Abstracts of the talks
Undergraduate StudentsDawit Adnew: Molecular Evolution of Candidate Genes for Caste Divergence in Apis
Jasmine Alexander-Floyd: Modeling the effect of fires on red-tailed leafhoppers and sedentary moths
Nicole J. Bader, Terri Johnson, Meghan R. Kent, Jenna Rice: Modeling the Dynamics of Corporate Credit Ratings
Crystal Bennett: TBA
Adam Boseman: On the zeros of Riemann’s zeta function displaced by a real number in the interval [0,1)
Amanda Brown: An Agent-Based Simulation of Insecticide Treated Nets and Their Impact on Populations at Risk for Malaria
Meghan Fitzgerald: Stealing spiders and clever hosts: a model of kleptoparasitism in an arachnid system
Kristen Gulledge, Emily Wisner: Has the Housing Market Hit Bottom?
Alan Guo: Lattice point methods for combinatorial games
Christy Minor: Agent-Based Simulations of Acquired Immunity in Malaria Endemic Populations
Dan Popescu: Stochastic signal transduction by G-protein coupled receptors
Erin Raspet: Kleptoparasitism in Onthophagus taurus: The role of density and major males
Shunda Rushing: How Field Conditions Influence Brood Ball Production By the Dung Beetle Onthophagus taurus
Jeremy Semko: Coarsening Droplets: Statistical Approaches to Evolution Dynamics Using ODE Simulations
Christine Wu and Ryland Pigg: The "Failures" of Risk Management
Graduate StudentsAbraham Abebe: Iterated Norms of Nikolskii-Besov Type Spaces with Generalized Smoothness
Heather Allmond: Model of the effect of dung age on the kleptoparistic behavior of the dung beetle Onthophagus taurus
Ricky Farr: Solving an Integral Equation by Hierarchical Matrix Methods
Robert Gove: A cause-effect network model of Arabidopsis lyrata
Luis G. Hercilla-Heredia: Mathematical Principles of Military Combat and Counter-Insurgency
Nels Johnson: Ecosystem Monitoring Using Mathematical Models, an Example
Abstract: Most social insects, like honey-bees, live in colonies with a reproduction division of labor between a queen and her female workers. This division of labor entails morphological specialization. The queen is larger and contains significantly larger ovaries than the workers. However, this caste divergence varies among species. To investigate a possible molecular basis of this variation we selected five functional candidate genes and studied their molecular evolution. We compared existing Apis mellifera sequences with their homologs that were obtained from Apis dorsata by high throughput sequencing. We hypothesized that genes that are relevant for caste divergence show signs of positive selection. From the aligned sequences we determined synonymous and non-synonymous substitutions between the two species and calculated relative rates of evolution in the five candidate genes. One gene (GB14535) showed significant excess of non-synonymous substitution. This gene is similar to the "abnormal oocyte" gene in Drosophila, which is a regulator of transcription, interacts with heterochromatin, and influences germ band length as a maternal effect gene. We conclude that this gene may have played an important role in caste divergence in Apis.
Abstract: Experiment (Panzer, 2003) showed that the post-fire recovery of the endangered red-tailed leafhopper ( Aflexia rubranura ) and sedentary moth (Papaipema eryngii ) depends on the number, quality, and proximity of neighboring unburned patches. We use a mathematical model to describe the local colonization-extinction dynamics of these species before and after the fire. We test our model's validity using the data from the experiment as well as using simulations.
Abstract: Kleptoparasitism among the dung beetle Onthophagus taurus involves the beetle stealing a brood ball of another beetle and either destroying the existing egg and replacing it with their own or destroying the brood ball completely. This is an extreme form of kleptoparasitism since it involves a total loss for one beetle, and yet only a small gain for the other. We develop a mathematical model to see how the aging effect of dung affects the kleptoparasitism of dung beetles. We will use the model to investigate which strategy is the most beneficial. Each strategy involves the beetle landing on new, 1, or 2 day old dung, deciding to steal or not steal, and deciding to guard or not guard. This results in 12 different strategies being investigated.
Modeling the Dynamics of Corporate Credit Ratings
Nicole J. Bader, Terri Johnson, Meghan R. Kent, Jenna Rice, North Carolina State University and Meredith College, Raleigh, NC
mentored by Dr. Peter Bloomfield and Dr. Jacquelin Dietz
Abstract: Since the early 1900s agencies like Standard and Poor’s and Moody’s have developed corporate rating systems to evaluate the quality of various investments. Investors have come to rely upon these ratings when purchasing corporate bonds. Given the recent volatility of the financial market, it is particularly necessary to assess whether or not an investment will be lucrative in the long run. Our team used historical credit ratings data, from 1981-2002, along with macroeconomic variables to create a predictive ratings model. Using information from S&P’s 2002 Ratings Performance report, along with unemployment and credit spread data from the Federal Reserve, our model takes into account a bond’s current rating to predict any upgrades or downgrades as it matures. An awareness of ratings transition probabilities will help investors make more informed decisions regarding an “appropriate” level of risk. We hope to use this model to explore ratings momentum and the ways in which that may complicate the use of a Markov model. We also plan to explore whether or not current ratings data will be consistent with the historical model. Monte Carlo methods will confirm the goodness of fit of our bond rating simulations. This research is based upon work supported in part by the National Science Foundation (NSF) under grant number DMS0703392.
Abstract: We investigate paths created by the zeros of the displacement of Riemann’s zeta function by a real number in the interval [0,1). A right bound and left barrier are given when the real number is equal to 1. We also give regions of the complex plane where no zeros of this function exists.
Abstract: One of the major diseases in sub-Saharan Africa, malaria causes millions of death across the continent every year. The current study explores malaria from a dynamical systems point of view, creating an agent-based simulation based on a diff erential model. The agent-based simulation, allows for investigation of the use of Insecticide Treated Bed Nets (ITNs) as a prospective preventative measure for malaria. Using mosquitoes and humans as interacting agents under set parameters, the simulation provided time-based graphs showing the infection rates of the infected human and mosquito populations. The findings showed that protecting 60% of a population with ITNs will provide almost 90% infection reduction among that population, and increasing protection to 70% will almost entirely eliminate infection.
Abstract: Argyrodes spiders are life-long kleptoparasites. Rather than building a web of their own, they live on the web of a larger host spider, stealing small food items and small quantities of web silk for nutrition. The host in our model system is Nephila clavipes, the golden web spider. Nephila build large orb webs with smaller barrier webs in front of the sticky surface. These barrier webs provide an ideal habitat for Argyrodes spiders, which congregate in those areas until a food item is available to be stolen. In order to further understand the behavior of the host spider we have built a model studying choice behavior in the system; specifically we focus on the point at which Nephila would abandon a web and rebuild in a new location to reduce kleptoparasite theft. This choice is related to food consumption which can be reduced with the presence of a large number of kleptoparasites.
Abstract: Hierarchical matrices are efficient data-sparse representations of certain densely populated matrices. The basic idea is to split a given matrix into a hierarchy of rectangular blocks and approximate each of the blocks by a low-rank matrix. Based on this structure, hierarchical matrices are often used as a numerical approximations for matrix arithmetic, inversion etc. In this presentation we demonstrate the use of hierarchical matrices on a simple motivational problem where we show how they can exploit continuity of the kernel function in integral equations.
Abstract: Cause-effect relationships are important for understanding the underlying nature of a system. Here we wish to understand how traits in the perennial herb Arabidopsis lyrata are affected by genetic variation of certain traits. We present a cause-effect network model which we use to generate simulated trait correlation data and compare the results to field data. We found that estimated correlations from the field data tend to follow the model data under hypotheses of genetic variation seen in field populations. We conclude that cause-effect networks provide a useful tool for understanding trait variation.
Abstract: The recent boom and bust of the housing market has been a focal point in the economic world. Many economists believe that the housing market is a major element in the nation’s recession. Once housing prices reach their minimum value, the number of homes being sold will increase. Eventually this will result in a climb of house prices. We strongly believe that this will jump start the overall economy and help improve the status of the United States economy. The purpose of this project was to use various models to estimate when the housing market will bottom out. Using data from the Case-Shiller Home Indices of twenty major US cities, along with data from the Federal Reserve Bank of St. Louis, autoregressive integrated moving average models were created. These models determined the month and year when the housing market as a national whole would reach bottom and were also used to create simulations. Now that the housing market appears to possibly be turning around, the simulations will be used to find the probability of the market turning downward again. One likely reason for the recent turnaround of the market is the $8,000 tax credit allotted for first time home buyers. Methods of intervention analysis will eventually be used to take the tax credit into account. This project previously focused on the nation’s housing market in its entirety, but the trends of the individual cities will be examined as well to discover any patterns that may exist. The research is based upon work supported in part by the National Science Foundation (NSF) under grant number DMS0703392.
Abstract: We encode arbitrary finite impartial combinatorial games in terms of lattice points in rational convex polyhedra. Encodings provided by these lattice games can be made particularly efficient for octal games, which we generalize to squarefree games. These additionally encompass all heap games in a natural setting, in which the Sprague-Grundy theorem for normal play manifests itself geometrically. We provide an algorithm to compute normal play strategies. The setting of lattice games naturally allows for mis\`ere play, where 0 is declared a losing position. Lattice games also allow situations where larger finite sets of positions are declared losing. Generating functions for sets of winning positions provide data structures for strategies of lattice games. We conjecture that every lattice game has a rational strategy: a rational generating function for its winning positions. Additionally, we conjecture that every lattice game has an affine stratification: a partition of its set of winning positions into a finite disjoint union of finitely generated modules for affine semigroups.
Abstract: This research involves the use of mathematical techniques to derive theoretical criteria for the outcome of military combat and counter-insurgencies. The mathematical techniques used include Dynamical Systems, Theory of Linearized Stability, and Non-Linear Differential Analysis. Generalized Mathematical models are constructed depicting various scenarios of military combat involving insurgent armies and guerrilla forces. In particular this work is more, realistic and quantitative than previous existing models and analyses. For each of the military models, explicit operational outcomes are calculated and analyzed for exploitable advantages. The research also shows mathematically the validity of some essential principles of combat derived in the past and used at present.
Abstract: For decades biologists have been using differential equation models to monitor nutrient and energy cycles in ecosystems. I believe the mathematical knowledge required for such modeling is at an undergraduate level, making it an excellent choice for interdisciplinary undergraduate research in math and biology. There are a number of sampling and parameter estimation issues that could be of interest to budding statisticians as well. I will motivate discussion with a simple example of monitoring leaf litter (biomass) in a small stream - building the model, the biological factors involved, its usefulness, and some data collection issues.
Abstract: Mathematical models can be used to predict how a disease will spread through a population. The use of a model allows for a better understanding of how specific factors will accelerate or reduce the spread of a disease. Malaria is a serious disease affecting 40% of the world's population, and killing over 1 million people each year. Creating a model of the spread of malaria through a population could aid in the control of this devastating disease. Through this research an agent-based simulation will be created that can model the spread of malaria in a population that has acquired immunity. Acquired immunity is gained through repeated exposure to malaria and can either prevent a person from contracting malaria or allow them to only have a milder form of the disease. The agent-based simulation will model the individual interaction of members of the different populations involved in the spread of malaria and keep track of the changes in the populations as a result of these interactions.
Abstract: We study the reaction-diffusion properties of the G-protein coupled receptors (GPCRs), the largest superfamily of receptors. They are the target for approximately half the world’s medicines. The GPCRs are characterized by biochemical reactions within a spatial structure. In many studies the spatial structure and thus the diffusion of molecules is ignored, with the reaction becoming the focus point. To overcome the limitation of using only a reaction-based approach, we apply the Reaction-Diffusion Master Equation (RDME) to GPCRs. The RDME incorporates not only the reaction-diffusion properties but also the discrete and stochastic nature of the GPCRs. The first part of the project focuses on the behavior of the GPCR’s output signal when four major parameters of the receptor system (diffusion, affinity of reaction, effector re-inactivation rate and the angular frequency the input signal) vary. For different values of the frequency, one can observe counterintuitive results like the output amplitude independence of affinity or resonance between the affinity and the re-inactivation rate. The second part of the projects deals with the relationship between the fluctuations of the input signal and those of the output signal. The system turns out to be robust (i.e. the input fluctuations do not have a strong influence on those of the output).
Abstract: Kleptoparasitism, the stealing of resources, has been documented throughout the animal kingdom. Within the dung beetles species Onthophagus taurus, the most commonly observed form of kleptoparasitism involves the stealing of reproductive resources; specifically, the stealing of each other's brood balls in which a female either destroys the brood ball created by another female or kills the existing egg to then replace it with her own. The purpose of this study was to test the mathematical model describing this behavior as well as to examine the factors that influence the frequency of kleptoparasitic behavior within this species. We hypothesized that factors that increased the difficulty of reproduction (i.e. increased density) as well as elements that increase the desire to reproduce (i.e. the presence of a major male) would increase the observed stealing behaviors. Indeed, a positive relationship between density as well as the presence of a major male and kleptoparasitic behavior was observed. Interestingly, egg replacement was not seen under circumstances of either increased difficulty or increased need to reproduce while egg destruction was seen with a significantly higher frequency under both conditions. This indicates that the motivation to kleptoparasitize is governed by more than relative brood ball production time, as was originally hypothesized; since engaging in these behaviors is necessarily time consuming without producing an obvious reproductive advantage.
Abstract: We are interested in looking at the conditions that affect brood ball production in the field by the dung beetle Onthophagus taurus. Conditions that could influence brood ball production include competition, both inter- and intraspecific, temperature, or the condition of the dung (being fresh or old). We looked at the production of brood balls in the field with two experiments. We followed eight pats over a 72 hr period at 12 hr intervals, so that we could document when O. taurus, entered the dung pat, and started to produce brood balls. We also looked at 19 different pats for brood ball production at three depths below the surface: 0-5cm, 5-10cm, and 10-15cm, to see where our beetles are most likely to be found and produce brood balls. The results suggest that O. taurus enter the cow pat within 12 hrs of initial creation and that the burying of dung began within 12 hrs and continued over the 72 hr period, with the highest amount of buried dung at 48 hrs. We found that the greatest amount of buried dung was found within 10 cm of the soil surface while most of our adults and brood balls were found within the first 5 cm.
Abstract:The dynamics of a slow-moving viscous fluid which coats a water-repellent surface is a simple physical system with a rich and complex mathematical structure. Using these mathematics, we can observe that, over time, stabilizing forces shape a fluid layer into an array of discrete droplets separated by an ultra thin layer of fluid. We can then analyze this system by utilizing paired ODEs which have been reduced from Reynolds’ PDE. Numeric solutions to the ODEs via MATLAB show that the droplets interact with one another to produce movement and mass exchange, occasionally giving rise to coarsening events. Such events are signified either by a droplet collapsing into the ultra thin layer or by two droplets colliding, and thus, merging. Using various methods, we aim to gain a better understanding of the dynamics of this system including the factors that influence coarsening events and persistence such as different parameters, initial conditions, and boundary conditions.
Abstract: The recent events of the stock markets have brought Nassim Nicholas Taleb’s “Black Swan Theory” to the forefront. The “Black Swan Theory” argues that the possibility or importance of rare, unpredictable events should not be ignored. With the recent economic situation of the United States becoming an important subject matter, this has led us to consider Risk Management techniques used by companies to prepare for potential losses in capital. While looking at strictly the risk measures involved with portfolio managing, the project aims to determine whether the Value-at-Risk is an adequate measure of risk or if an alternative measure such as Expected Shortfall, might prove to be a better measure of risk.
Historical stock prices provided by Yahoo Finance were used to compile different portfolios for analysis. Using the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model and Monte Carlo simulations, a Value-at-Risk and an Expected Shortfall were calculated for the stock portfolios. In addition, we will use our results to test our hypothesis concerning types of distributions used with assessing risk management and further evaluate the impacts of Value-at-Risk versus Expected Shortfall calculations.
The research is based upon work supported in part by the National Science Foundation (NSF) under grant number DMS0703392.