Since 2004 the mathematics department has taken part in the University of Mary Washington’s Summer Science Institute. Several students at the junior or sophomore level are chosen to work with professors on research projects. Each student receives a $3000 stipend in addition to free room and board for the entire ten-week period. The students may then choose to continue doing research during the following school year. Many students in the program have gone on to complete theses and graduate with Honors. Also many students have presented their work at national conferences.

## 2019

**Hannah Frederick** worked with Dr. Randall Helmstutler on a research project in the area of non-commutative cryptography. In her project, Hannah devised a method of altering the well-known “suitcase with two locks” scheme so that it could be implemented with a non-abelian group of matrices. Hannah designed a way for such a scheme to work using the group of invertible matrices over a finite field, using her new method as a key exchange protocol for the Hill cipher. In order to adapt this new idea to this group of matrices, Hannah had to prove many results about the ring of circulant matrices, and she performed experimental computations to test and confirm her conjectures (and eventually prove them). Hannah won the first place award in the presentation category at the Jepson Summer Science Symposium, and she looks forward to presenting her work at external conferences this coming year.

In her SSI project, **Makayla Ferrell** considered the dynamics of the Human Immunodeficiency Virus (HIV). She used both analytical and computational methods to investigate her model under the direction of Dr. Leo Lee. Specifically, she derived analytical solutions of some simplified versions of the model, and applied four different numerical schemes to approximate solutions to the model. Her analysis provided an interpretative diagram of how healthy cells react to HIV and how HIV spreads without treatment.

**Ashley Scurlock** and **Isabella Gransbury** worked with Dr. Jeb Collins on two separate projects this summer. Ashley investigated the properties of non-commutative number systems. In particular, she examined quaternions, a non-commutative generalization of the real and complex numbers that is four-dimensional. Ashley explored how some of the well-known functions and identities change when commutativity is taken away. She also examined an anti-commutative version of Pascal’s triangle and proved a relationship to the original Pascal’s triangle. Isabella’s work involved solving differential equations using Bertini, a software package used to solve systems of polynomial equations. Isabella considered approximating a nonlinear boundary value problem with the finite element method with linear elements, and then solving the resulting polynomial system with Bertini. She then expanded the work by developing a similar procedure for quadratic elements, for which the complexity of the polynomial system is significantly increased. She tested each of these algorithms and compared their results to determine which is more effective.

## 2018

**Emily MacIndoe** and **Amy Creel** worked together to solve the Susceptible-Infected-Virus (SIV) model for the Human Immunodeficiency Virus (HIV) infection mathematically and computationally under the direction of Dr. Leo Lee. Emily presented analytical solutions to two simplified versions of the model giving exact results. Her results not only give insight into the roles of birth and death rates in the SIV model, but also contribute to our understanding of HIV dynamics. On the other hand, Amy constructed algorithms using various numerical techniques and created computer codes to approximate solutions of the model. Throughout her numerical experiments, she was able to see how rapidly a patient would progress to Acquired Immune Deficiency Syndrome (AIDS). Her results could be used to determine parameters in the model that were estimated from patient data and develop better treatment options for a given AIDS patient.

Under the guidance of Dr. Julius Esunge, **Creigh Brigman** conducted an investigative project on “The Beauty of Complex Analysis” during the 2018 Summer Science Institute. The project allowed him to highlight differences between real and complex analysis, identify some of the key tools and techniques in the field and show how these simplify otherwise intractable problems from calculus and real analysis. Creigh gave an oral presentation of his work at the closing SSI symposium.

**Riley Anderson** and **Makenzie Clower** worked under the supervision of Dr. Jeb Collins on implementing and refining a machine learning algorithm to improve performance of a software called Bertini. The Bertini software works to solve polynomial problems and can be very slow when using the default settings. Riley and Makenzie used machine learning techniques to determine the optimal settings for Bertini to optimize the computational efficiency. Riley used neural networks to determine when a polynomial is singular, as different settings are required to solve singular polynomial equations. Makenzie examined the various other settings of Bertini, and used neural networks to determine their optimal values for a particular polynomial. Both of them had to generate sufficiently large datasets for the machine learning algorithms, and then used those data sets to find the best machine learning algorithm for their particular tasks. Makenzie presented a poster at the SSI conference at the end of the summer, and Riley presented a talk. Riley won the best oral presentation out of all SSI speakers. They will be continuing their work during the semester, and presenting at three conferences in the next few months, including the Joint Mathematics Meetings in Baltimore.

## 2017

**Ekta Kapoor** and **Gail Crunkhorn** worked with Dr. Debra Hydorn on developing several measures for assessing interrater reliability (IRR) for a situation with more than one rating criteria. Evaluating is important for assessing the rating criteria and how well raters understand and consistently apply those criteria, but the traditional measures for IRR are useful for situations where only one rating is made by two or more raters. Gail and Ekta focused their work on Fleiss’s kappa, which “corrects” overall agreement between raters based on the probability of agreement just due to chance. They wrote several programs in R to conduct simulations to study the distribution of their kappa modifications assuming uninformed raters (ratings produced at random). They used the results of their simulations to evaluate the use of their modified kappa criteria on an example set of ratings which had four raters rating 17 “objects” using nine different criteria. This project was a continuation of the work Gail and Ekta had done as part of the spring PIC Math course (MATH 361).

**Shannon Haley** worked with Dr. Randall Helmstutler on the project “A Massey-Omura Cryptosystem with Disjoint Permutations.” The original encryption scheme was patented by Massey and Omura in the 1980s, its algorithm being based on the well-understood algebraic properties of modular arithmetic. Shannon’s project was an attempt to modify their scheme in a non-commutative setting, wherein the algebra would hopefully be sufficiently more complicated to better conceal information. Shannon worked on finding ways to adapt the original Massey-Omura system so that it could be implemented over permutation groups. These groups are known to be non-commutative and very large, two desirable properties in cryptography. However, this move to a noncommutative setting requires an entire rebuilding of the system. Shannon worked out one way to accomplish all of this, replacing modular exponentiation with an action by conjugates in a symmetric group, even developing measures of security for the resulting system. Shannon is continuing her research in this area over the next academic year, looking at different ways of building non-commutative Massey-Omura schemes in her honors thesis research.

Under the direction of Dr. Julius Esunge, **Jack McMinimy** undertook a statistical analysis project on a massive dataset with over six million entries of airline travel information. (Yes, that’s right, six million.) Using the statistical analysis program R, Jack focused his attention on analyzing departure delays based on airline, geographic location, month, and time of day. His culminating poster was presented at the SSI symposium in July.

## 2016

**Nora Benedetto** and **Kelley Swenson** (1st place winner for the poster presentation) worked on statistical projects with Dr. Melody Denhere, analyzing UMW’s historic data provided by the Office of Institutional Analysis and Effectiveness. Kelley’s project, Predicting Enrollment using Time Series Models, was motivated by the PIC Math course that Dr. Denhere taught in the spring. The purpose of the project was to find a statistical model that could accurately predict enrollment for UMW. Four statistical models were considered, including the Holt-Winters and AR(1) models. By way of cross-validation, Kelley determined that the Holt-Winters model was the most accurate for short-term enrollment projections. She received a first place award for her poster presentation at the SSI symposium. Nora analyzed course evaluation completion rates at UMW using different statistical tools. This work continued work she had started in the spring and the resulting project was titled Cluster Analysis of Course Evaluation Response Rates. The goal of the project was to determine if significant trends existed in the response rates at UMW. ANOVA tests and hierarchical and k-means cluster analysis methods were used to determine these trends. She concluded that in the fall semesters, upper level students and the College of Education were the different groupings with the highest response rates.

**Rachelle Dambrose **(1st place winner for the oral presentation) and **Aaron Thomas** worked together to solve Poisson’s Equation analytically and computationally under the direction of Dr. Leo Lee. In particular, their projects focused on heat flow through a two dimensional square domain. Their projects were titled *Numerical Approximation of Poisson’s Equation* and *Mathematical Solution to Poisson’s Equation*, respectively. In Aaron’s project, he found the mathematical solution of the equation using the method of eigenfunction expansion and wrote computer programs to graph/simulate his solutions. On the other hand, Rachelle derived several numerical models of Poisson’s equation with the Taylor series expansions of functions. She also developed a computer code based on her algorithm to solve numerical model problems using various input parameters. Then Aaron and Rachelle performed together an experiment to find the temperatures through a heated piece of aluminum foil with a chemistry professor. Their experimental data was compared to both analytical and computational solutions of Poisson’s equation demonstrating that their models and computer programs could be used to predict real world phenomenon.

## 2015

Dr. Julius Esunge led a research project working with UMW undergraduates **Rebecca Revercomb** and **William Scheid**. Both students worked on projects in computational actuarial science and their project titles are *Yield Curves and Interest Rate Modeling *and* The Yen/Dollar Match Using GARCH Modeling*, respectively. Their efforts culminated in presentations at the Summer Science Symposium, where Rebecca gave an oral presentation and William gave a poster presentation.

## 2014

**Marlene Caceres** and **Victoria Moore** worked on statistical projects in the emerging field of functional data analysis with Dr. Melody Denhere. By way of simulation study and application to real world data, Victoria carried out a comparative study on the effect of the basis choice in the functional linear regression model. The study tested the effects of choosing between three of the widely used basis functions – the Fourier basis, the B-spline basis, and the monomial basis. Using R for simulations, her results from the simulation study and the real data supported the fact that the Fourier basis works best with periodic data, while the other two basis functions were ideal for the non-periodic data.

Marlene’s project was titled The Curse of Dimensionality in Functional Data. Marlene’s project focused on reducing the dimension of the functional data model by applying some multivariate statistical methods to a reduced form of the functional regression model. These methods were principal component analysis (PCA), factor analysis (FA), and stepwise regression. The study consisted of investigating applicable methods that could be used for dimension reduction and elimination of multicollinearity, generating functional data with a distinctive pattern, then applying the dimension reduction methods and determining how well the reduced data retained the original trend. Marlene used measures of model efficiency to determine that on average, PCA was the best at retaining the variation in the data followed by FA with the stepwise method being the least effective.

## 2013

**Casey Howren** analyzed a mathematical model of communicable diseases describing the susceptible, infected, and recovered populations under the direction of Dr. Leo Lee. Her project was titled *The S I R Model for Evaluating the Impact of Epidemics on a Population*. In her project, she derived the exact solution for the model under some appropriate assumptions, developed an algorithm using Euler’s Method, and wrote the corresponding computer code to compute numerical solutions. Also, she applied her method to real world data in order to model the behavior of H1N1, as well as polio and HIV. Her work seeks to show that using mathematical methods to model epidemics can accurately predict the behavior and severity of a disease.

**Katie Jones** and **Kyle Genovese** worked with Dr. Julius Esunge on projects in applied statistics. The statistical tools of multiple regression and principal component analysis were introduced to the students, alongside the proper organization of data, and the use of the R software. Thereafter, these techniques were applied to different sets of data, to model economic growth in developing countries, analyze total energy consumption in the USA, and to study nationwide individualized education plans.

**Dane Lawhorne** spent his summer studying the digital line and digital circles with Dr. Randall Helmstutler. These topological spaces have been known to computer scientists since the 1970s, who used them as discrete or “pixelated” versions of the real line and unit circle. Dane proved that although the digital line is countable and digital circles are always finite, they still share many of the same properties and relations as the ordinary real line and circle. For instance, the symmetry groups of the digital line and the real line have related algebraic structures, and all the relations between the real line and circle found in homotopy theory carry over to the homotopy theory in the digital cases. Dane and Dr. H. plan to travel to Longwood University to present Dane’s work at the sectional meeting of the MAA in November.

**Kimberly Hildebrand** worked with Dr. Debra Hydorn to examine gender bias in hiring practices. The number of women hired for a fixed number of positions was modeled using the sum of independent Bernoulli random variables, each with a different proportion of female applicants. The proportion of female applicants for each position was modeled using a series of different Beta distributions, with the average proportion of women less than, equal to, or more than the proportion of men. Kimberly derived the expected value and variance for the number of women hired and the probability that one or fewer women are hired. She wrote a program in R to run simulations which she compared to the results for the Binomial distribution. Her results suggest that when the average proportion of female applicants is around 0.60 the probability that one or fewer women is hired is around 15% but when the average proportion of women applicants is around 0.40 the probability that one or fewer women is hired is around 2%. Depending on the proportion of female applicants for a given situation, these results could be used as evidence to support a gender bias in hiring practices.

## 2012

**Peter Slattery** and **Morgan Brown** worked together on the description of waves by moving objects under the direction of **Dr. Leo Lee**. Peter’s project was titled * Brave the Wave*. In his project, he found a mathematical expression for the wave of a vibrating string with fixed, motionless endpoints. He also wrote computer programs to simulate his findings. Then he applied his analytical results and computer simulations to win carnival balloon popping games.

*was the title of Morgan’s portion of the project. She investigated numerical wave models with the Taylor series expansions of functions. After deriving and analyzing the models, she developed her own codes to give both a numerical and a visual representation of the object’s motion. Then she used her models with their computer animations to find optimal times to shoot at a balloon attached to a vibrating string.*

**How to Win Every Time****Kwadwo Brobbey** and **Benjamin Tuxbury** worked with **Dr. Julius Esunge** on a series of problems in the field of stochastic programming and optimization. The perennial desire to maximize profit and minimize cost lies in literally every field. As such, mathematical models that accomplish the stated objectives are extremely desirable. One method of optimization, stochastic programming, has become increasingly useful as computers are being developed with greater processing power. There are a multitude of potential applications of stochastic equations and Monte-Carlo simulations (exhaustive simulations). They offer the ability to minimize risk, in order to maximize long run profits in almost any imaginable sector. In agriculture they can be used to model weather patterns, so farmers have a better idea of how to plant crops. Also, in any commercial setting, stochastic models can predict how many customers will show up given changing circumstances. When written correctly, simple programs have the ability to establish the most favorable decision given unknown variables. These programs can then be adapted to suit different contexts as they become more and more complex. Their project captured a

number of real-world applications of stochastic optimization.

## 2011

** Profiting with Options Using the Black-Scholes Equation by Kathryn Dillinger under Dr. Lee:** Katie Dillinger derived and analyzed numerical models of the Black-Scholes equation using the explicit, implicit, and Crank-Nicolson methods attained through finite difference equations. She also developed her own codes to determine which numerical method was best by comparing her computational results with analytical output from Becca’s work. After finding both analytical and numerical solutions, the team gathered data from the real-life examples of the S&P 500 index and its European option chain for the month of June 2011. The data allowed them to compare the accuracy of each solution in a real-life scenario and to analyze the result.

** Explorations of the Laplace Transform by Marianne Dubinsky and Applications of the Laplace Transform by Catherine O’Doherty under Dr. Esunge:** Marianne Dubinsky and Catherine O’Doherty

worked on a project with Dr. Esunge focusing on properties and applications of the Laplace Transform in analysis, probability and differential equations. It was interesting to see how the construction and properties of some important functions flow naturally from determining the transforms of certain base functions. The project was presented during the closing symposium of the Jepson Summer Science Institute, and both students presented talks at MathFest in Lexington, KY in August. In January 2012, they will present some of their work at the Joint Mathematics meetings in Boston and both will be completing honors theses in the spring with Dr. Esunge.

** Mathematical Analysis of Option Pricing by Rebecca Presor under Dr. Lee:** Rebecca Presor worked on an option pricing model under the direction of Dr. Leo Lee. In her project, she examined the economic phenomenon of option pricing through mathematical means using the Black- Scholes model. She derived the analytical solution to the model based on given input data such as terminal and boundary conditions. She then wrote computer programs to simulate her analytical solutions.

** Homotopy Theory of Finite Topological Spaces by Ryan Vaughn under Dr. Helmstutler:** Ryan Vaughn worked with Dr. Helmstutler on a project titled The Homotopy Theory of Finite Spaces.

Their project attempted to understand how finite topological spaces may be used to define groups, thereby providing a formal link between finite topology and abstract algebra. The idea for the project came from the observation that so far only infinite spaces have been used to form groups in topology: no one had figured out how to use finite spaces for the same purpose. It turns out there is a good reason for this, as Ryan eventually proved that no finite space can give the

right kind of algebraic structure in topology. Ryan’s talk at the Jepson Summer Science Symposium took the second place award for best presentation. Ryan and Dr. H plan to travel to Boston to present their work at the AMS-MAA Joint Mathematics Meetings in January.

## 2010

**Geometric Brownian motion:** **A safe assumption?**** by Snyder-Beattie and Kevin Groat under Dr. Esunge:** Andrew Snyder-Beattie and Kevin Groat worked on a project dealing with financial markets. On the heels of the recent global financial crisis, this project sought to examine the leading model for pricing of financial derivatives and expose the students to the underlying mathematics. Actual data from traded securities was examined within the context of the Black-Scholes pricing mechanism with a view to deciding how well this data follows the model. The conclusions of this 10 week study were presented to SSI participants and faculty at the closing symposium on July 21 at UMW and also at the summer meeting of the MAA on August 6 in Pittsburgh, PA. At both gatherings, this project received a commendation for “outstanding presentation”.

** Concentration of a Chemical Pollutant Modeled by a Fourier Series by Teresa Yao under Dr. Lee:** Teresa worked on analyzing equations modeling the diffusion of a chemical pollutant in one and two dimensional regions. Based on given input data such as initial and boundary conditions, she derived the Fourier series-that is, a combination of infinite sums of sine and cosine terms-that models the solution of the equation in both one and two dimensional regions. She then developed computer programs to simulate each Fourier series in different dimensions.

**On Numerical Models of Chemical Pollutant Diffusion by Erin Strange under Dr. Lee:** Erin used numerical models to analyze the diffusion of a chemical pollutant in a rod. She first derived a mathematical model that describes how the chemical pollutant disperses in the rod over time. She then derived three different numerical models using a centered difference in space and forward, backward, and averaged differences in time, respectively. After she derived the numerical models, she wrote computer programs for each model to see the chemical concentration at each time step in the form of vectors, graphs, and animated graphs. Finally she compared her numerical output from the exact output from the mathematical equation and determine which numerical model is best.

## 2009

** A Comparison of Remedial Measures for Multicollinearity in Multiple Regression Analysis by Sarah Ball, under Dr. Hydorn:** Sarah Ball is majoring in both mathematics and economics. As a result, when Dr. Hydorn had the opportunity to work with Sarah this summer, she found a topic that would be of interest in both majors. Sarah worked on a project in regression analysis, investigating remedial methods for data sets with multicollinearity, or strong associations, among the independent variables. Sarah investigated a new method for the situation in which there are two highly correlated independent variables, which they are calling the “2 point” method. In this method two data points are added to the data set to stabilize the regression estimates while leaving the estimates basically unchanged. Sarah continues with us this fall and plans to graduate in May of 2010.

**Numerical Estimates of Temperature Changes Using the Finite Difference Method by Elizabeth Bernat, under Dr. Lee:** Liz used the finite difference method to approximate solutions to Laplace’s equation in order to find the temperature distribution over the interior of the same domain. After creating a code to run the numerical scheme, she verified that it produced results that were physically

reasonable for the given boundary conditions.

** Generalized Dihedral Groups and Geometry by Barbara Brown, under Dr. Helmstutler:** Barbara Brown worked with Dr. Helmstutler on generalizations of the classical dihedral groups. These groups are known to model the symmetries of regular polygons, and Barbara examined ways of generalizing their construction in order to produce new groups with similar algebraic features. She was able to prove a structure theorem on the commutativity of such groups and computed many of these generalized dihedral groups of low order. She will continue her work in the fall as part of an honors project in mathematics. Barbara presented her work at UMW’s Summer Science Symposium, where she won first place in the presentation category.

** An Exact Solution to Laplace’s Equation Inside a Rectangle by Kathryn Christian, under Dr. Lee:** Kathryn worked on modeling heat conduction in a two-dimensional solid, which is one in which the same-shaped top and bottom surfaces are parallel and insulated, and the factors contributing to heat conduction do not depend on z-coordinates. She studied an exact solution obtained by the method of separation of variables and then created a computer program to calculate approximations for such solutions.

## 2008

Catherine Castleberry and Katie Hunsburger worked with Dr. Mellinger on a project titled Coding and Cryptography with Hyperovals of PG(2,2^s). A hyperoval of the projective plane PG(2,q), q even, is a collection of q+2 points, no three collinear. In this project, we use hyperovals to construct several families of secret sharing schemes and binary linear codes. The secret sharing schemes use either the nucleus of the underlying conic (used to create the regular hyperoval) or the coefficients of the quadratic form as the secret. The codes are generated by incidence matrices arising from the points and the secant or skew lines. We are able to prove many results about minimum distances and dimensions of our codes.

Jonathan Stallings and Thomas Wolfe worked with Dr. Hydorn on two separate projects. Jonathan’s project was motivated by the need for analyzing the error in GPS data. Assuming an underlying bivariate normal distribution for the longitude and latitude, he used improved estimates of the eigenvalues of the sample covariance matrix to produce an improved estimate of the covariance matrix. This new estimate of the covariance matrix was then used to produce a confidence ellipse for the longitude and latitude. Jonathan wrote a computer simulation to generate random data to investigate the effectiveness of his new error ellipse in capturing the true longitude and latitude. Thomas worked on developing a statistical tool for choosing a best model from a set of competing models, when some of those models are for the natural log of the dependent variable (Y) instead of Y. The method he used is the Akaike Information Criteria, and he derived the AIC for the log-normal distribution (the distribution of Y when the log of Y is normal). Thomas also wrote a computer simulation to determine the effectiveness of his AIC in identifying the correct model from among four competing models (linear, exponential, logarithmic and power).

## 2007

Christopher Triola worked with Dr. Lehman on research related to recursive sequences of integers, taken modulo primes. They generalized results about the periodicity of second-order linear homogeneous recurrence relations, such as the Fibonacci sequence, to higher orders, using methods from number theory and abstract algebra. The main new result was a criterion for the factorization of cubic polynomials modulo prime numbers in terms of the period lengths of related third-order recurrence relations.

Roberto Palomba worked with Dr. Helmstutler on problems related to categories. Category theory is essentially one level of abstraction higher than, say, abstract algebra. Specifically, the project was an attempt to classify certain kinds of categories which arise in parts of algebraic topology. They made use of known structural theorems from semigroup theory to classify parts of the structure of these categories. The main result is that such categories must exhibit a very strong type of homogeneity. Roberto presented his findings at the national meeting of the MAA in San Jose, CA in August 2007.

## 2006

Erin Keegan and Bob Carrico worked with Dr. Hydorn on finding the distribution of the number of shared items in the top n portion of three randomly ordered lists of N numbers, and on estimating the eigenvalues of a covariance matrix using confidence interval estimates of the characteristic polynomial.

Gardner Marshall and Ryan Platt worked with Dr. Helmstutler on two distinct research projects in topology. Both projects utilized methods of homotopy theory and abstract algebra to examine certain geometric problems in higher dimensions. Ryan worked on extending a theorem on spheres (the Borsuk-Ulam Theorem) to higher dimensions, and then used the results to solve some classical partitioning problems on spheres. Gardner gave an in-depth analysis of some exotic topological objects known as the “spin groups” and learned how they model strange rotational phenomena in quantum physics.

## 2005

Jared Moon and Allison Piccolo worked with Dr. Edmunds on basins of attraction.

Sean Droms and Chris Meyer worked with Dr. Mellinger on geometric constructions of error-correcting codes.

## 2004

Keith Manion and Lisa Song worked with Dr. Edmunds on toy competition models.

Amanda Passmore and Jennifer Stovall worked with Dr. Mellinger on constructions of LDPC codes.