32nd Annual Mathematics Symposium - Western Kentucky University
October 12 - October 13, 2012
*= student presentation
st
Friday 4:30pm, Registration and refreshments begin, 1 floor of Snell Hall (SN)
Friday 4:50 pm, ROOM 1108 SN
Welcome by Associate Dean and Interim Head of the Mathematics Department: Bruce Kessler
Friday 5:00-6:00 pm, ROOM 1108 (SN)
INVITED TALK
Carlos Castillo-Chavez
(Arizona State University & MIT)
The role of mobility, transportation systems and adaptive behavior on
dynamics: from homeland security to disease
The speed and quality of public information play a critical role in the selection of policies aimed at mitigating
the impact of disease dynamics through the deliberate disruption of the cohesiveness of social networks with
the help of controlled measures that include "local social distancing," travel restrictions, closing of schools
and public places, and more. Selected challenges posed by the dynamics of emergent and re-emergent
diseases will be formulated with the help of mathematical models. We will first discuss the pioneering
modeling work that physicians-theoreticians Ross, Kermack and McKendrick carried nearly a century ago.
The lecture will conclude with the analyses of epidemics that include the most recent H1N1 pandemic
followed by a discussion of the challenges posed to homeland security with emphasis on the deliberate
release of biological agents in public venues.
PARALLEL SESSIONS, Friday
Friday 6:00 - 6:20 pm
ROOM 1101 • Melanie Autin, Ngoc Nguyen (WKU)
The Applied Statistics Center at WKU
Many researchers and companies need to make data-driven decisions, using statistically valid analyses of
their data. Currently there is a lack of on-campus statistical support for university researchers. The new
Applied Statistics Center at WKU aims to provide student-led, faculty-mentored statistical consulting. The
Applied Statistics Center will provide hands-on experience for students studying applied statistics by allowing
them to work on real projects and to participate in the different stages of research and statistical consulting.
Students will also develop marketable skills, all while being paid. We will present information for those
interested in working in the Applied Statistics Center and for those interested in utilizing its services.
ROOM 1102 • Bruce Kessler (WKU)
Modeling the Mathinator
The “Mathinator”, a Nerf dart gun mounted on a rig where I can measure, adjust, and lock the firing angle,
has been a teaching “toy” that I have used for a number of years, both in my trigonometry class and in talks
for area math and physics classes. Lately, in an effort to make the school talk more engaging and
accessible to a younger crowd, I have changed my presentation to include data collection and modeling in
order to determine the velocity of the launcher. However, on my first attempt at this new presentation, wind
resistance, which I had previously been able to ignore, snuck back into my calculations, with a vengeance.
This talk will provide some background on the trajectory model, where I went wrong in my last attempt, and
the mathematical “fix” for my presentation.
ROOM 1103 • Mustafa Atici (WKU)
How a simple problem can be turned into a very difficult one?
Let a set [n]= {1,2,...,n} be given. Finding a subset
with minimum cardinality such that, for
any two distinct elements
, there exists disjoint subsets
such that
is called the extremal set problem. Such a set
is very easy to solve. On the other hand if
is called a separating set of [n]. if
, then problem
, then the problem is very difficult to solve.
Friday 6:30 - 6:50 pm
ROOM 1101 • Claus Ernst (WKU)
On the curvature of a random walk in confinement
Assume that I am prisoner in a confinement sphere of radius R. To kill time I walking around where each of
my steps is one unit long. In my walk I am choosing a random direction for each step, that is I walk like a
drunk! What is the average turning angle for each step ?
ROOM 1102 • Veronica Bunn * (University of Kentucky)
Chaotic Dynamics of an Epidemic Model with Periodicity
We analyze the bifurcation behavior of an SIS model with 2-periodic constant demography. A traditional nonperiodic SIS model does not result in period doubling bifurcations; however, when adding periodicity, the
model undergoes a period doubling route to chaos. We numerically detect the first period doubling
bifurcation as a function of one and multiple parameters.We simplify the original transcendental equation by
using the Taylor approximation of the transmission rate. The total population is globally attracted to a 2-cycle,
so we must use the 2-fold composition of the infected class equation to encompass both population values.
By using the Period Doubling Bifurcation Theorem, we are able to analytically find parameter values that give
rise to these bifurcations. The two major conditions in this theorem reduce to two cubic equations in I, the
infected class, and the 5 model parameters. Using conditions imposed on the parameters in our model
together with one other reasonable condition on the parameters, we establish that each of the cubic
equations has one real root. Equating these real roots gives an equation in terms of our parameters that,
when satisfied, results in a period doubling bifurcation.
ROOM 1103 • Robert Cass*, Benjamin Braun (University of Kentucky)
Spherical Geometry and Linear Fractional Transformations
This expository talk will introduce spherical geometry and reflections in great circles. A stereographic
projection of the Riemann sphere will then be used to study these reflections in the extended complex plane.
This analysis will motivate one definition of a Möbius transformation. Finally, all Möbius transformations will
be described algebraically using linear fractional transformations.
Friday 7:00-7:40pm Food and Refreshments!
Friday 7:40 - 8:00 pm
ROOM 1101 • Daulton Cockerell* (Gatton Academy), Uta Ziegler (WKU)
Knot Theory: An Introduction to 2D Embedding of a 3D Polygon (Work in Progress)
The DNA packaging process in virus capsules is rather intriguing. When researchers open capsules of
mutated P4 viruses to examine the DNA, it is often knotted. In this research project, virus DNA is modeled by
a 3D polygon in space. The goal of this project is to collect data about the knotting properties of these
polygons. Before examining the knotting, however, it is often helpful to simplify the polygon. This talk
addresses this simplification step. More specifically, the talk explains the algorithm which was developed by
the presenter to modify a given 3D polygon into an abstract 2D embedding describing the knotted polygon.
This talk describes work in progress.
ROOM 1102 • Ryan Anderson*,Yulia Babenko (Kennesaw State University)
Simultaneous Approximation of a function and its derivative by linear splines
Linear splines, in particular interpolating splines, are used to approximate a function given on a discrete set
of values of the function. Linear splines are widely used in many applications targeting geometric modeling of
curves and surfaces as piecewise linear functions are generally easy to work with. The concept of linear
splines have been extended to bilinear (linear in each variable) and further to polylinear splines with many
results having been proved. In this talk, I will introduce the concept of spline interpolation and discuss new
results on simultaneous approximation of a multivariate function (of certain smoothness) and its derivatives
by linear splines as well as present some results on the error of approximation.
ROOM 1103 • Tyler Ghee*, Dominic Lanphier (WKU)
The Grey Area of Rook Polynomial
We introduce and generalize rook polynomials. These polynomials are used to count the number of ways
rooks can be placed on a various chessboards so that none attack any of the others. We will consider a
variation of rook polynomials, by considering the cases where at least some of the rooks are under attack.
Friday 8:10 - 8:30 pm
ROOM 1101 • Sam Saarinen* (Gatton Academy), Claus Ernst (WKU)
Establishing Geometric Properties of a New Planar Traveling Salesman Route Heuristic
Planar geometry is applied to studying solutions of the NP-Hard Euclidean Traveling Salesman Problem. To
start, basic properties of solutions to the Euclidean Traveling Salesman Problem are established. Then a
new polynomial-time heuristic algorithm is developed, and the solutions it produces are analyzed for the
geometric properties of curvature, enclosed area, and length of the route. For each observed trend, a
heuristic argument is given that lends support to the observation. An upper bound on the length of any
constructed planar route is derived, and used to support an empirically derived formula for the upper bound
on the length of outputs of the new heuristic. The use of this formula as a pruning heuristic in a branch-andbound algorithm is discussed, which may be easier to implement than other optimal search algorithms.
ROOM 1102 • Chris Donovan* (Clayton State University)
Numerical Experiments in Determining Animal Coat Pattern Formation
In 1952, Alan Turing proposed a model for the formation of animal coat patterns using the reaction-diffusion
equation. Since his discovery there has been great leaps in the ability to solve such large systems
computationally. In this presentation I will develop the foundation for solving partial differential equations,
such as the reaction-diffusion equation, using finite difference methods. I will further introduce the more
sophisticated Peaceman-Rachford finite difference scheme needed to solve the necessary system more
efficiently, and show how it can be used to generate animal coat patterns.
ROOM 1103 • Donna Daulton*, Richard Schugart (WKU)
Optimal control of oxygen therapy for treatment of a bacterial infection in wound healing
A mathematical model was formulated to use optimal control theory for the analysis of treatment for a
bacterial infection in a wound using oxygen therapy. An optimal control example will be given with both its
analytical and numerical solutions. Preliminary findings, with analysis, will also be presented in this talk.
____________________________________________________________________________________
Saturday from 8:00 am - Registration and refreshments, First Floor of Snell Hall (SN)
REGISTRATION continues until 11:00am SATURDAY
PARALLEL SESSIONS, Saturday
Saturday 8:30 - 8:50 am
ROOM 1101 • Ayman Alzaatreh, Carl Lee and Felix Famoye (Austin Peay State University)
A new method for generating families of continuous distributions
In this talk, a new method is proposed for generating families of continuous distributions. A random variable
X, “the transformer”, is used to transform another random variable T, “the transformed”. The resulting family,
the T-X family of distributions, has a connection with the hazard functions and each generated distribution is
considered as a weighted hazard function of the random variable X. Many new distributions, which are
members of the family, are presented. Several known continuous distributions are found to be special cases
of the new distributions.
ROOM 1102• Aaron Young* (WKU)
Ultimate Insanity
Instant Insanity 2 is a variation of the poplar, classic 1967 puzzle, Instant Insanity, in which there are five
different colors instead of four and the tiles are on a cylinder. A graph theoretical approach is used to find the
number of solutions to the Ultimate level of Instant Insanity 2.
ROOM 1103• Zachary Bessinger*, Rong Yang (WKU)
Comparative Study of Common Community Detection Algorithms
Network science has been a popular growing area of research in the last couple of decades. Networks can
be found almost everywhere in the form of biological, information, and social networks which can be
represented in the form of graphs. In these networks, vertices can be grouped into communities by having
many relationships between them. Finding the precise members of these communities is a non-trivial task.
However there are algorithms for community detection that rely on a metric called modularity to determine
how good a community partition is. In this study, we analyze and compare three popular community
detection algorithms based on their runtimes, modularity scores and applicability of the algorithms on data
sets of different sizes. The results from conducting tests on several popular graph data sets will be
presented.
ROOM 1108 • Dominic Lanphier (WKU)
The Expected Order of an Element of a Group and Apery's Constant
We study the expected size of the subgroup generated by a randomly chosen element of a finite group. We
determine certain properties of these numbers and how large such a subgroup should be, on average, for
cyclic groups. The methods that we use are number theoretic.
Saturday 9:00 - 9:20 am
ROOM 1101 • Indranil Ghosh, Ayman Alzaatreh (Austin Peay State University)
A study of the Gamma-Pareto (IV) distribution and its applications
Pareto distribution and their close relatives and generalizations provide very flexible families
of heavy-tailed distributions which may be used to model income distributions as well as a wide
variety of other social and economic distributions. On the other hand, gamma distribution
has a wide application in various social and economic spheres such as survival analysis, to
model aggregate insurance claims and the amount of rainfall accumulated in a reservoir etc.
Combining the above two heavy tail distributions, using the technique by Alzaatreh, et al. (2012
a), we define a new distribution, namely, gamma-Pareto (IV) distribution, hereafter called as
GPD(IV ) distribution. Various properties of the GPD(IV ) are investigated such as limiting
behavior, moments, mode and Shannon entropy . Also some characterizations of the GPD(IV )
distribution are mentioned in this paper. Maximum likelihood method is proposed for estimating
the model parameters. For illustrative purposes, real data sets are considered as an application
of the GPD(IV ) distribution.
ROOM 1102 • Fang Wu*, Ferhan Atici (WKU)
Existence Results in Discrete Fractional Calculus
We first introduce discrete nabla fractional calculus. Then we consider up to first order nabla fractional
difference equations with initial conditions. Such problems are called initial value problems (IVP). We use
some fixed point theorems to prove the existence of solutions for the discrete fractional IVP. By defining
upper and lower solutions for nonlinear fractional difference equations, we prove the existence of solutions of
the IVP between upper and lower solutions. Some examples will be given to illustrate our results.
ROOM 1103 • Rachel Aldrich* (Otterbein University), Sarah Drummond (Eastern Illinois University)
The Varieties of One-Sided Loops of Bol-Moufang Type
In this presentation I will extend the work of J.D. Phillips and P. Vojtechovsky, who found the relationships
among 26 different varieties of quasigroups and 14 different varieties of loops of Bol-Moufang type. I will
show that there are 20 varieties of one-sided loops of Bol-Moufang type, and also which of these varieties
have the left and right inverse properties of quasigroups. All proofs and counterexamples have been aided
by the automated theorem prover, Prover9, and the finite model builder, Mace4.
ROOM 1108 • Andrew Ledoan (University of Tennessee at Chattanooga)
On the difference between consecutive prime numbers
In 1976, Gallagher proved that the Hardy-Littlewood prime k-tuple conjecture implies that, for the primes up
to x, the number of primes in the interval (x, x + λ log x] follows a Poisson distribution with mean λ, where λ is
any fixed positive constant. Using inclusion-exclusion, Professor Daniel. A. Goldston and I recently proved
that the number of consecutive primes with difference λ log x has the Poisson distribution superimposed on
the conjectured asymptotic formula for pairs of primes with this difference. In this talk, I will present an
extension of Gallagher's theorem and more precise asymptotic formulas if λ goes to 0 as x approaches
infinity. In order to establish these asymptotic formulas, we also proved new singular series average results.
Saturday 9:30 - 9:50 am
ROOM 1101 • Kristen Knight*, Ayman Alzaatreh (Austin Peay State University)
On the gamma-half normal distribution and its applications
We propose and study a new statistical distribution, namely, the gamma-half normal distribution. Various
structural properties of the gamma-half normal distribution are derived. The shape of the distribution may be
both unimodal and bimodal, as well as reverse-J shaped. Results for moments, limit behavior, mean
deviation, Shannon entropy, Fisher information matrix and variance covariance matrix are provided. The
method of maximum likelihood estimation is proposed to estimate the model parameters. Three real data
sets are used to illustrate the applications of the gamma-half normal distribution.
ROOM 1102 • Rasitha Jayasekare *, Ryan Gill, Kiseop Lee (University of Louisville)
Application of Finite Mixture Models involving Poisson distribution
An application of a mixture model involving two Poisson distributions for stock prices is discussed. The
parameters are estimated using the Expectation – Maximization (EM) algorithm with a constant mixing
probability as well as mixing probabilities which depend on order size. Five years of stock data is used to
illustrate the method.
ROOM 1103 • Heather Hunt *, Allison Perkins, Prasanna Sahoo (University of Louisville)
A functional equation on groups
Let G be any arbitrary group and F a field of characteristic not equal to two. Using the general solutions of
the functional equations f(xy) + f(yx) = 2f(x) + 2f(y) and f(xy)+f(yx) = 2f(x)f(y), we present all functions f, g : G
F that satisfy the functional equation f(xy) + f(yx) = 2g(x) + 2g(y) + 2λg(x)g(y) for all x, y ∈ G.
ROOM 1108 • Mark Robinson (WKU)
Discrete Least Squares Approximation Using Models of Various Types
Discrete least squares approximation is known best through its application with polynomial models, in
particular the least squares linear model or "line of best fit." In many cases, however, polynomial models are
not appropriate for the given data. The utilization of various types of models (including linear, polynomial,
and exponential) is considered, as well as the comparison of models.
Saturday 10:00 - 10:20 am
ROOM 1101 • Allison Perkins*, Prasanna Sahoo (University of Louisville)
On a sine functional equation with involution on groups
Let G be a 2-divisible group and ℂ be the field of complex numbers. We present all functions
that satisfy the functional equation
is an involution.
. For all x,y in G, where
ROOM 1102 • Lucas Hoots* (University of Louisville)
A simple extension of quota systems
Taylor, Young, and Zwicker generalized simple majority rule for two alternatives to a quota system. We
extend this further by considering the n alternative case for n ≥ 3.
ROOM 1103 • Katey Bjurstrom* (University of Louisville)
A generalization of Arrow’s Theorem
Let be a finite set of at least four candidates, and let
be the number of voters. Any mapping of the
form
where
is the set of all linear orders on and
is the set of all complete
binary relations on is called a collective choice function. Arrow's classic theorem shows that any collective
choice function satisfying independence of irrelevant alternatives (IIA) and Pareto (P), where the range is a
subset of weak orders, is based on a dictator. We will show how this result can be generalized in such a way
that the collective choice function is instead based on a weak dictator.
Saturday 10:30-11:00 am, Refreshments
Saturday 11:00 - 12:00 pm, ROOM 1108 (SN)
INVITED TALK
John Ringland
(State University of New York at Buffalo)
Nurturing with ulterior motives: the use of refuges in agricultural pest control
When nonlinear dynamical processes occur in a heterogeneous environment, complexity and surprises are
often in store. A case in point is in the reproduction and survival of an agricultural pest species in a habitat
consisting mainly of crops that have been made strongly pesticidal by genetic engineering, but partly of
benign "refuges" for the pests. I will show how nurturing pests can actually be an effective means of
controlling them, but that there are balancing acts involved. If we get it right, the greatly feared development
of pesticide resistance in the pest population can be not only delayed but prevented indefinitely. If we get it
wrong, we can actually ignite this process we are trying to prevent.
Funding for the 2012 Symposium at WKU is provided by NSF grant DMS-0846477 through the
MAA Regional Undergraduate Mathematics Conferences Program, www.maa.org/RUMC, by
Ogden College of Science and Engineering, WKU, by Carol Martin Gatton Academy of
Mathematics and Science, WKU, and by the Department of Mathematics, WKU.