Abstracts of the talks
- Charles Allee and Ivanti Galloway: The Effects of Mutation Levels on the Evolution of Cooperation
- Sudesh Baral: Improving the Accuracy of Least Squares Problem
- Yahnick Barclay and Marwah Jasim: Math Modelling of Soap Film
- Lindsay Bradley: A Mathematical Model of Chronic Myeloid Leukemia with Chemotherapy Treatment
- Casey Carter and Stephanie Rhoten: Modeling the Effect of Voluntary and Involuntary Quarantine on the Spread of Infectious Diseases
- John Caughman and Dillon Ruthland: Implementing a GUI Platform to Study Linear Viscoelastic Materials
- Taylor Cesarski: Symmetries of Degree 7 Polynomials
- Torie Chomko and Jamie Shive: Odd Dominating Sets in the Strong Product of Graphs
- Chasity Dorsett: Mathematics behind Anamorphosis
- Matthew Farmer: Triangles in Cayley Graphs
- Nina Galanter and Dennis Silva, Jr.: The Territorial Raider Model with Strategic Movement and Multi-Group Interactions
- Taylor Harbold, Michelle Page, and Courtney Rasmussen: Zip Codes and Neural Networks: Machine Learning for Handwritten Number Recognition
- Kristin Hinson: On Bond Percolation in the Infinite Knight Graph
- Rhoni Moffit: Hurricane Evacuation using Game Theory
- Peter Jakes: Degree Six Polynomials and Their Solvability by Radicals
- Michael Keenan: Symmetries of Quartic Polynomials
- Aaleah Lancaster and Qaleelah Smith: Optimization of Border Patrol Strategies
- MaLyn Lawhorn and Rachel Schomaker: Comparative Analysis of Transcriptomic Data Accounting for Variation in Gene Flexibility
- Kevin McCall and Alex McCleary: A Generalization of Total Perfect Codes in Graphs
- Emily Nance: Analysis of Perceptions of an Emergency Nursing Academy
- Anh Nguyen and Jay Saini: Cooperation in Finite Populations: Being Alone Helps
- Roberto Perez: An Examination of Factors Affecting Incidence and Survival in Respiratory Cancers
- Kelly Reagan: A Mathematical Model of the Spread of Dengue Fever Incorporating Mobility
- David Reynolds: Eigencurves of the Singular Sturm-Liouville Problem
- Sara Rodgers: Modelling the Optimal Cost of Prevention Methods of Dengue Fever
- Stephen Steward: Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
- Jessica Weed: Galois Groups of Degree 15 p-adic Polynomials
- Phillip Andreae: Analytic Torsion: Generalized Metric Invariance
- Susan Beckhardt: Geometric Group Theory and Asymptotic Property C
- Francis Bilson Darku: Accuracy in Effect Size Estimation for IID Observations
- Asim Dey: Modeling Extreme Aviation Accident using the Generalized Pareto Distribution
- Ryan Grove: Comparison of Nonlinear and Linear Stabilization Schemes for Advection-Diffusion Equations
- Kareem Hamdan: Manipulation of DT Codes
- Joey Hart: Variable Importance for Scattered Data
- Andrew Kane: A New Analysis Strategy for Designs with Complex Aliasing
- Ksenia Kyzyurova: Linking Statistical Emulators of Complex Scientific Computer Models
- Jeff Lail and Angie Larsen: Model II Regression Coefficient Confidence Intervals
- Austin Lawson: Stability for Multi-radii Persistence
- Gary Larson: Site Dependent Transitions in Hidden Markov Models for Bayesian Protein Structure Alignment
- Nikolai Lipscomb: A Semi-Lagrangian Method for Systems of Time-dependent Partial Differential Equations
- Bin Luo: Robust Variable Selection Using Penalized Adaptive Weighted Least Square Regression
- Abdullah Al Manum: Numerical Analysis of Temperature Dependent Viscosity on a Conjugate Heat Transfer Process for a Vertical Flat Plate
- Joshua Martin: A Vietoris-Rips Lemma for Multi-parameter Persistence
- Jeong Sep Sihm: A Modified Binary Optional Randomized Response Technique Model
- Raymond Smith: A Mathematical Investigation of Vaccination Strategies to Prevent Measles Epidemic
- Byungjae Son: Positive Radial Solutions to Classes of Singular Problems on the Exterior of a Ball
- David Sykes: Optimal Aggression in Kleptoparasitic Interactions
- Tanja Zatezalo: Assessing the Adequacy of First Order Approximations in Ratio Type Estimators
- Panpan Zhang: Distributions in a Class of Poissonized Urns with an Application to Apollonian Networks
Undergraduate Students
Graduate Students
Analytic Torsion: Generalized Metric Invariance
Phillip Andreae, Duke University, Durham, NCmentored by Dr. Mark Stern
Abstract: We study the Ray-Singer analytic torsion T associated to a flat vector bundle with hermitian metric h over an odd-dimensional compact manifold with Riemannian metric g. In the acyclic case (and, with the appropriate interpretation, more generally), T is known to be independent of the metrics h and g, i.e., T is a topological invariant. We frame the metric independence of T in terms of a certain closed one-form on the space of metrics, and we prove that furthermore T is independent of the metric on the exterior bundle, which may be chosen independently of g.
The Effects of Mutation Levels on the Evolution of Cooperation
Charles Allee*, Mercer University, Macon, GA and Ivanti Galloway*, UNCG, Greensboro, NCmentored by Dr. Jonathan Rowell
Abstract: Although cooperation can be advantageous for groups, a non-cooperative individual in a cooperative group can increase their own fitness. Since natural selection acts against the less fit we would expect to see no cooperative individuals, but in the environment we can observe many populations in which cooperation is evident. Here, we examine how mutations within populations lead to a transition from selfishness to cooperation. We examine a system comprised of codependent individuals with behavioral mutations and investigate under which conditions coexistence is possible or when one population will invade and exclude the other. We show when both populations are allowed to mutate, coexistence is possible due to mutation selection balance. Also we show coexistence can be achieved at lower mutation levels when initial populations are relatively high and movement in the environment is allowed.
Improving the Accuracy of Least Squares Problem
Sudesh Baral*, Benedict College, Columbia, SCmentored by Dr. Naima Naheed
Abstract: Least squares problems arise repeatedly in scientific and engineering computations. The term least squares describes a recurrently used approach to solve an overdetermined or inexactly specified systems of equations in an approximate sense. Instead of solving exactly, we seek only to minimize the sum of the squares of the residuals. In this project, we constructed a matrix A formed by the first n columns of the 10x10 Hilbert matrix. We used normal equations and QR methods to solve the least squares problem for A. We observed that the more sophisticated QR method is superior to the normal equations method.
Math Modelling of Soap Film
Yahnick Barclay* and Marwah Jasim*, Bennett College, Greensboro, NCmentored by Dr. Ajanta Roy
Abstract: Soap films are thin layers of water based liquid surrounded by air. If two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Plateau borders. Films are used as model systems for minimal surfaces, which are widely used in mathematics. Dipping a wire frame into soapy water produces an iridescent soap film that clings to the frame. Such soap films are related to minimal surfaces, which are beautiful geometric objects that minimize surface area locally. Visually, minimal surfaces can be thought of as saddle surfaces that bend upward in one direction in the same amount that they bend downward in the perpendicular direction. We are using complex analysis to present a nice way to describe minimal surfaces and to relate the geometry of the surface with this description.
Geometric Group Theory and Asymptotic Property C
Susan Beckhardt, SUNY University at Albany, Albany, NYmentored by Dr. Boris Goldfarb
Abstract: Geometric group theory allows us to discover algebraic properties of an infinite group by investigating the geometric features of the group or metric spaces on which it acts. In the first half of this talk, I will give a brief introduction to geometric group theory and coarse geometry. In the second half, I will explain my result (arXiv:1508.01265), which gives a condition under which a group has Asymptotic Property C, and give a sketch of the proof.
Accuracy in Effect Size Estimation for IID Observations
Francis Bilson Darku, University of Texas at Dallas, TXmentored by Dr. Bhargab Chattopadhyay
Abstract: Effect sizes are widely used quantitative measures of the strength of a phenomenon, with many potential uses in psychology and related disciplines. In this article, we propose a general theory for a sequential procedure for constructing sufficiently narrow confidence intervals for effect sizes (such as correlation coefficient, coefficient of variation, etc.) using smallest possible sample sizes, importantly without specific distributional assumptions. Fixed sample size planning methods, which are commonly used in psychology, economics and related fields, cannot always give a sufficiently narrow width with high coverage probability. The sequential procedure we develop is the first sequential sampling procedure developed for constructing confidence intervals for effect sizes with a prespecified width and coverage probability. As a general overview, we first present a method of planning a pilot sample size after the research goals are specified by the researcher. Then, after collecting a sample size as large as the estimated pilot sample size, a check is performed to assess if the conditions to stop the data collection have been satisfied. If not, an additional observation is collected and the check is performed again. This process continues, sequentially, until the specified conditions are satisfied. Our method ensures a sufficiently narrow confidence interval width along with a specified coverage probability.
A Mathematical Model of Chronic Myeloid Leukemia with Chemotherapy Treatment
Lindsay Bradley*, Winthrop University, Rock Hill, SCmentored by Dr. Kristen Abernathy
Abstract: Chronic Myeloid Leukemia (CML) is a prevalent type of cancer, where the presence of cancer stem cells is well studied. In this talk, we modify existing Gompertzian growth models to study the dynamics of CML and the effects of CML under chemotherapy treatment. We present numerical simulations and preliminary stability results. We conclude with future work and remaining open questions.
Modeling the Effect of Voluntary and Involuntary Quarantine on the Spread of Infectious Diseases
Casey Carter*, Northern Kentucky University, Highland Heights, KY and Stephanie Rhoten*, Gonzaga University, Spokane, WAmentored by Dr. Jonathan Rowell
Abstract: The imposition of quarantine has been included among several approaches employed in combating the spread of infectious diseases. Quarantine protocols were partially credited with the prevention of outbreaks of Ebola within the United States in 2014, and they were historically important in preventing the spread of various communicable diseases during the immigration waves of the 1800�s. In this paper, we extend the standard SIR model of disease transmission and incorporate multiple features such as exposure and quarantine classes and the risks posed by the disposition of infected bodies and cleaning of contaminated sites after death. We examine the effects when quarantine rates are both voluntary (social withdrawal) and involuntary (governmentally imposed), and we integrate game theoretic and dynamical analyses of the optimal responses for both individuals and government agencies in avoiding exposing or reducing the prevalence of the disease, respectively. Our model considers both closed populations (appropriate to rapid diseases such as Ebola) and open populations where disease is possibly endemic. We conclude with a comparison of quarantine protocols imposed within the population as diseases arise and at immigration bottlenecks before arrival to the main population (e.g. an Ellis Island model).
Implementing a GUI Platform to Study Linear Viscoelastic Materials
John Caughman* and Dillon Ruthland*, University of South Carolina, Columbia, SCmentored by Dr. Paula Vasquez
Abstract: Complex fluids are ubiquitous in nature with growing applications in the medical, pharmaceutical, and chemical industries. In addition, biopolymers are central constituents of the human body; among other things they are responsible for molecular sensing, catalysis, and cellular motion. Our work focuses on the study of pulmonary mucus. Because the mechanical properties of this material are pivotal to its proper function, understanding its responses to different deformations is critical. In this talk we will discuss our progress towards the implementation a Graphical User Interface (GUI) application, guided by experimental results, which will aid in the study of changes in the properties of lung mucus due to disease progression or treatment. The GUI takes as input experimental results and provides fitting parameters for several linear viscoelastic models.
Symmetries of Degree 7 Polynomials
Taylor Cesarski*, Elon University, Elon, NCmentored by Dr. Chad Awtrey
Abstract: Finding solutions of polynomial equations is a central problem in mathematics. Of historical 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 radicals (square roots, cube roots, etc.). Polynomials of degree four or less have been shown to be solvable by radicals, while the same is not true for higher degrees. How do we determine which polynomials are solvable by radicals? To answer this question, we study an important object that is associated with 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 symmetries of the corresponding polynomial's roots, and these symmetries can determine whether or not the polynomial is solvable by radicals. We discuss a new algorithm for determining the Galois group of a degree seven polynomial which improves upon prior research. In particular, previous methods rely on factoring two or more auxiliary polynomials while ours requires only one.
Odd Dominating Sets in the Strong Product of Graphs
Torie Chomko* and Jamie Shive*, Virginia Commonwealth University, Richmond, VAmentored by Dr. Dewey Taylor
Abstract: An odd closed r-dominating set in a graph is a subset of the graph's vertices with the property that the closed r-ball centered at any vertex in the graph contains an odd number of vertices in the subset. This talk explores the existence of odd closed r-dominating sets in the strong product of graphs. The n-fold strong product of simple graphs has an odd closed r-dominating set if and only if each factor has an odd closed r-dominating set. We also explore an application of this idea to the popular "lights out" game on graphs.
Modeling Extreme Aviation Accident using the Generalized Pareto Distribution
Asim Dey, University of Texas at Dallas, TXmentored by Dr. Kumer Pial Das
Abstract: Air travel is considered a safe means of transportation. But when aviation accidents do occur they often result in fatalities. Fortunately, the most extreme accidents occur rarely. However, 2014 was the deadliest year in the past decade causing 111 plane crashes, and among them worst four crashes cause 298, 239, 162, 116 deaths. In this study we wish to assess the risk of the catastrophic aviation accident by studying the number of fatalities from aviation accidents in the 33-year period from 1982 to 2014. Applying a generalized Pareto model that allows the distribution of extreme fatal injuries, we estimate the probabilities of serious aviation accidents. We also predict the maximum fatalities from an aviation accident in future. And the uncertainty in the inferences is quantified using simulated aviation accident series, generated by Monte Carlo simulation and bootstrap resampling.
Mathematics behind Anamorphosis
Chasity Dorsett*, Bennett College, Greensboro, NCmentored by Dr. Ajanta Roy
Abstract: The purpose of this study is to present the mathematics behind anamorphosis. The mathematics includes complex analysis, Taylor series, conformal mappings, Laplace equations, and Mobius transformation. This study uses Complex Tool software, which will allow mathematical functions, known as conformal mappings to demonstrate the transformation of an image, known as anamorphosis by incorporating different polynomial functions using the java programming. History of the two types of anamorphosis, perspective and oblique will be given as well as different pictures will be used to display visual aid for each example. Also, the images will also include optical illusion, which is another form of anamorphosis. The overall results of this analysis will include images of different anamorphosis from the starting point of the unit disk under the map f(z) and transform as the f(z) transforms. The anticipated outcome of this research is to display the connection of mathematics behind anamorphosis by using the Complex Tool software. The findings may be useful in creating fun projects or additional research.
Triangles in Cayley Graphs
Matthew Farmer*, Winthrop University, Rock Hill, SCmentored by Dr. Jessie Hamm
Abstract: Cayley Graphs were introduced by Author Cayley in 1878. Since then, they have been studied extensively due to their connection to group theory, graph theory, computer science, and other fields. In this presentation, we will briefly review Cayley graphs and then introduce two new parameters for Cayley Graphs: Cay^n (G) and Cay_n (G). When n=3, these parameters tell us about the existence of triangles within Cayley Graphs. We find Cay^3 (G) for all groups G and gives some results for Cay_3 (G) along with future directions.
The Territorial Raider Model with Strategic Movement and Multi-Group Interactions
Nina Galanter*, Grinnell College, Grinnell, IA and Dennis Silva, Jr.*, Worcester Polytechnic Institute, Worcester, MAmentored by Dr. Jan Rychtar and Dr. Jonathan Rowell
Abstract: We analyzed a territorial raider game, a graph based competition for resources, with strategic movement. First we investigated the game in which players are treated as individual organisms. Utilizing a machine learning algorithm, we discovered that the only strict Nash equilibrium strategy sets occur when all players raid one another and players do not compete with one another. This indicates equilibria are generated by derangement functions of a graph. Thus, we found that a graph will permit a derangement if and only if it permits a strict Nash equilibrium. We then extended the game to the case where players are �hives� or �armies� which can divide themselves among multiple territories. We examined this division of armies both in the discrete and continuous cases. Our results include Nash equilibria for regular graphs and regular bipartite graphs in both of these cases. Our results suggest that while group entities defend in more cases than in the single player game, the portion of a group defending varies based on the degree of vertices and the advantage given to owners in protecting resources.
Comparison of Nonlinear and Linear Stabilization Schemes for Advection-Diffusion Equations
Ryan Grove, Clemson University, Clemson, SCmentored by Dr. Timo Heister
Abstract: Accurately solving advection-diffusion equations that appear in the finite element discretization of a mantle convection simulation is an important computational issue to the computational geoscience community because it allows for users studying mantle convection to create reliable simulations for something as small and simple as a 2D simulation on their personal laptop to something as complex as a massively parallel 3D simulation on their university supercomputer. 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. Using the open source scientific library ASPECT, the SUPG and Entropy Viscosity schemes are compared using stationary and non-stationary test equations. Differences in maximum overshoot and undershoot, smear, and convergence orders are compared to see if improvements can be made to the existing numerical method existing in ASPECT.
Manipulation of DT Codes
Kareem Hamdan, University of Texas at Dallas, TXmentored by Dr. Noureen Khan
Abstract: In this work, we do an analysis of Dowker's codes of an alternating knot. Then we construct a bijective function which generates a set that can be partitioned into two classes, that is, a class of classical knots and a class of virtual knots, up to mirror images.
Zip Codes and Neural Networks: Machine Learning for Handwritten Number Recognition
Taylor Harbold*, Michelle Page*, and Courtney Rasmussen*, University of North Carolina at Wilmington, Wilmington, NCmentored by Dr. Cuixian Chen
Abstract: Neural Network is an idea taken from artificial intelligence that utilizes an oversimplification of the synapse processes that occur in the brain. By mimicking these processes, we create a simpler method for solving complex problems. We applied these techniques to predict the true value of hand-written digits from zip code data using prediction models generated in MATLAB and the RSNNS package in the program R-Studio.
Variable Importance for Scattered Data
Joey Hart, NC State University, Raleigh, NCmentored by Dr. Pierre Gremaud
Abstract: Many problems in science and engineering are plagued by high dimensionality that limits mathematical analysis. One way to combat this is to compute a measure of the importance of the variables. However, in many applications the underlying function is only known through scattered data. In this case, a statistical model must be constructed and variable importance computed from it. There are a variety of statistical models and variable importance measure that may be chosen. How to choose them remains a topic of active research. In this talk, we present a new algorithm to compute variable importance from scattered data and compare it with existing methods.
On Bond Percolation in the Infinite Knight Graph
Kristin Hinson*, Winthrop University, Rock Hill, SCmentored by Dr. Arran Hamm
Abstract: For a graph G=(V,E), let G_p=(V,Bin(E,p)) where Bin(E,p) keeps edges from E with probability p independently (and discards an edge with probability 1-p). For a locally finite graph G (i.e. |V| is infinite and the degree of each vertex is finite), let C(p) be the event that G_p contains an infinite path. Our objective is to identify p_c, the critical probability, which has the property that if p > p_c, then Pr(C(p)) = 1 and if p < p_c, then Pr(C(p))=0 (that p_c exists is a standard fact in the study of bond percolation). Now suppose we take G to be the infinite knight graph which has vertex set in the integer grid such that (x0, y0)~(x1,y1) if and only if |x1-x0|+|y1-y0|=3 and x0 is not equal to x1 and y0 is not equal to y1. We allow B to be the subgraph created by placing the knight at (0,0) and only allowing the knight to move horizontally two units and one unit vertically from each position. We are interested in the case when H is taken to be the union of B and the graph created by reflecting B over the line y=x. In this case, we obtain a nontrivial upper and lower bound on pc for H, the former via coupling with bound percolation on the integer grid and the latter by an appropriate union bound.
Hurricane Evacuation using Game Theory
Rhoni Moffit*, Bennett College, Greensboro, NCco-authored by KeeAera Hood*, Bennett College, Greensboro, NC
mentored by Dr. Hyunju Oh, Dr. Jan Rychtar and Dr. Joon-Yeoul Oh
Abstract: During the hurricane season, residents in southeast coast area experience frequent warnings for hurricanes. The residents need to be evacuated to safety at least 20 to 50 miles away from the impacted area. With a mass evacuation, even 24 hour notice may not be enough since necessities such as lodging are limited and the actual evacuation distance can easily be more than 100 miles. When a hurricane is approaching, the residents prepare with installing blocks on the windows, buying gas/food and deciding if and when to evacuate. In general, if they are getting ready too early, the probability of the hurricane hitting a certain area is not yet reliable. However, if the residents wait almost until the end, their lives get threatened (short term predictions are relatively accurate). Moreover, when everybody evacuates at the same time, there will be logistical issues such as traffic congestions and no fuel in gas stations. The variation of the individual risk for the person to stay throughout the hurricane can cause a natural variation of an appropriate optimal evacuation time ahead of the hurricane. The goal of this project is to find an optimal time of evacuation. The optimal time depends on an individual circumstances and risks (for example, a family with young children is in a different situation than a single healthy young person) and the objective is to find the time as a function of the individual risk and the risk distribution within the population.
Degree Six Polynomials and Their Solvability by Radicals
Peter Jakes*, Elon University, Elon, NCmentored by Dr. Chad Awtrey
Abstract: For about 500 years, formulas have existed to find exact solutions to quadratic, cubic and quartic polynomials. However, it was proven later that not all solutions to quintic polynomials can be found exactly, or solved by radicals. As a result, a method was created in the 20th century using a property of each function called its Galois group in order to determine which degree five polynomials could be solved exactly and which could not. This project expands upon this discovery by exploring degree six polynomials. By using computer software, the Galois group of a degree six polynomial can be determined by only using two resolvent polynomials, improving upon prior methods. From this information, it can then be determined whether or not the polynomial is solvable by radicals. Further research can explore higher degree polynomials as well as reducible polynomials, as the current method is only viable for irreducible polynomials.
A New Analysis Strategy for Designs with Complex Aliasing
Andrew Kane*, University of Georgia, Athens, GAmentored by Dr. Abhyuday Mandal
Abstract: Non-regular designs are often used in designing industrial experiments for run-size economy. However, these designs often give rise to partially aliased effects, where the effect of one factor cannot be estimated without making additional assumptions on the model. The existing methods available in the literature for analyzing data from such designs are not satisfactory. In this paper we propose applying adaptive lasso regression techniques for variable selection.
Symmetries of Quartic Polynomials
Michael Keenan*, Elon University, Elon, NCmentored by Dr. Chad Awtrey
Abstract: In the 1500s mathematicians discovered that all quartic polynomials are solvable by radicals, meaning we can find a quartic polynomial's roots using only using the coefficients of the polynomial, the basic arithmetic functions (addition, subtraction, multiplication, and division), and radicals (square roots, cube roots, etc.). It wasn't until the 1800s when mathematicians showed why quartic polynomials are solvable by radicals and why not all polynomials of degree greater than four are. By attaching a group structure to a polynomial (called the polynomial's Galois group), we can determine whether the polynomial is solvable by radicals. We can also see the relationships among the roots. Naturally, a branch of mathematical research has emerged to develop methods to determine Galois groups of polynomials. Previous methods for determining Galois groups of quartic polynomials involve factoring and creating larger polynomials (called resolvent polynomials); a process which can be computationally inefficient. We will discuss how to compute the Galois group of a quartic polynomial that does not rely on factoring large-degree resolvents. Instead, we use only two pieces of data about the polynomial: (1) the number of roots in the field extension it defines, and (2) its discriminant. We will also compare the efficiency of this method to the efficiency of resolvent-based methods.
Linking Statistical Emulators of Complex Scientific Computer Models
Ksenia Kyzyurova, Duke University, Durham, NCmentored by Dr. Jim Berger and Dr. Robert Wolpert
Abstract: Gaussian processes, together with an objective Bayesian
implementation of the processes, have become a common tool for emulating
(approximating) complex computer models of processes. Sometimes more
than one computer model needs to be utilized for the predictive goal.
For instance, to model the true danger of a volcano pyroclastic flow,
one might need to combine the flow model (which can produce the flow
size and force at a location) with a computer model that provides an
assessment of structural damage, for a given flow size and force.
Direct coupling of the computer models is often difficult, for computational
and logistical reasons. In this work, we focus on coupling two such computer
models by coupling separately developed Gaussian process emulators of
the models. The research involves both developing the overall coupled
emulator, and then evaluating its performance as an emulator of the true
coupled computer models.
A key issue in developing the coupled emulators
is to produce an accurate way of quantifying the uncertainty in the emulator.
The fact that we utilize to couple the emulators is that in certain parametrization
one can give closed form expressions for the overall mean and variance
of the coupled emulator.
With analytic expressions available for
the mean and variance of the coupled emulator, we form confidence intervals
for the coupled prediction assuming normality. The coupled emulator is
not actually normally distributed, so part of the study is to assess
the effect of this assumption. This is done both analytically with Taylor�s
series approximations, and via simulation and examples.
The initial
investigation has been with simple test functions as the simulators,
to see if the approach being taken makes sense. The application to complex
computer models of the types mentioned at the beginning of the abstract
is demonstrated as well.
Model II Regression Coefficient Confidence Intervals
Jeff Lail and Angie Larsen, UNCG, Greensboro, NCmentored by Dr. Scott Richter
Abstract: Our project seeks to derive and test confidence intervals for Model II Regression in a multivariate system. Previous research proposed a method for estimating multivariate functional relationships between sets of oceanographic data, and in the process, proposed a method for using Model II Regression in the multivariate space. Our project supplements that work by running repeated Model II Regression tests on multivariate data, using combinations of covariance matrices and distributions to test the confidence intervals proposed by previous research. Our project uses bootstrapping R hard coding to test these confidence intervals.
Optimization of Border Patrol Strategies
Aaleah Lancaster* and Qaleelah Smith*, Bennett College, Greensboro, NCmentored by Dr. Hyunju Oh and Dr. Jan Rychtar and Dr. Joon-Yeoul Oh
Abstract: U.S. Customs and Border Protection (CBP), is responsible for securing the border between U.S. ports of entry and has divided the 2,000-mile U.S. border with Mexico among nine Border Patrol sectors. CBP reported spending about $3 billion to support Border Patrol's efforts on the southwest border in fiscal year 2010 alone, and apprehending over 445,000 illegal entries and seizing over 2.4 million pounds of marijuana. The number of border patrol officers has been increased but because of the limitation of patrolling personnel and budget, it is critical to allocate resources appropriately. The goal of this project is to optimize border patrol routes for the border with Mexico. The infiltrators' goal is to enter US successfully while patrols intend to capture infiltrators to prevent illegal cross-border activities. We developed different scenarios to find the optimal solution using grids. By using a grid and applying analytical techniques, we were able to develop a close approximation of the number of intruders caught with each possible patrol strategies. Eventually we were able to also find the probability of an immigrant being caught. Using the probability we found an optimal solution would be the patroller moving along the border in an S motion. Our findings in the future should be tested in the real world to see if it works. This research serves as a reference to possibly solving the border patrol problem.
Site Dependent Transitions in Hidden Markov Models for Bayesian Protein Structure Alignment
Gary Larson, Duke University, Durham, NCmentored by Dr. Scott Schmidler
Abstract: Statistical models for protein alignment typically treat insertion and deletion processes at each site along the protein backbone as identically distributed. Although this provides advantages in computational efficiency, the genetic and biophysical processes giving rise to real sequences often violate this assumption. For example, amino acid insertions and deletions may be easily accommodated in flexible surface loops of a protein, but strongly unfavorable when they disrupt core internal structures or side-chain interactions. Existing probabilistic models for multiple sequence alignment can account for site-dependent rates, but rely on large data sets for accurate inference. Motivated by problems in multiple alignments of protein structural families, we develop a simple hierarchical model for inference of site-dependent transition matrices in hidden Markov models for multiple alignments. We demonstrate that this approach enables estimation of site-dependent rates using smaller data sets in the context of both multiple sequence alignment and in the recently developed Bayesian structure alignment framework. We demonstrate the potential for the hierarchical structure to allow fitting smaller data sets in the sequence-only framework.
Comparative Analysis of Transcriptomic Data Accounting for Variation in Gene Flexibility
MaLyn Lawhorn*, Winthrop University, Rock Hill, SC and Rachel Schomaker*, Florida Southern College, Lakeland, FLmentored by Dr. Olav Rueppell and Dr. Jonathan Rowell
Abstract: Comparative transcriptomic data, or gene expression profiles, can be used to examine similarities and differences in the gene expression of organisms or cells, allowing scientists to conclude whether two different experiments address phenomena that share a biological mechanism. This approach can be beneficial when trying to discover treatments for cancer or mental health diseases. Typically, two independent experiments are performed that each identifies differences in gene expression between two experimental conditions, tissues, or organisms. The differences are compared between experiments to test for significant overlap in differentially expressed genes. Traditionally, all genes are treated equal without taking into consideration that genes may have varying numbers of transcription factor binding sites, leading to different variability in gene expression. Thus, conclusions from these analyses may overstate the degree of genetic overlap due to the simplifying assumption that all genes are equal. The purpose of this study was to create a computer simulation that takes into consideration varying numbers of transcription factor binding sites of genes and generates random gene expression profiles. Thus, the simulations generated a null expectation of transcriptome overlap for genes with different genetic architectures. When these distributions were compared to empirical data, results suggested that traditional methods for measuring gene overlap may need to be reevaluated and more stringent criteria need to be applied. Overlap may have been systematically overestimated in the past due to inherent differences in gene expression flexibility among genes.
Stability for Multi-radii Persistence
Austin Lawson, UNCG, Greensboro, NCmentored by Dr. Greg Bell and Dr. Cliff Smyth
Abstract: We investigate various notions of stability for networks constructed as Rips complexes over finite sets of points with variable radii. We prove a classical stability result as well as a radius-stability result that is unique to this situation. This is joint work with Greg Bell, Joshua Martin, James Rudzinski, and Cliff Smyth.
A Semi-Lagrangian Method for Systems of Time-dependent Partial Differential Equations
Nikolai Lipscomb, University of North Carolina at Wilmington, Wilmington, NCmentored by Dr. Daniel X. Guo
Abstract: We will study a semi-Lagrangian method for numerically solving a general system of time-dependent partial differential equations. Topics include convergence, stability, and error analysis. Numerical simulations using the method will be presented.
Robust Variable Selection Using Penalized Adaptive Weighted Least Square Regression
Bin Luo, UNCG, Greensboro, NCmentored by Dr. Xiaoli Gao
Abstract: In high-dimensional settings, penalized least squares approach can lose its efficiency in both estimation and variable selection due to the existence of either outliers or heteroscedasticity. In this manuscript, we propose a novel approach, penalized adaptive weighted least squares (PAWLS), for high-dimensional robust analysis. Our main idea is to relate the irregularity of each observation to a weight vector. The estimation of such a weight vector results in the simultaneous robust estimation, variable selection and outlier detection. The proposed PAWLS is justified from both Bayesian understanding and penalized redescending M-estimation points of view. The performance of the proposed estimator is evaluated in both simulation studies and real examples.
Numerical Analysis of Temperature Dependent Viscosity on a Conjugate Heat Transfer Process for a Vertical Flat Plate
Abdullah Al Manum, University of Texas at Dallas, TXmentored by Dr. Felipe Perreira
Abstract: The present work describes the results obtained by the numerical simulation of conjugate heat transfer process for a vertical flat plate. In this analysis, the viscosity of the fluid is considered to be varied inversely with the temperature of the fluid within the boundary layer. The equations and related boundary conditions representing this flow are first transformed into dimensionless non-linear equations by using a set of non-similar transformations. The resulting equations obtained by introducing the dimensionless stream function with similarity variable into the non-dimensional governing equations, are solved numerically using the implicit finite difference method known as Keller box-scheme. Comprehensive effects of the conduction parameter, viscosity variation parameter and Prandtl number are examined on the boundary layer flow and thermal fields. Eventually, the numerical results for the velocity profiles, temperature profiles, skin friction coefficient, rate of heat transfer and the surface temperature distributions are exhibited graphically for different values of the above mentioned parameters.
A Vietoris-Rips Lemma for Multi-parameter Persistence
Joshua Martin, UNCG, Greensboro, NCmentored by Dr. Greg Bell and Dr. Cliff Smyth
Abstract: Recent progress has been made in wireless sensor networks using tools from computational homology. In this talk, we investigate the coverage problem in sensor networks with variable radii. We construct generalizations of the classical Cech and Rips complexes corresponding to the variable radius case and prove a version of the classical Vietoris-Rips lemma that relates these complexes. This is joint work with Greg Bell, Austin Lawson, James Rudzinski, and Cliff Smyth.
A Generalization of Total Perfect Codes in Graphs
Kevin McCall* and Alex McCleary*, Virginia Commonwealth University, Richmond, VAmentored by Dr. Dewey Taylor
Abstract: A total perfect code in a graph is a subset of the graph's vertices with the property that the open neighborhood of each vertex in the graph contains exactly one vertex in the subset. Total perfect codes in the direct product of graphs have been classified. In this talk, we introduce a generalization of this idea. Let R be a ring with identity. A total perfect ring code is a map from the vertex set of a graph to the ring such that for each vertex in the graph, the ring assignments of its neighbors sum to the identity. We explore the existence of total perfect ring codes in the direct product of graphs. The n-fold direct product of simple graphs has a total perfect ring code if and only if each factor has a total perfect ring code.
Analysis of Perceptions of an Emergency Nursing Academy
Emily Nance*, UNCG, Greensboro, NCmentored by Dr. Sat Gupta
Abstract: Cone Health created the Emergency Nursing Academy in 2011 where newly hired nurses are educated in all aspects of emergency nursing. In 2014, preceptors and graduates of the program were given a survey to assess their perceptions of the development of professional practice behaviors, cultivation of interpersonal relationships, and job satisfaction during and immediately after orientation. The main goal of the study is to do comparisons between graduates' and preceptors' survey responses. We also examine if the graduates who felt more welcomed and part of the unit or confident in their skills were more satisfied with their job. Results of this study will help Cone Health determine if any changes to the Emergency Nursing Academy are needed.
Cooperation in Finite Populations: Being Alone Helps
Anh Nguyen*, Texas Christian University, Fort Worth, TX and Jay Saini*, UNCG, Greensboro, NCmentored by Dr. Jan Rychtar and Dr. Jonathan Rowell
Abstract: We consider the evolution of cooperation in finite populations and we model a scenario where two individuals can interact only if both intend to do so with their counterpart. This feature allows a possibility for individuals to remain alone for a given round and not interact with anybody. Such an individual receives a baseline payoff rather than one based upon a matrix game. We provide sufficient conditions on the payoff matrix that will guarantee fixation probabilities to be monotone relative to the baseline payoff. We then apply the findings to the Prisoner�s Dilemma and Hawk-Dove games. In both cases, the possibility that an individual might remain alone increases the chances that cooperation or non-aggression fixes within the population. Moreover, weak selection models overlap with our model, and we consider how one can generalize our model even further.
An Examination of Factors Affecting Incidence and Survival in Respiratory Cancers
Roberto Perez*, University of Puerto Rico - Mayaguez, Mayaguez, Puerto Ricomentored by Dr. Kate Cowles
Abstract: In the United States and worldwide, respiratory cancers are a leading cause of cancer-related deaths for both men and women. According to the American Cancer Society, lung cancer in particular accounts for approximately 27% of all cancer deaths. The Surveillance, Epidemiology, and End Results (SEER) Program provides data on all incident cancer cases from 1973 to 2012 in certain regions of the United States. The goal of this study is to use data provided from SEER to (1) determine which demographic variables distinguish between types of respiratory cancers and (2) determine which variables affect survival of patients with various types of respiratory cancers. In addition, we combine the SEER data set with an additional data set from the National Cancer Institute�s Small Area Estimates for Cancer Risk Factors and Screening Behaviors which contains smoking prevalence data by county for the years 1997 to 1999 and 2000 to 2003 to determine what characteristics of county populations affect respiratory cancer incidence rates. In this study, we use machine learning techniques and ecological and survival analysis on SEER cancer data and National Cancer Institute data to estimate survival for respiratory cancer patients and to approximate the prevalence of respiratory cancers by age, sex, and race. We observe that the most commonly reported respiratory cancer is lung cancer for both men and women. Elderly people are at a higher risk for developing respiratory cancers, and their survival prognosis is worse. In addition, males have a higher incidence of most respiratory cancers than females, and survival rates for males are lower than females for most respiratory cancers. One of the best predictors for survival analysis is tumor grade. The SEER data contains a race category called Other which consists of American Indian/AK Natives and Asian/Pacific Islanders. Blacks and Other are at a higher risk for all respiratory cancers than whites. Furthermore, blacks have the worst survival prognosis, while Other has the best survival prognosis for all respiratory cancers.
A Mathematical Model of the Spread of Dengue Fever Incorporating Mobility
Kelly Reagan*, Elon University, Elon, NCmentored by Dr. Karen Yokley
Abstract: Dengue fever is transmitted by day-biting mosquitoes in tropical urban areas. There is currently no vaccine for dengue fever or for the fatal form of the disease, dengue hemorrhagic fever and dengue shock syndrome, so preventative measures are the only solution to slow the spread of the disease. Mathematical modeling can be used to help investigate methods of intervention. In order to model dengue fever, an SIR model can be used to describe the dynamics of the susceptible (S), infected (I) and recovered (R) humans and mosquitoes. Previous research by Lourdes Torres-Sorando and Diego J. Rodr�guez incorporated the framework of an SIR model to create a visitation model for malaria. The model simulates malaria spreading between two patches, which is similar to showing how dengue fever spreads when individuals in a community leave their area to go to work in a different area. Additional analysis was conducted on an SIR dengue transmission model developed by Lourdes Esteva and Cristobal Vargas. The visitation concepts of the Torres-Sorando and Rodriguez model were then incorporated into the Esteva and Vargas model in order to show humans traveling between two areas and being infected with dengue. A preliminary system of ordinary differential equations has been generated using both of the models and future research will be conducted on the biting rate as a function of time as well as on the model�s fit for dengue.
Eigencurves of the Singular Sturm-Liouville Problem
David Reynolds*, Wake Forest University, Winston Salem, NCmentored by Dr. Stephen Robinson
Abstract: This project deals with a certain set of Singular Sturm-Liouville Problems. The major concerns lie in the properties of the associated eigencurves. Essentially this expands the work found in Eigencurves for two-parameter Sturm-Liouville Equations by Dr. Binding and Dr. Volkmer in order to include a larger class of problems. This is done largely through variational methods, as well as pulling from A Proof of the Existence of Weak Solutions to Some Differential Equations by Andrew Frey. By incorporating certain conditions on our problem we guarantee solutions, and are able to achieve a characterization of the first eigencurve, the derivative of the eigencurves, as well as some concavity properties. The major interest lies in the divergence from the work of Drs. Binding and Volkmer due to the inclusion of a singular point. Our results differed from theirs in a uniform manner that arose from the inclusion of the conditions that guaranteed our solutions. This provides an interesting link between the Regular Sturm-Liouville Problem and the Singular case.
Modelling the Optimal Cost of Prevention Methods of Dengue Fever
Sara Rodgers*, Elon University, Elon, NCmentored by Dr. Crista Arangala
Abstract: Dengue fever is becoming the most prevalent mosquito-borne illness, endangering 2.5 billion people worldwide. With no vaccine, dengue fever has a huge social and economic burden on those affected, due to high costs of prevention and treatment of the disease. In this paper, we will use a series of differential equations relating the density of mosquitoes and infected mosquitoes in an area, number of infected people, and cost of prevention and treatment to model the economic consequences of dengue fever in a specific area. This model will provide a way to compare economic burdens faced by affected countries with the spread of dengue fever in that specific area.
A Modified Binary Optional Randomized Response Technique Model
Jeong Sep Sihm, UNCG, Greensboro, NCmentored by Dr. Sat Gupta
Abstract: We propose a modified two-stage binary optional randomized response technique model with an additional optionality question. In this model, the optionality is estimated not by splitting the sample into two, but by asking one additional randomized question regarding the main question's sensitivity to the same sample. A similar approach was explored in a previous study in which the unrelated question model was used for both the additional optionality question as well as the main sensitive question. In this study, we propose using the indirect response model for the additional optionality question and using the two-stage binary optional model for the main question, which we assume would be highly sensitive to some respondents. Computer simulation and theoretical proof show that this new approach produces more accurate estimator.
A Mathematical Investigation of Vaccination Strategies to Prevent Measles Epidemic
Raymond Smith, NCA&T University, Greensboro, NCmentored by Dr. Nicholas S. Luke and Dr. Liping Liu
co-authored by Elijah MaCcarthy and Aleah Archibald
Abstract: The purpose of this project is to quantitatively investigate vaccination strategies to prevent measles epidemics. A disease model which incorporates susceptible, vaccinated, infected, and recovered populations (SVIR) is used to investigate the process of how an epidemic of measles can spread within a closed population where a portion of the population has been vaccinated. The model is used to predict the number of infections and resulting reproductive number for the measles based on a variety of initial vaccination levels. The model is further used to investigate the concept of herd immunity, which states that if a certain percentage of the population is vaccinated then it will provide protection for the entire population. Results generated from these modeling efforts suggest that approximately 95% of the population should be vaccinated against the measles in order to establish a herd immunity.
Positive Radial Solutions to Classes of Singular Problems on the Exterior of a Ball
Byungjae Son, UNCG, Greensboro, NCmentored by Dr. Ratnasingham Shivaji
Abstract: We study positive radial solutions to singular boundary value problems of a steady state reaction diffusion equation on an exterior of a ball with a nonlinear boundary condition on the interior boundary. We discuss existence, multiplicity and uniqueness results for the case f(0)>0 and existence and non-existence results for the case f(0) <0. We prove existence and multiplicity results by the method of sub and supersolutions.
Modeling the Dynamics of Glioblastoma Multiforme and Cancer Stem Cells
Stephen Steward*, Winthrop University, Rock Hill, SCmentored by Dr. Kristen Abernathy and Dr. Zach Abernathy
Abstract: In this talk, we extend the work of Kronik, Kogan, Vainstein, and Agur (2008) by incorporating the cancer stem cell hypothesis into a treatment model for Glioblastoma Multiforme. Cancer Stem Cells (CSCs) are a specialized form of tumor cell with normal adult stem cell properties. CSCs are believed to be one of the primary reasons for cancer recurrence since they are more resilient to current treatment practices and are able to repopulate the tumor. We present a system of nonlinear ordinary differential equations that describes the interaction between cancer stem cells, tumor cells, and alloreactive cytotoxic-T-lymphocytes (CTLs). Under the assumption of constant treatment, we present conditions on the treatment amount that leads to a locally stable cure state. We also explore a more biologically accurate treatment schedule in which CTLs are injected periodically. In the case of periodic treatment, we numerically establish treatment schedules that lead to cancer persistence, cancer recurrence, and cancer eradication. We conclude with a sensitivity analysis of our parameters and a discussion of biological implications.
Optimal Aggression in Kleptoparasitic Interactions
David Sykes, Texas A&M University, College Station, TXmentored by Dr. Jan Rychtar
Abstract: Kleptoparasitism is the resource gathering behavior where one individual steals resources from another individual. Mathematical models that describe the expected costs and benefits associated with a kleptoparasitic strategy help us explain the variation observed in kleptoparasitic interactions. Here we analyze a model that is a variant of what is often called a �producer-scrounger� model. Our results show that evolutionarily stable strategies in the simultaneous game (where individuals choose strategies simultaneously) differ from previously found evolutionarily stable strategies in the extended form game (where individuals choose strategies in sequence). Specifically, we find that if sufficiently large energy investments from both competing individuals are allowed then there is an evolutionarily stable pair of strategies whereby both individuals invest equal amounts of energy in a contest. If the maximum energy investment is sufficiently limited for either individual then we find evolutionarily stable pairs of strategies whereby the individuals invest unequal amounts of energy.
Galois Groups of Degree 15 p-adic Polynomials
Jessica Weed*, Elon University, Elon, NCmentored by Dr. Chad Awtrey
Abstract: Polynomials whose coefficients are p-adic numbers play a central role in abstract algebra and number theory. A classical result states that given a prime number p and a positive integer n, there exist only finitely many �distinct� degree n polynomials with p-adic coefficients. Researchers have therefore focused on methods for counting the number of such polynomials as well as computing useful characteristics of each polynomial. One of the most important such characteristics is the polynomial's Galois group, an object which encodes arithmetic information concerning the polynomial's roots. The most difficult cases arise when the prime p divides the composite degree n. In this case, past research has dealt with all degrees less than or equal to 14. Therefore, our research focuses on our newly-developed methods for computing Galois groups of degree 15 polynomials with 5-adic coefficients.
Assessing the Adequacy of First Order Approximations in Ratio Type Estimators
Tanja Zatezalo, UNCG, Greensboro, NCmentored by Dr. Sat Gupta
co-authored by: Subhash Kumar Yadav, Department of Mathematics and Statistics (A Centre of Excellence), Dr. RML. Avadh University, Faizabad, India
and Javid Shabbir, Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan
Abstract: In many papers first order approximation for the theoretical mean square error has been used, on ratio type estimators for the population mean and variance. The main focus of this paper is on examining the adequacy of the first order approximation. We have calculated the theoretical mean square errors for many ratio type estimators, based on the first order approximation, and the corresponding empirical mean square errors. We observed that the first order approximation for the ratio type mean and variance estimators generally works well. The discrepancy between the two depends on sampling fraction. The first order approximation does not work well when the sampling fraction is big.
Distributions in a Class of Poissonized Urns with an Application to Apollonian Networks
Panpan Zhang, George Washington University, Washington, DCmentored by Dr. Hosam Mahmoud
Abstract: We study a class of Polya processes that underlie terminal nodes in a random Apollonian network. We calculate the exact first and second moments of the number of terminal nodes by solving ordinary differential equations. These equations are derived from the partial differential equation governing the process. In fact, the partial differential equation yields a stochastic hierarchy of moment equations, which can be bootstrapped to get higher moments from the equations that have been solved for lower moments. We also show that the number of terminal nodes, when appropriately scaled, converges in distribution to a gamma random variable via the method of moments.