Here are the technical reports written by the participants of AMSSI 2005, 2006, and 2007. All the reports can be downloaded in Adobe Portable Document Format (PDF). Click here to download Adobe Acrobat Reader, free software that will allow you to view and print PDF files.
Versions of four of the Technical Reports below have been published:
- Jeannine Abiva, Erika Camacho, Edna Joseph, Arpy Mikaelian, Charles Rogers, John Shelton, Stephen Wirkus, "Alcohol's Effect on Neuron Firing," The Mathematical Scientist, p. 32, vol. 32, (2007).
- Robert Bewernick, Jeremy Dewar, Eunice Gray, Nancy Rodriguez, Randall Swift, "On the Representation of Birth-Death Processes with Polynomial Transition Rates," Journal of Statistical Theory and Practice, p. 227, vol. 1, (2007).
- J.M. Anderson, A.A. Byrne, R.P. Fields, L.M. Segovia, R.J. Swift, "A Simple Stochastic Epidemic Process with Total Quarantine of Infectives," Applied Mathematical Sciences, p. 753, vol. 2, (2008).
- A. Gallegos, T. Plummer, D. Uminsky, C. Vega, C. Wickman, M. Zawoiski, " A mathematical model of a crocodilian population using delay-differential equations," Journal of Mathematical Biology, p.737-754, vol. 5, (2008).
Megan Armentrout, Whitworth University
Amber Goodrich, Central Washington University
Jennifer Nguyen, California State Polytechnic University
Lizette Ortega, University of Arizona
With data collection efforts underway in over 45 states, racial profiling in police practice is an issue of national concern. This study focuses on Los Angeles because of its diverse racial composition and the large quantity of data collected by the Los Angeles Police Department. Under a consent decree with the United States Department of Justice, the Los Angeles Police Department is required to make this data available. Using records for over 600,000 traffic stops, we analyze racial disparities found in stop and search rates. Logistic regression models are used to determine which variables are significantly related to disparities in search rates and other police practices. Based on our findings, the possibility of racial profiling cannot be ruled out.
A Mathematical Model of Political Affiliation -- 2007
Carol Ambrose, Illinois Wesleyan University
Jennifer Jones, Colorado State University
Kurt Larson, California State Polytechnic University, Pomona
Lucy Orozco, Loyola Marymount University
This work explores voting trends by analyzing how individuals form their political affiliations during a presidential campaign. Using a variation of the traditional epidemiological model, we construct an ODE model that represents the transition of potential voters through various levels of political interest in either the Republican or Democratic Party. We analyze variations of our model to understand the impact of various interactions between potential voters, such as those between politically-charged and apathetic individuals during a presidential campaign. Finally, we calculate and interpret threshold values to determine the stability of the steady state solutions.
Mathematical Models of Reptile Populations Using Delay Differential Equations -- 2007
Tenecia Plummer, Albany State University
Cinthia Vega, Loyola Marymount University
Clare Wickman, University of Maryland, Baltimore County
Michael Zawoiski, City College of New York
A well-known phenomenon among reptile species is temperature-dependent sex determination (TSD) in which the temperature of egg incubation determines the sex of the hatchlings. Based on previous work in (Murray, 2002) and (Woodward and Murray, 1993) we develop a delay differential equation (DDE) model describing the nesting habits of crocodilians; the delay accounts for some of the dependence of birth and death rates on the age of the population members. We are able to reasonably account for age structure in the population while finding a nonzero stable equilibrium. Stability of this equilibrium allows us to compare this to the biological data of Smith and Webb (1985, 1987); we obtain very strong agreement. Additionally, we solve our model numerically using a modified Runge-Kutta solver in Matlab and investigate the effects of environmental changes on the population.
Permutations in Concatenated Zigzag Codes -- 2007
Shawn Abernethy Jr., Elizabethtown College
Cindy Lee, Loyola Marymount University
Jasmin Uribe, University of Arizona
Sai Michael Wentum Jr., College of Charleston
Coding theory is a branch of mathematics, computer science and electrical engineering that explores the transmission of information across noisy channels. Coding theory is used in data transmission, data storage, and telecommunications. The focus of this project is on concatenated zigzag codes, which are constructed using permutations. We study the effects of permutations on the error-correcting capabilities of the coding scheme. In conjunction, we explore the behavior of average dispersion in order to further our understanding of randomness of a permutation and find correspondence with error-correction.
Algebraic Interleavers in Turbo Codes -- 2006
Jason Dolloff, Southwestern University
Zackary Kenz, Concordia College-Moorhead
Jaquelyn Rische, Whittier College
Danielle Ashley Rogers, Western Washington University
Coding theory is the branch of mathematics and electrical engineering that involves transmitting data across noisy channels via clever means. While the transmission can be very error-prone, there are various methods used to send the data so that a large number of the errors can be corrected. Applications in communications, the design of computer memory systems, and the creation of compact discs have demonstrated the value of error-correcting codes. Our research examines a specific class of codes called turbo codes. These high performance error-correcting codes can be used when seeking to achieve maximal information transfer over a limited-bandwidth communication channel in the presence of data-corrupting noise. In order to gain a deeper understanding of turbo codes, we study a particular component, interleavers. An interleaver is a device that scrambles the sequence of data bits before transmission. In order to study this component and how it affects the performance of turbo codes, we examine two of its properties: spread and dispersion. Using computer simulations, we will not only study how these properties affect the error rates, but also work toward creating new properties that will aid in examining the effectiveness of interleavers.
Deterministic and Small-World Network Models of College Drinking Patterns -- 2006
Lorenzo Almada, University of Georgia
Roberto Rodriguez, North Carolina State University
Melissa Thompson, Central Washington University
Lori Voss, University of Missouri-Rolla
Drinking on college campuses, especially binge drinking, contributes to numerous unintentional injuries, sexual assaults, and poor performance in classes. We are interested in modeling college drinking in order to guide policymakers in the creation of laws that will help decrease college binge drinking and its effects. In order to model the manner in which college drinking spreads, we have created two different models. Our first model modifies a five equation, deterministic, homogeneous, compartmental model of college drinking developed by Scribner, et.al. In order to consider the dynamics of college drinking, our model assumes that social interactions, social norms, and individual risks are most influential in students' decisions to consume alcohol. We focused our attention on binge drinkers by combining light and moderate drinkers into one class and by incorporating the social interactions between social drinkers and problem drinkers and between bingers and problem drinkers. As part of our investigation, we simulated different alcohol environments by varying the parameters. We analyze the implications of this model from both mathematical and sociological perspectives. However, we recognize that a homogeneous model of drinking does not accurately represent the individual-to-individual interaction, connection, and influence of members of the same group. Hence, we created a second model based on graph theory and small-world networks. This model considers a heterogeneously mixed population, where students are represented as unique individuals. In particular, the effects of clusters, or clique groups, are analyzed in relation to the dynamics of the system. Finally, we compare our deterministic model with our network model and give some recommendations.
On Solutions to a Nonlinear Elliptic Equation -- 2006
Michael Grigsby, California State Polytechnic University, Pomona
Cathy Ho, Boston University
Adriana Melgoza, Loyola Marymount University
Hernan Osco, California State Polytechnic University, Pomona
Some Simple Epidemic Models -- 2006
Jill Anderson, South Dakota School of Mines & Technology
Adrienne Byrne, Buffalo State College
Ruthie Fields, Boston College
Linda Segovia, Honors College of Florida Atlantic University
In this paper, a brief overview of the simple deterministic susceptible/infective (SI) epidemic model is detailed. The deterministic model is appropriate for large populations, where random interactions can be viewed as being averaged out. In smaller populations however, random interactions play a significant role. For example, N.T.J. Bailey (1963) considered a stochastic SI epidemic model. With the help of previous work by Gani and Swift (2006), Bailey's solution is modified to include a preventative quarantine of the susceptible population. From this new equation, the associated probability generating function, the transient probabilities, steady state probabilities, expected value, and expected epidemic duration can be found. Computer simulations support the results for this stochastic SI model.
Alcohol's Effect on Neuron Firing -- 2005
Jeannine T. Abiva, Loyola Marymount University
Edna S. Joseph, University of the Virgin Islands
Arpy K. Mikaelian, University of California Santa Barbara
Charles R. Rogers, North Carolina State University
Neurons are responsible for transmitting messages throughout the body via long distance electrical signals known as action potentials. These depend on the active transport of sodium and potassium ions across the cell membrane. The effect of various drugs on the process of neuron firing is a current research interest. The Hodgkin-Huxley equations, a system of four nonlinear ordinary differential equations, mathematically model the influx and efflux of these ions across the cell membrane. In the presence of alcohol, the release of potassium ions is accelerated. We propose a modified version of these equations, which incorporates the effect of alcohol, and examine its implications through mathematical analysis in dynamical systems. We investigate the qualitative behavior and interpret the results of the steady-state solutions in the fast and fast-slow phase planes.
A Continuous Model of Gene Expression -- 2005
Joseph Hunt, Lamar University
Lissette Laplace, University of the Virgin Islands
Elizabeth Miller, Ferris State University
Jason Pham, California State Polytechnic University, Pomona
Gene expression is the process by which a gene makes its effect on a cell or organism. Linear differential equations have been explored as a model for gene expression. We discuss the shortcomings of this model, and we propose a system of nonlinear differential equations to mathematically model gene expression in prokaryotes, specifically bacteria. We investigate this biological system using explicit functions that describe the processes of protein synthesis which includes transcription, translation, degradation, and feedback in hope of shedding light on their associated rates. We analyze the transient and steady state solutions of the model and give a biological interpretation of these results.
Exploring Interleavers in Turbo Coding -- 2005
Benjamin Moreno, California State Polytechnic University, Pomona
Laura Smith, Western Washington University
Andrea Viteri, The College of St. Catherine
Kouadio David Yao, University of Arkansas at Little Rock
The primary aim of coding theory is the successful transmission of information across noisy channels. For half a century, coding theory has been used in a variety of applications such as communications, the design of computer memory systems, and compact discs. Our research focuses on a class of codes called turbo codes, which are currently used in deep-space and satellite communications. In particular, we examine one component of these codes called an interleaver; this component permutes data before transmission. We study properties of interleavers such as spread, dispersion, and cyclic decomposition. The project focuses on the effectiveness of turbo codes, examining how the abovementioned characteristics of interleavers affect the error rates. We use computer simulations to test our theoretical findings.
Population Processes -- 2005
Robert L. Bewernick, University of California, Los Angeles
Jeremy D. Dewar, Loyola Marymount University
Eunice Gray, Sam Houston State University
Nancy Y. Rodriguez, Loyola Marymount University
Deterministic population models describe population sizes and their dynamics. However, random chance plays a large part in the growth of real-life populations. In this technical report, birth-death formulations for single and competing populations are developed. It is shown that these stochastic processes have expected values that agree with the corresponding deterministic models. A representation for the partial differential equation that a probability generating function of a birth-death process with polynomial transition rates is derived. This representation is in terms of Stirling numbers and is used to develop some of the properties of these processes. The analysis in this report uses both analytical methods and simulations.
