31st Annual Mathematics Symposium - Western Kentucky University
October 28 - October 29, 2011
Mathematics and Sports!
*= student presentation
Friday 4:30pm, Registration and refreshments begin, Snell Hall (SN)
Friday 4:50 pm, ROOM 2113 SN Welcome by Dean Blaine Ferrell
Friday 5:00-6:00 pm, ROOM 2113 (SN)
INVITED TALK
Ron Gould
(Emory University)
Rating Player Performance - The Old Argument of Who is Best
Sports statistics have often resulted in arguments about who is really the best player. Beginning with the
NFL quarterback rating formula, we consider quantitative methods of rating players performance including
quarterbacks, hitters in baseball and basketball players. Some of the history of such ratings will be discussed
along the way. Special attention will be paid to linear weighting schemes. We look for the strengths and
weaknesses of each method, including ESPN's new quarterback index.
PARALLEL SESSIONS, Friday
Friday 6:00 - 6:20 pm
ROOM 1101 • Stuart A. Foster (WKU), Jonathan Quiton (WKU), Rezaul Mahmood (WKU)
A framework for automated adaptive spatial modeling of near-surface atmospheric variables using
the Kentucky Mesonet
Weather is critical to operational decision making across many economic sectors. In situ dense networks,
such as the Kentucky Mesonet, are capable of providing samples of the near-surface atmosphere from a
finite number of observation sites. The challenge of spatial interpolation is to use observed data for
temperature, wind speed, and other variables to estimate values at locations where no observations are
available. In an environment where weather conditions vary smoothly over space, the task is straightforward
and can be readily implemented using any of a number of established methods. However, the complexity of
near-surface atmospheric processes in areas of diverse, even chaotic terrain poses significant challenges for
the dynamic spatial interpolation of near-surface atmospheric variables. This presentation describes a
framework for adaptive modeling to interpolate near-surface atmospheric variables using weighted
combinations of observations based on geographic and topographic proximity of Kentucky Mesonet sites.
ROOM 1102 • David Sekora* (WKU), Claus Ernst (WKU)
Manipulation and Analysis of Three Dimensional Knotted Polygons in Confinement
In this research, we use programs developed with Mathematica to model DNA in a viral capsid as an
equilateral polygon in confinement. The problem is that these polygons are exceedingly complex and
compact, and so various simplification and relaxation routines are required to reduce the polygon to a more
manageable form. The programs created and discussed are Rotator, which aims to simplify compact
polygons by expanding them through repeated isomorphic rotations, and Dowker, which converts a twodimensional projection of a polygon to a sequence of integers in a topologically (but not geometrically)
reversible process. These programs are part of a larger software package by the research group, entitled
“Polygon Toolkit.”
ROOM 1103 • Peter Dragnev (Indiana University-Purdue University Fort Wayne), J. Brauchart (U New
South Wales, Australia), E. Saff (Vanderbilt University), C. van de Woestijne (Montanuniversitat Leoben,
Austria)
A fascinating polynomial sequences arising from electrostatics problem on the sphere
A positive unit point charge approaching from infinity a perfectly spherical positively charged isolated
conductor will eventually cause a negatively charged spherical cap to appear. The determination of
the critical distance when this occurs ρ ( d ) ( d is the dimension of the unit sphere) is known as Gonchar's
1 + ρ (d ) is equal to the largest positive zero of a certain sequence of monic
polynomials of degree 2d − 1 with integer coefficients which we call Gonchar polynomials. Rather
surprisingly, ρ (2) is the Golden ratio and ρ (4) the lesser known Plastic number. But Gonchar polynomials
problem. We show that
have other interesting properties. We discuss their factorizations, investigate their zeros and present some
challenging conjectures.
Friday 6:30 - 6:50 pm
ROOM 1101 • Funda Ekiz* (WKU), Ferhan Atici (WKU)
The Rational Expectations: A New Formulation of the Single-Equation Model on Time Scales
In this talk we first give a short introduction to time scale and stochastic calculus. Then we talk about history
of the rational expectations (REs) and economist, John F. Muth , behind the REs idea. In addition, we
introduce a single-equation rational expectation model (REM) on isolated time domains which include
periodic time domains as well as non periodic time domains. Finally, we discuss the Cagan’s Hyperinflation
Model on isolated time scales and the method of solving this model.
ROOM 1102 • Jeremy Phelps* (WKU), Jonathan Quiton (WKU)
Multiple objective linear programming with stochastic coefficients and
a quota constraint with applications to stock trading and sports
In this talk we present our current investigation on multiple objective linear programming as applied to stock
trading and sports where we allocate stocks or player’s playing time based on a fixed investment allocation
or fixed total playing time. The general idea is identify and maximize performance-enhancing objectives
(PEOs ) subject to performance-damaging objectives (PDOs) such as stock (or player) inconsistent
performance as reflected in some volatility metric. More specifically, we are using the covariance matrix of
the performance coefficients both as a constraint and in constructing a confidence bound to our optimal
solution. Finally, we demonstrate our approach using SAS.
ROOM 1103 • Sunny Potay* (WKU), Mustafa Atici (WKU)
What is secret sharing scheme in Cryptography?
It refers to a method through which secret key K can be shared among a group of
authorized participants, such that when they come together later, they can figure out the secret key K to
decrypt the encrypted message. Any group which is not authorized cannot determine secret key K. In some
situations, it may not be safe to give secure key K to an individual person. Even if this person is an
authorized one.
Friday 7:00-7:40pm Food and Refreshments!
Friday 7:40 - 8:00 pm
ROOM 1101 • Jeremy B. Maddox (WKU)
Bipolar decompositions for stationary scattering states of the Schrödinger equation
We analyze the stationary state wave functions of the Schrödinger equation for one-dimensional scattering
problems in terms of so-called “bipolar” components. These bipolar components represent counterpropagating traveling waves associated with system’s incident, transmitted and reflected probability
amplitude. The asymptotic behavior the bipolar components and the necessary relationships between them
are developed. Finally, we formulate a set of coupled equations of motion from which these can be
numerically determined for a certain class of scattering problems with asymptotically convergent potential
energy.
ROOM 1102 • David Benko (University of South Alabama)
The Grandest of all Grand Slams
Have you ever wondered which one of the four tennis grand slam tournaments is the best? Wimbledon,
Roland Garros, US Open or Australian Open? They are played on different surfaces: grass, clay, acrylic hard
court, synthetic hard court, and under different weather conditions: sun, wind, rain, snow. :) How can we pick
a winner? As always, mathematics can help us to find the answer. Our method can also be applied to some
other sports.
ROOM 1103 • Uta Ziegler (WKU)
Viral DNA and Knots
Researchers want to understand how viral DNA is packed inside the ‘heads’ during the assembly process of
viruses. One thing which is known from experiments is that the viral DNA is often knotted. In our research,
viral DNA is modeled as freely-joined, unit-length polygons, and the ‘head’ is a sphere. This talk focuses on
the process of identifying the knot(s) in randomly generated polygon models of the viral DNA. (Note: Other
presentations address how the polygons may be generated and manipulated.)
Friday 8:10 - 8:30 pm
ROOM 1101 • Lukas Missik* (WKU), Dominic Lanphier (WKU)
Asymptotics and Probability in Iterative Duels
Game theory is used in various branches of science and sports to study the interaction of players in a
competition. The different choices that players make during a game can have an influence on the outcome.
Probability theory often plays a fundamental role, in that the likelihood of a choice can often be quantifiable.
We investigate a specific class of games that resemble duels and analyze the outcomes (i.e. which player
wins) of those games.
ROOM 1102 • Tucker Joyce* (WKU), Richard Schugart (WKU)
Analyzing a Mathematical Model of the Interaction of Bacteria and Neutrophils in a Chronic Wound
It is estimated that 5-10 billion dollars are spent each year in the United States on the treatment of chronic
wounds. A wound is classified as chronic when the body fails to remove the bacteria from the wound,
prolonging the inflammation period indefinitely. Neutrophils are a major body cell type involved in fighting the
bacteria. Currently no model exists that models the dynamics of how neutrophils interact with bacteria in this
process. The goal of this project is to analyze a system of ODEs that models the interaction of these cell
types. This is done using both analytic and numerical methods, and the goal of my work was to verify and
adjust the model as necessary.
ROOM 1103 • Madeline O. Oldham* (WKU)
Introduction to Park Factors
An overall description of park factors is provided as well as a look into those factors that contribute to
needing a park factor. The park factors at five various fields have been analyzed based upon home runs,
triples, doubles and singles. Those five fields being considered are: Coors Field, AT & T Park, Turner Field,
Wrigley Field and Citi Field.
____________________________________________________________________________________
Saturday from 8:00 am - Registration and refreshments, First Floor of Snell Hall (SN)
REGISTRATION continues until 12:00pm SATURDAY
Saturday 8:30-8:50 am
ROOM 1101 • Anthony Montemayor* (WKU)
Generating Random Polygons in Spherical Confinement
To test biological models of DNA packing such as in the capsid of a bacteriophage, it is important to have a
fast algorithm to generate random polygons in confined spaces. This talk will discuss the development of
such an algorithm in the case of spherical confinement using elementary results in statistics.
ROOM 1102 • Nihan Acar* (WKU), Ferhan Atici (WKU)
Data Fitting for Tumor Growths with Discrete Fractional Calculus
As well as the doctors seek the methods of treatments for the various types of cancers, mathematicians have
started to cooporate with them to find a better outcome as a treatment. The aim of our project is, to develop
Discrete Fractional Models of tumor growth for given data and to estimate parameters of these models in
order to have better data fitting.
In this talk, we first introduce Discrete Fractional Calculus and Nabla Fractional Operators. Then, we give
some properties in relation to these operators. Additionally, we solve a
order, discrete nabla fractional
equation with initial condition. We interpret our result, and finally, we use some tumor growth data and
estimate the parameters for a fractional sigmoidal model to obtain better data fitting.
ROOM 1103 • Elizabeth Haynes* (Southern Illinois University)
Smale Flows on Three-Manifolds
3
We will discuss how to realize Smale flows on S . We will examine one template, and the possible links and
knots formed by the entrance and exit sets. This talk will avoid technical details and present examples, with
models.
Saturday 9:00-9:20 am
ROOM 1101 • R. Drew Pasteur (The College of Wooster)
Resources for Undergraduate Research in Sports Analytics
Over the last decade, sports-related research in the mathematical sciences has grown
significantly, partially because of widespread access to large amounts of data. Research
questions on sports topics are of interest to many undergraduates, and often such projects are
quite accessible. For students who are considering work in this area and the faculty members
advising them, we will discuss data sources, types of problems and approaches, key pieces of
literature, and venues for presentation and publication.
ROOM 1102 • Mark Robinson (WKU)
Differentiability of Functions of Several Variables
In a course in single-variable calculus, students are introduced, by way of a limit definition, to the derivative.
Later, in multivariable calculus, students again encounter limits (this time of functions of several variables)
and then learn (sometimes to their great surprise) that differentiability of a function of several variables
requires more than simply the existence of partial derivatives. The concept of differentiability of functions of
several variables is examined in this presentation, with accompanying examples.
ROOM 1103 • Mikhail Khenner (WKU)
Analysis of liquid film stability using differential equations
Thin liquid films with flat surface resting on a smooth solid support become unstable due to variety of factors,
such as heating, vibration, and intermolecular forces. As a result of the instability development, a film
agglomerates into droplets. In this talk I will describe the analysis of the instability using linearized equations
for small perturbations of the flat equilibrium and computations of the nonlinear phase of the instability.
Saturday 9:30 – 9:50 am
ROOM 1101• Dominic Lanphier (WKU)
Heartless Poker
The probabilities, and hence the rankings, of the standard poker hands are well-known. But what happens to
the rankings in a game where a deck is used without a suit (heartless poker, for example), or with an extra
suit, or extra face cards? Does it ever happen that two or more hands will be equally likely? In this talk we
examine these and other questions, and show how probability, some analysis, and even some number
theory can be applied.
ROOM 1102• Christopher S. MacMahan* (University of South Carolina), Joshua Tebbs (USC), Christopher
Bilder (University of Nebraska-Lincoln)
Two-Dimensional Informative Array Testing
Array-based group testing algorithms for case identification have been widely considered for use in infectious
disease testing, drug discovery, and genetics. In this work, we generalize previous statistical work in array
testing to account for heterogeneity among individuals being tested. We first derive closed-form expressions
for the expected number of tests (efficiency) and misclassification probabilities (sensitivity, specificity,
predictive values) for two-dimensional array testing in a heterogeneous population. We then present two
``informative" array construction techniques which exploit population heterogeneity in ways that can
substantially improve testing efficiency when compared to classical approaches which regard the population
as homogeneous. Furthermore, a useful byproduct of our methodology is that misclassification probabilities
can be estimated on a per-individual basis. We illustrate our new procedures using chlamydia and gonorrhea
testing data collected in Nebraska as part of the Infertility Prevention Project.
ROOM 1103• Sinem A. Karatas* (WKU), Thomas Richmond (WKU)
Ordered compactifications
The fundamental and useful notions of finiteness in topology, algebra and analysis is the compactness. Of
course, many spaces of interest (for instance, the real line) are not compact. The natural motivation lies in
the concept of compactifying spaces. Thus, in order to state a compact space in applications,
mathematicians construct many different compactification processes. In the mathematical discipline of
geneal topolgy, Stone–Čech compactification βX of a topological space X is the one of the most important
universal property. The Stone–Čech compactification βX of a topological space X is the largest compact
Hausdorff space generated by X in a unique way.
In this talk we start with an elementary perspective of general topology, then provide properties and
examples for βX-X if X is totally ordered space, also we discuss a technique for satisfying a construction
X=βX-X..
Saturday 10:00 –- 10:20 am
ROOM 1101 • Sri Harsha Vege* (WKU), Huanjing Wang (WKU)
An Empirical Study of Ensemble Feature Selection Techniques
Feature selection technique is an important data preprocessing step. It has been a focus of much research
in data mining and machine learning where datasets containing hundreds of thousands of features are
available. The main objective of feature selection is to remove irrelevant and redundant features and further
improve the predictive performance of the predictors using the reduced datasets. We will be studying various
feature ranking techniques and prepare an ensemble method which combines multiple feature ranking
techniques using rank ordering of features. The reduced datasets will be used in building the classification
models using well-known classifiers. The results obtained will be evaluated using AUC (Area under receiver
operating characteristic curve) performance metric.
ROOM 1102 • Tyler Clark* (WKU), Thomas Richmond (WKU)
Continued Radicals and Cantor Sets
We will construct several continued radicals and look at their
convergence properties. Furthermore, we will look at some conditions in which
a continued radical creates a set homeomorphic to the cantor set. Finally, we
will observe a continued radical formed by the set {1,2} and different properties of the cantor set obtained.
ROOM 1103 • Ian Burchett* (WKU), Rong Yang (WKU)
Quantifying Computer Network Security
Simplifying network security data to the point that it is readily accessible and usable by a wider audience is
increasingly becoming important, as networks become larger and security conditions and threats become
more dynamic and complex, requiring a broader and more varied security staff makeup. In this talk, we will
provide some background information about network security evaluation and introduce a new metric
framework to measure the overall “health” of a given network.
Saturday 10:30-11:00 am, Refreshments
Saturday 11:00 - 12:00 pm, ROOM 2113 (SN)
INVITED TALK
Ken Massey
(Carson-Newman College)
Sports Ratings by Linear Regression with Lp norms
The college football schedule graph is very sparse, and game results exhibit substantial variation. Many
mathematical models have been constructed to objectively evaluate team strengths. I will explain the popular
least squares regression (L2) approach, and then generalize to other norms. We will discuss computational
algorithms and properties of the solution. Results from recent college football seasons will be used for
illustration.
Funding for the 2011 Symposium at WKU is provided by NSF grant DMS-0846477 through the
MAA Regional Undergraduate Mathematics Conferences Program, www.maa.org/RUMC and by
Ogden College of Science and Engineering, WKU.