probabilities inside the cereal boxes. How many purchases are required, on average, in order to get a complete collection? How many are needed to complete multiple collections? Probabilistic and empirical results are used to describe the probability distribution of the number of purchases necessary for a full collection as coupon number and collection number increase. We present a simulation program that allows us to examine the empirical results.

### One step at a time: Modeling ant movement

Amanda Traud, NC State University, Raleigh, NC
mentored by Dr. Alun Lloyd and Dr. Rob Dunn

Abstract: Ants live in large colonies interacting with many individuals. A first step towards understanding these interactions is to understand individual ant movement paths. We study the movements of Formica subsericea workers, large North American ants that live in colonies containing upwards of 10000 individuals, to gain insight into movement mechanisms and interactions. Using image analysis techniques, we gather individual movement data from video recordings and model these movements using a two state continuous time Markov chain mixed with correlated random walks. We compare data and models for single ant movements to models and data for ant pairs and reveal whether movements are significantly affected by ant interaction.

### Classification of Network Traffic using Random Forest Approach

Manasa Tumkur Vijayendra and Neha Singh, UNCG, Greensboro, NC
mentored by Dr. Shanmugathasan Suthaharan

Abstract: Classification of network traffic is one of the important processes that help improve the quality and management of computer networks. However the high dimensional network traffic system recording large number of observations combined with varieties of traffic types makes the task of classification more complex. Recent studies indicate that the Machine Learning technique called Random Forest can assist in the classification of complex dynamical systems by extracting low dimensional structures from high dimensional spaces. Random Forest achieves this by constructing decision trees that provide classification model by forming a forest of related tree structures. In our research we studied the effectiveness of Random Forest approach to classification of network traffic using Lawrence Berkeley National Laboratory (LBNL) data sets. We mainly focused on classifying Simple Mail Transfer Protocol (SMTP) and File Transfer Protocol (FTP) available in the LBNL data sets using the popular Random Forest classification technique. The simulation results and findings will be presented at the conference.

### A Maximum entropy approach to the numerical approximation of invariant densities

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

Abstract: Let the Frobenius-Perron operator P : L^1 (0, 1) → L^1 (0, 1), related to a non-singular transformation S : [0, 1] → [0, 1], have an invariant density function g* . We propose a piecewise quadratic maximum entropy method for the numerical approximation of the invariant density function g* . The role of the partition of unity property of the quadratic functions for the numerical recovery of g* has been depicted. The pro-posed algorithm overcomes the ill-conditioning shortage of the traditional maximum entropy method which only employs polynomials, so that any number of moments can be used to increase the accuracy of the computed invariant density. The convergence rate of the method depends on the approximation order of the exact invariant density via linear combinations of the quadratic moment functions. Numerical results are presented to justify the theoretical analysis of the method.

### Evaluation of Human Odor Suppression by L-2 Norm

Christopher Vanlangenberg, UNCG, Greensboro, NC
mentored by Dr. Haimeng Zhang
co-authored by Shamitha Dissanayake, Glenn Crisler, Bronson Strickland, Steve Demarais, Todd Mlsna

Abstract: Human scent is one of the most complex mixtures available in the human body and influenced by various internal and external factors. A method was developed to extract maximum human body odor with minimum non-skin odor contaminations with minimum subject discomfort. We collected VOCs in the human scent of 65 human subjects, and approximately 5280 unique compounds were found among the subjects. The method was further developed to identify and compare the VOC profile produced by humans to determine the effect of 4 different scent control products. We proposed an ad-hoc method to identify the best compounds and a ranking method to shortlist the covariates (compounds) for the study. Discriminant Analysis (DA) and Principal Component Analysis (PCA) were used to simplify the complex outcomes associated with competitive scent elimination mechanisms of various scent elimination products. Decision Tree diagrams were used identify important covariates contributing to each product. Finally an L-2 norm approach on the principle components was proposed to evaluate the overall scent reduction of each human subject and hence the 4 different scent elimination products were compared.

### Analysis of Sequential Statistical Learning for Signature Discovery in Network Intrusion Detection System

Laxmi Vishnubhotla, UNCG, Greensboro, NC
mentored by Dr. Shan Suthaharan

Abstract: Signature Discovery (SD) algorithms have been used in network intrusion detection systems for intrusion traffic classification. In SD algorithms, statistical and mathematical relationships between different features are learned sequentially and signatures are discovered to characterize different types of regular and attack network traffic for classification. Recently a new SD algorithm has been proposed and it uses the following statistical techniques with single-pass learning: variance-based, randomness-based, and statistical distribution-based and Lagrangian Support Vector Machine based learning. However the single-pass learning may ignore the learning sequence that provides the best classification. In our research we analyzed multiple-pass learning scenario by selecting different combinations of the statistical learning approaches adopted in the single-pass learning technique. For this analysis, we used the publicly available network intrusion data set [NSL-KDD], a cross-validation technique and an ensemble approach. The cross validation technique allows the detection of lowest false positive error rate for all the sequences while the ensemble approach can help the selection of the best sequence (or a combination of sequences) for generating robust classifiers. The results and observations will be presented at the conference.

### Zero-Inflated Poisson (ZIP) Distribution: Parameter Estimation and Applications to Model Data from Natural Calamities

Quintel Washington, Southern University at New Orleans, New Orleans, LA
mentored by Dr. Nabendu Pal

Abstract: This work deals with estimation of parameters of a Zero-Inflated Poisson (ZIP) distribution as well as using it to model some natural calamities� data. First, we compare the maximum likelihood estimators (MLEs) and the method of moments estimators (MMEs) in terms of standardized bias (�SBias�) and standardized mean squared error (�SMSE�). We then proceed to show how datasets from recent natural disasters can be modeled by the ZIP Distribution.

### Approximations of Random Dispersal Operators/Equations by Nonlocal Dispersal Operators/Equations

Xiaoxia Xie, Auburn University, Auburn, AL
mentored by Dr. Wenxian Shen

Abstract: This paper is concerned with the approximations of random dispersal operators/equations by nonlocal dispersal operators/equations. It first proves that the solutions of properly rescaled nonlocal dispersal initial-boundary value problems converge to the solutions of the corresponding random dispersal initial-boundary value problems. Next, it proves that the principal spectrum points of nonlocal dispersal operators with properly rescaled kernels converge to the principal eigenvalues of the corresponding random dispersal operators. Finally, it proves that the unique positive stationary solutions of nonlocal dispersal KPP equations with properly rescaled kernels converge to the unique positive stationary solutions of the corresponding random dispersal KPP equations.

### Risky Sexual Behaviors among College Students � Predictors of STD

Qi Zhang, UNCG, Greensboro, NC and Tracy Spears, UNC, Chapel Hill
mentored by Dr. Sat Gupta

Abstract: It is well known that indulging in risky sexual behavior increases one�s risk of sexually transmitted diseases (STD). Previous research shows that high-risk sexual behaviors are common among college-students, even in an age of increased public education about the consequences of such behavior. Since most students may consider sexual behavior as a private and sensitive question, measuring prevalence of such behaviors is not easy. A Randomized Response Model is helpful in estimating such prevalence. Gupta et al. (2013) estimated prevalence of some risky sexual behavior among UNCG students using an Optional Unrelated Question RRT Model. Their results suggest that this model effectively reduces under-reporting of such behaviors. In this follow-up work we focus on other variables in the data. We estimate the relationship between STD history and several factors, such as age, gender, and the number of sexual partners.

### 3D Hydrodynamic phase-field Models and Simulations for Biofilms

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

Abstract: Biofilm is ubiquitous in our daily life, such as in dental plaque and sewage pipes. It is a micro-organism, where bacteria stick together by secreting extracelluar polymeric substances (EPS). Significant attentions have been attracted into biofilm research, due to the fact that it is useful in sewage, as well as, it is the cause for many chronic disease. In this talk, a three dimensional hydrodynamic model has been derived, by phase-field approach. The hydrodynamical effects on biofilm formation and functioning have been analyzed. Later, by distinguishing various types of bacteria, the quorum sensing and antimicrobial effects in biofilm would also be discussed. Our modeling approach appears to be an effective tool for understanding the mechanism of biofilm formation and functioning and provides an insight on biofilm treatment.