Since 2004 the mathematics department has taken part in the University of Mary Washington’s Summer Science Institute. Two students at the junior or sophomore level are chosen to work with a professor on a research project. Each student receives a $2800 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, and several students in the program have gone on to complete theses and graduate with Honors. Several students have also presented their work at national conferences.
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.
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. How to Win Every Time 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.
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.
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.
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.
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.
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).
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.
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.
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.
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.