# 11/06/10

## Abstracts of the talks

Roshonda Barner: Fisher's Exact Test and It's Applications
Crystal Bennett: Winning strategy for Chomp Grid with 0, 1, 2, or 3 pieces in the 3rd row
Ian Duncan and Adam Eury: The effect of vector-host coupling on the dynamics of vector-borne diseases
Katherine Grzesik: A Poisson Approximation for the Number of kl-Matches
Matthew Jester, Jasmine Davidson: Detecting & Modeling Genes influenced by Natural Selection in Drosophila ananassae
James Kniffen Jr., Nicole Mack: Collapsing Building Blocks: Genotyping, Haplotyping, and Epistatic Modeling
Brandi Luongo: Some work on a problem by Marco Buratti
Rachel Marceau, Kristopher Hoover: Effectiveness of Grammatical Evolution Decision Trees in Identifying Disease Causing Polymorphisms
Stephen Meier: Analyses of the relative contributions of multiple mating, and recombination rate to intracolonial genetic diversity in honey bees
John Patterson: Estimating P-values for Randomization Test
David Schuchart: Measuring Behaviors of Peromyscus Mice from Remotely Recorded Thermal Video Using a Blob Tracking Algorithm Analysis
Keli Sikes: On the edge-balance index set of complete bipartite graphs
Wesley Stewart, Olivia Bagley, Rachael Beckner, Allison Huber: Comparison of Analytical Methods for Genetic Association Studies
Candace Swords: Fractal Basin Boundaries and Newton's Method
Peichun Wang: Bifurcation Analysis of the Tubuloglomerular Feedback System in the Loop of Henle

Jasmine Alexander-Floyd: Modeling the Effect of Fires on Red-tailed Leafhoppers
Matt Bowen: A numerical method for solving cardiac ionic models
Ricky Farr: Evaluating Derivatives of the Riemann Zeta Function using Euler-Maclaurin Summation
Meghan Fitzgerald: Cost-benefit analysis of stealing insects (i.e. a how-to make a comparison model of incomparable behaviors)
Yair Goldberg: The Derivatives of the Riemann Zeta Function, and Methods of Finding Zeros of Analytic Functions
Robert Gove: Predicting GUI Test Case Feasibility
Nels Johnson: Measurement Error in 1-1 Matched Case-Control Studies
Yi Li: Computing Stokes Flows Driven By Open Immersed Boundaries
Dani Moran: Voronoi diagrams - definition and applications
Brian Sinclair: A Survey of Group-like Binary Systems
Matthew Wilhelm: The use of confocal microscopy and rheometry to assess the validity of the semi-flexible polymer theory of collagen gels and associated image analysis challenges

### Modeling the Effect of Fires on Red-tailed Leafhoppers

Jasmine Alexander-Floyd, Drexel University, PA
mentored by
Dr. Maya Chhetri

Abstract: Experiment (Panzer, 2003) showed that the post-fire recovery of the endangered red-tailed leafhopper (Aflexia rubranura) depends on the number, quality, and proximity of neighboring unburned patches. We use a mathematical model to describe the local colonization-extinction dynamics of this species before and after the fire. We test our model’s validity using the data from the experiment as well as using simulations.

### Fisher's Exact Test and It's Applications

Roshonda Barner, NCA&T, Greensboro, NC
mentored by
Dr. Mingxiang Chen

Abstract: Fisher’s Exact Test is a statistical test to determine if there is a non-random association between two categorical sets of data. This test was developed by statistician and geneticist Ronald Fisher. This test is used in place of Chi-squared test because the chi-squared test is a test for approximating large samples while this test is for finding an exact probability for small samples. A popular example of the use of Fisher’s Exact Test is the lady tasting tea. A lady claimed that she could distinguish whether her milk was added to her cup before or after her tea. She was given 8 cups of tea, 4 of which milk was added before tea and 4 of which milk was added after her tea. The lady was asked to determine which cups where which. The guesses that the lady gave were tested against the actual results. If the lady’s guesses had a non-random association with the actual results then it would be safe to say that she could actually taste the difference. Similarly, this test can be applied to many fields of study including biology. We used test to conduct what is known as a gene ontology analysis. If two sets of genes are identified as being different it is important to look at how the attributes of the two sets of genes differ. For each attribute, we conducted a Fisher’s Exact Test to determine the probability that the attribute differs between the two sets strictly by chance.

### Winning strategy for Chomp Grid with 0, 1, 2, or 3 pieces in the 3rd row

Crystal Bennett, NCA&T, Greensboro, NC
mentored by
Dr. Kenneth Berg

Abstract: The objective of this summer REU project portion is to design and implement a winning strategy for chomp grid with up to three pieces in the 3rd row. The take-away game Chomp is played on a rectangular grid with m number of rows and n number of columns. The grid is divided into squares. Two players alternate removing pieces from the grid. The lower right hand piece is poison and the player who removes this piece loses. By expanding the standard version of Chomp of 2 rows and 5 columns to 3 rows and indefinitely many columns, we would like to investigate if we can come up with a way to find all the losing positions of a chomp grid when there are 3 rows and 0, 1, 2, or 3 pieces in the top row. Losing positions are members of the set L. All other positions are considered winning and a member of the set W. If a player can force the game to alternate from a position in W to a position in L and back to a position in W making the game go “W, L, W, L”, the other player will lose. We have developed theorems for common conditions in chomp so that the player can manipulate the game as such.

### A numerical method for solving cardiac ionic models

Matt Bowen, Duke University, Durham, NC

Abstract: In many experiments and simulations, the observation has been made that, under rapid period pacing, cardiac cells undergo a period-doubling bifurcation in which the duration of action potentials alternates between a long value and a short value. While in a single cardiac cell this bifurcation is understood, even in a one dimensional cable of cardiac cells, its exact nature is still unclear. A model introduced by Mitchell and Schaeffer exhibits this so called alternan behavior. Like most models of action potentials, this model has both "fast" and "slow" time-scales, resulting in a relatively stiff numerical problem. In this talk, I will discuss the application of scheme known as spectral deferred correction to the reaction-diffusion system in the model.

### The effect of vector-host coupling on the dynamics of vector-borne diseases

Ian Duncan and Adam Eury, UNCG, Greensboro, NC
mentored by
Dr. Gideon Wasserberg and Dr. Clifford Smyth

Abstract:The role of the host is often ignored when modeling vector-borne diseases. This project investigated the effect of the vector’s host-dependence (hereafter, vector-host coupling) on disease dynamics. Specifically, we examined how disease prevalence in host populations changes with host or vector abundance. We used an object-oriented-programming approach to simulate three vector-host coupling scenarios: uncoupled, using random movement of the vector (hypothetical), semi-coupled where vectors seek hosts only for blood-meals (e.g., mosquitoes), and totally-coupled where the vector requires contact with the host throughout its life-cycle (e.g., ticks). In all scenarios, decrease in prevalence with host abundance was observed resulting from decrease in the vector-to-host ratio. In the uncoupled scenario, these relations occurred throughout the host abundance range. In contrast, in the totally-coupled scenario prevalence first increases and later decreases. In the semi-coupled scenario, prevalence remains constant at low host abundance and then decreases. These relations result from the vector’s host-seeking behavior which increases the connectivity of the host population at low densities, which buffers the decrease in the vector-to-host ratio. Based on a preliminary literature analysis, the majority of papers addressing the effect of host abundance found a positive association, which is partially consistent with the novel predictions of our model.

### Evaluating Derivatives of the Riemann Zeta Function using Euler-Maclaurin Summation

Ricky Farr, UNCG, Greensboro, NC

Abstract: A method of evaluating the derivatives of the Riemann Zeta Function will be discussed based on Euler-Maclaurin summation. This method enables evaluation at all complex numbers, where it is defined, particularly in the left half plane. Real world implementation issues will be presented and results given. Root finding of the derivatives of the Riemann Zeta function will also be discussed.

### Cost-benefit analysis of stealing insects (i.e. a how-to make a comparison model of incomparable behaviors)

Meghan Fitzgerald, University of Wisconsin Madison, WI
mentored by
Dr. Jan Rychtář

Abstract: Kleptoparasitism, the stealing of prepared resources, is a common behavior in many species of insects. For instance, a female dung beetle may steal a brood ball which was carefully prepared by another and place their own egg inside, thrips steal housing and food catches from other species of thrips, and an obligate kleptoparasitic spider lives inconspicuously on a host spider's web watching for unnoticed prey in an attempt to avoid building a web of their own. These systems are often analysed theoretically, but it can be difficult to bring this theory together with empirical data. I will review possible methods for cost-benefit behavioral analysis, using a predator-prey system and a symbiotic mutualism for demonstration. I will also discuss potential theoretic and empirical uses for these models in kleptoparasitic systems.

### The Derivatives of the Riemann Zeta Function, and Methods of Finding Zeros of Analytic Functions

Yair Goldberg, UNCG, NC
mentored by
Dr. Sebastian Pauli and Dr. Filip Saidak

Abstract: The Riemann zeta function and its zeros are at the center of the 150-year-old question of the Riemann Hypothesis, and as such, has been the subject of numerous (though finitely many) articles, papers, and books. Less well studied are its derivatives, and their zeros. Unlike the zeros of the zeta function itself, the derivatives' zeros are distributed seemingly randomly (within bounds determined by the individual derivative functions). We will focus on some strategies for finding zeros of analytic functions, with a focus on methods that seem particularly well suited to these functions.

### Predicting GUI Test Case Feasibility

Robert Gove, The University of Maryland at College Park, College Park, MD
mentored by
Dr. Atif Memon and Dr. Lise Getoor

Abstract: One approach to assess software quality is to use a model of the graphical user interface (GUI) to automatically generate test cases, which are sequences of events, which may detect faults in the software. However, a test case may be infeasible if one or more of the events in the sequence are disabled or made inaccessible by a previously executed event (e.g. a button is grayed out). Existing techniques focus on repairing sequences to make them feasible, but we propose avoiding infeasible sequences altogether. We will discuss our approach using machine learning to predict which test cases are infeasible and remove them from the test suite.

### A Poisson Approximation for the Number of kl-Matches

Katherine Grzesik, SUNY Oswego, NY
mentored by Anant Godbole

Abstract: Consider a lecture class with a population $N$. Suppose a student keeps track of the order of students called upon to answer a question. Each student on the roster has $l$ friends before his/her name and $l$ friends after his/her name. A kl-match occurs when two students, who are in each other's list of $2l$ friends or are themselves, are called upon within the $k$ previous questions. Let $X_n$ denote the number of {\it kl-matches}. The definition of the random variable $X_n$ assumes that each student has a full window of $2l+1$ friends and a full window of $k$ previous questions. This scenario is built off of Burkhardt, Godbole, and Prengman's (1994) paper about the distribution of $k$-matches. The distribution of $X_n$, in an equiprobable case, is approximated by a Poisson random variable if $lk^2 =$ o$(N)$. In the non-equiprobable case, the distribution is also approximately Poisson. A coupling could decrease the amount of total distance variation incurred in the Poisson approximation.

### Detecting & Modeling Genes influenced by Natural Selection in Drosophila ananassae

Matthew Jester and Jasmine Davidson, UNCG, NC
mentored by Dr. Malcolm Schug and
Dr. Roland Deutsch

Abstract: Evolutionary biologists are interested in detecting genes that are targets of natural selection in the genome of organisms in natural populations. Many factors affect the ability to detect natural selection including, genetic drift, migration, mutation, and recombination. We are currently developing a mathematical model using a coalescent approach to identify genes that have been targets of natural selection. The coalescence model predicts the time at which two or more loci have a common ancestor. This model will give us a genealogy of a certain number of alleles at each locus that we will use to generate a neutral genealogy. We will compare data from natural populations with our neutral model to test the hypothesis that natural selection has affected a specific region of the genome. To test the strength of the model and determine appropriate parameter values for genetic drift, natural selection, migration, and mutation, we will use molecular markers called microsatellites distributed across the genomic region around the furrowed gene in D. ananassae which was previously shown to be a target of natural selection. We will present the genome markers and parameters of the model.

### Measurement Error in 1-1 Matched Case-Control Studies

Nels Johnson, Virginia Polytechnic Institute and State University, Blacksburg, VA

Abstract: In epidemiological research, matched case-control studies are popular. Measurement error of covariates is a modeling problem that often needs to be addressed. However, there is no statistical approach to handle both problems together. We propose a parametric Bayesian method for addressing measurement error models in matched case-control studies. We demonstrate our approach on a 1-1 matched case-crossover study in public health epidemiology.

### Some work on a problem by Marco Buratti

Brandi Luongo, Clayton State University, Morrow, GA
mentored by
Dr. Elliot Krop

Abstract: Marco Buratti’s conjecture states that if p is a prime and L a multiset containing p-1 non-zero elements from the integers modulo p, then there exists a Hamiltonian path in the complete graph of order p with edge lengths in L. Say that a multiset satisfying the above conjecture is realizable. We show a recursive construction that produces larger realizable sets from realizable sets.

### Collapsing Building Blocks: Genotyping, Haplotyping, and Epistatic Modeling

James Kniffen Jr. and Nicole Mack, NC State University, Raleigh, NC
mentored by
Dr. Alison Motsinger-Reif

Abstract: All genetic association analysis relies on a specific feature of the human genome called Linkage Disequilibrium (LD), which describes the nonrandom assortment of alleles (variants) across the genome. LD is seen through high levels of physical and statistical correlation, meaning much of a person’s genomic data is redundant. This redundancy allows large sets of single-variant genetic data to be collapsed into haplotype blocks, which identify all correlated variant sites in a particular genomic region. Using haplotypes instead of single-variant genetic data can increase the power of gene-mapping studies since data becomes denser and the number of tests performed is reduced. By using haplotypes in the association studies, the genomic regions that might be disease-related can be identified before looking for more complex models within those specific regions. Combining haplotypes in association studies with data-mining methods, such as Multifactor Dimensionality Reduction, can also detect complex predictive models in the genome. We will investigate this collapsing approach, with real and simulated data, for its performance and power to detect complex predictive models. We will use data simulation and analysis tools for empirical comparisons and real data analysis, implemented in the R language and Unix/Linux applications in a supercomputing environment. The research is based upon work supported by the National Science Foundation under CSUMS grant # DMS-0703392 (PI: Sujit Ghosh).

### Computing Stokes Flows Driven By Open Immersed Boundaries

Yi Li, Duke University, Durham, NC
mentored by
Dr. Anita Layton

Abstract: We present numerical methods for computing two-dimensional Stokes flow driven by forces singularly supported along open, immersed boundaries. Two second-order accurate methods are developed: one for accurately evaluating boundary integral solutions at a point, and another for computing Stokes solution values on a rectangular mesh. We first describe a method for computing singular or nearly singular integrals, such as a double layer potential due to sources on a curve in the plane, evaluated at a point on or near the curve. To improve accuracy of the numerical quadrature, we add corrections for the errors arising from discretization, which are found by asymptotic analysis. When used to solve the Stokes equations with sources on an open, immersed boundary, the method generates second-order approximations, for both the pressure and the velocity, and preserves the jumps in the solutions and their derivatives across the boundary. We then combine the method with a mesh-based solver to yield a hybrid method for computing Stokes solutions at N^2 grid points on a rectangular grid. Numerical results are presented which exhibit second-order accuracy.

### Effectiveness of Grammatical Evolution Decision Trees in Identifying Disease Causing Polymorphisms

Rachel Marceau and Kristopher Hoover, NC State University, Raleigh, NC
mentored by
Dr. Alison Motsinger-Reif and Dr. David Reif

Abstract: The ability to identify polymorphisms that predict complex diseases is essential to the field of genetics. Complex diseases are thought to be due to a myriad of factors including environmental exposures and complex genetic risk models. These epistatic models create an analytical challenge, requiring methods to perform both statistical estimation and variable selection to compare genetic model hypotheses. One such new method to detect these models is Grammatical Evolution of Decision Trees (GEDT). Decision trees are easily interpretable, but their power is limited in identifying highly complex models due to their hierarchical model building approach. Grammatical evolution, a type of evolutionary computation, is used to avoid this problem by mimicking natural selection and evolving decision trees to detect and model gene-gene interactions. Only the best fitting models are allowed to “breed” and produce new models. GEDT has shown initial success, but is still in its infancy. Currently, we are evaluating and optimizing different parameter settings such as the number of generations the method runs, the mutation rate used, etc. to improve the performance of GEDT. We want to specifically look at how these parameters impact the proportion of misclassified models. After analyzing the parameters we will examine GEDT’s performance compared to other competing methodologies. We hope to optimize GEDT and show how successful this easily interpretable model can be at identifying gene-gene interactions.
This material is based upon work supported by the National Science Foundation under the NSF-CSUMS project DMS-0703392 (PI: Sujit Ghosh).

### Analyses of the relative contributions of multiple mating, and recombination rate to intracolonial genetic diversity in honey bees

Stephen Meier, UNCG, Greensboro, NC
mentored by
Dr. Olav Rueppell and Dr. Roland Deutsch

Abstract: Genetic diversity is important in the Western Honey Bee (Apis mellifera). High levels of genetic diversity enhance disease resistance, division of labor, and decrease the risk of diploid drone production at the colony level. This project applies statistical simulations to evaluate the relative contributions of multiple mating and recombination rate to colony genetic diversity in Apis mellifera. Quantifying the genotypic variance among workers allows the re-evaluation of these two variables, assuming different distributions of paternity, loci number, individual loci effect sizes, chromosome length, and colony size. The simulation results will be compared to empirical findings from the literature, leading to conclusions about the evolutionary explanations of the excessive mating frequency and recombination rate of honey bee queens.

### Voronoi diagrams - definition and applications

Dani Moran, UNCG, Greensboro, NC
mentored by
Dr. Greg Bell

Abstract: The Voronoi diagram is a fundamental data structure in computational geometry. A simple model of the concept is the 'post office problem' - that is, how should a fixed set of n 'post offices' be processed in order to quickly determine the 'post office' closest to an arbitrary address. We will be discussing the more precise mathematical definition of the Voronoi diagram and some of its associated structures, such as the Delauney triangulation and the power diagram. We will also be discussing some of the applications in computer science, the natural sciences and other areas of mathematics.

### Estimating P-values for Randomization Test

John Patterson, UNCG, Greensboro, NC
mentored by
Dr. Scott Richter

Abstract: Randomization test can be a good way to estimate parameters of a population. When doing a Randomization test on a particular data set to estimate a particular parameter, a p-value is calculated. A p-value represents the probability of an occurrence equal to or greater than the observed occurrence, given that the null hypothesis is true). Many times samples collected are too large to obtain an exact p-value. In cases such as this, an estimated p-value is computed based on a sample of the possible occurrences. Once the estimate is computed, a confidence interval is constructed around the data to give the estimate a certain amount of error in which the true p-value may reside. This investigation is to test two methods of computing confidence intervals around the estimated p-value of a randomization test when the p-value is small: the Wald interval and the score interval. When the p-value is small and close to the decision rule, it is important that the confidence interval constructed around the estimate has the level of confidence prescribed. It was found that the Wald interval did not reach the prescribed level of confidence for more than half the cases examined. Not only did the Wald interval not reach the prescribed level of confidence, it also tended to underestimate the true p-value. The score interval reached the level of confidence prescribed for all except one extreme case, and did not show bias. These findings suggest that the score interval should be used when estimating p-values for a randomization test.

### Measuring Behaviors of Peromyscus Mice from Remotely Recorded Thermal Video Using a Blob Tracking Algorithm Analysis

David Schuchart, UNCG, Greensboro, NC
mentored by
Dr. Matina Kalcounis-Rueppell, Dr. Sebastian Pauli, Dr. Shanmugathasan Suthaharan

Abstract: Measuring behaviors of free-living, wild animals is difficult because the presence of an observer can impact the behaviors being measured. Additionally, measuring behavior of nocturnal animals is difficult because traditional methods of recording behaviors such as filming or observing from behind a blind are not possible in the dark. One solution that mitigates both of these difficulties is to use remote thermal (IR) video to record behaviors of free-living animals. Remote thermal video recording eliminates observer bias and allows for the study of animals in the dark. However, remote thermal video recording introduces a new problem in that it can generate massive amounts of image data, especially if recording is done continuously in real time, which takes a significant time investment to process by hand. We show that using modern video and signal processing techniques, it is possible to measure behaviors directly from video data in an automated way. The species we used in our analysis were two species of Peromyscus mice, P. californicus and P. boylii. Over 131 nights in 2008 and 2009, continuous thermal video of free-living wild mice was recorded as part of a study to examine the behavioral context of ultrasound production by these two species. We analyzed 265 videos totaling 172 gigabytes to quantify behaviors associated with locomotion. As our first variable, we examined running speed. We used a C++ library called OpenCV to apply Gaussian and median filters in order to isolate the mice from the background of the video. Once the mice were isolated, we used a blob-tracking library cvBlob, which recognizes "blobs" of similar pixels as foreground, to write the location of the mice to a data file. The data file containing the location information was analyzed against time in order to generate speed data. We examined running speed in relation to species and abiotic conditions. We found the average speed of P. californicus was 1.465 meters per second and the average speed of P. boylii was 1.710 meters per second. There was not a significant difference between the speeds of the two species. Additionally, we found that there was not a significant difference in speed between the sexes of each species. Ours is the first study to quantify running speed of free living nocturnal rodents and relate these speeds to biotic and abiotic factors. Further, our study provides a method for measuring other behaviors recorded remotely from thermal video recording. Our study demonstrates that is possible to remotely record and measure behaviors from free-living mammals using blob tracking algorithms in conjunction with other modern video processing techniques.

### On the edge-balance index set of complete bipartite graphs

Keli Sikes, Clayton State University, Morrow, GA
mentored by
Dr. Elliot Krop

Abstract: Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $Z_2=\{0, 1\}$. For $i\in Z_2$, let $e_f (i) =$ \textit{card}$\{e\in E(G) : f (e) = i\}$. A labeling $f$ of a graph $G$ is said to be \emph{edge-friendly} if $| e_f(0)-e_f(1) |=1$. An edge-friendly labeling $f$ : $E(G)\rightarrow Z_2$ induces a partial vertex labeling $f^+ : V(G)\rightarrow A$ defined by $f^+(x) = 0$ if the number of edges, incident on $x$ and labeled $0$, is more than the number of edges incident on $x$ with label $1$. Similarly, $f^+(x) = 1$ if the number of edges incident on $x$ and labeled $1$ is more than the number of edges incident on $x$ and labeled $0$. $f^+(x)$ is not defined if the number of edges incident on $x$ with label $1$ is equal to the number of edges incident on $x$ labeled by $0$. For $i\in Z_2$, let $v_f(i) =$ \textit{card}$\{v\in V(G) : f^+(v) = i\}$. The \emph{edge-balance index set} of the graph $G$, $EBI(G)$, is defined as $\{|v_f(0) - v_f(1)| :$ the edge labeling $f$ is edge-friendly$\}$. We continue the work by Kong, Lee, and Wang "On the edge-balanced index sets of some complete k-partite graphs, and determine the $EBU(K_{n,n-2})$.

### A Survey of Group-like Binary Systems

Brian Sinclair, UNCG, NC

Abstract: Groups are often presented as the fundamental structure of abstract algebra. However, we do encounter examples lacking of some necessary group property, be it a general ring under its multipication, the octonians, or a novel algebraic model for an application. We will define and see examples of group-like binary systems (magmas, semigrouops, monoids, quasigroups, loops) and explore basic properties and pitfalls inherient in these non-intuitive systems.

### Comparison of Analytical Methods for Genetic Association Studies

Wesley Stewart, Olivia Bagley, Rachael Beckner, Allison Huber, NC State University, Raleigh, NC
mentored by
Dr. Alison Motsinger-Reif and Dr. Sujit Gosh

Abstract: Detecting genetic variants that predict common, complex disease is a major goal of human genetics, but is a difficult challenge due to complex underlying etiologies. There are several potential models for the genetic etiology of complex traits such as heterogeneity, epistasis, and sources of noise in the data set that present challenges for statistical association methods. The effect of these different factors is largely unknown for data-mining approaches. The impact of these types of noise needs to be understood to properly apply and modify data mining approaches in human genetics. Multiple data simulations will be used to compare some of the commonly used novel, computer-intensive, and traditional association analysis approaches such as Multifactor Dimensionality Reduction, logistic regression, LASSO and Random Forests. The results of this study will be helpful to guide the analysis of real data. Freely available data simulation and analysis tools will be used for comparison and data analysis, using R language and Unix/Linux applications. The super computing cluster at NCSU will be used through the High Performance Computing (HPC) center to aid in the computationally intensive comparisons. This material is based upon work supported by the National Science Foundation under grant number NSF-CSUMS project DMS-0703392 (PI: Sujit Ghosh).

### Fractal Basin Boundaries and Newton's Method

Candance Swords, Clayton State University, Morrow, GA
mentored by
Dr. Chris Raridan

Abstract: Each of the three roots of $z^3-1=0$ has a basin of attraction in the complex plane defined by those $z_0$ for which the iterations of Newton's Method with initial guess $z_0$ converge to a root. The boundary of these basins of attraction is a fractal and corresponds to a Julia set, the structure of which can be described in detail by binary address. In this presentation, we provide an introduction into the study of fractals with focus on the dynamics of the Newton iterations for points in the Julia set of the cubic polynomial $f(z)=z^3-1$.

### Bifurcation Analysis of the Tubuloglomerular Feedback System in the Loop of Henle

Peichun Wang, Duke University, Durham, NC
mentored by
Dr. Anita Layton

Abstract: The tubuloglomerular feedback (TGF) system in the loop of Henle in the kidney mediates oscillations in tubular fluid pressure, flow rate, and NaCl concentration in the loop of Henle through a negative feedback system with certain time delay and feedback gain. In this study, extending previous research on the thick ascending limb (TAL), we developed a mathematical model of TGF system in the whole loop of Henle, which consists of both the thick ascending limb and the thick descending limb (TDL), with compliant walls. Numerical simulations were run for the model with different parameters (time delay and feedback gain) to investigate behaviors of the TGF system. Analytical solutions of the model equations were also obtained by deriving and solving the characteristic equations with linear approximation. And finally a bifurcation analysis will be done to display different behaviors on the whole parameter region.

### The use of confocal microscopy and rheometry to assess the validity of the semi-flexible polymer theory of collagen gels and associated image analysis challenges

Matthew Wilhelm, Columbia University, New York, NY
mentored by
Dr. Laura Kaufman

Abstract: Type I collagen is the most prevalent structural protein in mammalian tissues. It heavily influences both the mechanical and structural properties of the extracellular matrix. As such, there is considerable interest in making use of it in scaffolds for tissue engineering. Semi-flexible polymer theory potentially allows for us to understand the coupling between the network structural parameters and mechanical properties, thus guiding design of the scaffolds. However, the validity of the predictions made by semi-flexible polymer theory has not yet been fully established. We use a combination of confocal reflectance microscopy (CRM) and time sweep rheometry to probe the structure/mechanical property relationship in collagen gels. Characteristic mesh sizes and total fibril lengths are obtained from CRM images via contour and region-based algorithms, respectively. We tentatively find that both the Morse and MacKintosh models provide reasonable predications of dependence of elasticity on the network properties of collagen gels at 37C. While the MacKintosh model appears to be valid for gels prepared at lower temperatures as well. However, the analysis of the CRM images required to reach this conclusion is certainly nontrivial. The uniquely challenging features of the associated image analysis and possible future directions will be discussed.