## Abstracts of the talks

### Undergraduate Students

**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

### Graduate Students

**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.