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Friday
Mustafa Atici (WKU) - How can Mathematics and
Computer Science be so much Fun?
Graph Theory based combinatorial games can be a useful way of
teaching children science such as mathematics and computer science.
In this study we will introduce a new Graph Theory based game. Most
of the graph games are either /achievement game/ or /avoidance game/.
The game is called an /achievement game/ if the last person to move
wins; otherwise it is an /avoidance game/. Here are some games: 1)
Generalized Tic-Tac-Toe( Ramsey numbers). 2) Path construction. 3)
Hex-a-hop( Eulerian cycle). 4) Lights out. 5) Path removal. *Path
Removal Game*: There are two players, say A and B. Connected graph G
is given. The first player A removes a path from the graph G and the
second player B removes a path from the G. They will continue
removing paths from the G at their turns until all the links are
deleted. Whoever removes the last path is the winner.
David Benko (WKU) - The Harmony of Distant Cities
We will consider some problems involving the arc-length of curves and
the distances of sets. In particular we will discuss different ways
of measuring the distance between cities. When the problem is getting
complicated, harmonic analysis comes to our rescue. Prerequisite: the
knowledge of integrals.
Barry Brunson (WKU) - Toward a Lean and Lively Algebra
We, as mathematics faculty, generally want students to take more
math, not less. Nevertheless, I am advocating a considerably
trimmed-down college algebra course, one that concentrates on key big
ideas, that gives a glimpse of the underlying beauty and magic of
mathematics, and that doesn’t get mired in inert material. This
approach assumes both constant availability of (and an enlightened
use of) technology, and incorporation of generous amounts of
real-world data. (The underlying hope, naturally, is that more
students will be seduced into studying more math!)
Erin Capece (WKU) - Cauchy Integrals of Two-By-Two Matrices
The Cauchy Integral Formula will hold true for any complex function
f(z) that is analytic inside and on a simple closed contour C and a
point contained in C. Now, for a 2 by 2matrix A, a new matrix f(A)
can be defined with the help of the Cauchy Integral Formula. Using
the Cauchy Integral Formula, several interesting properties of f(A)
can be found in relation to f(z).
Kevin Coulton (Eastern Illinois University) - Mechanical
Integrals in Early Rockets
A brief description of the mathematical processes employed in the
guidance systems of the first modern rockets.
Patrick Coulton (Eastern Illinois University) - The
Besicovitch Conjecture for Equilateral Triangles
A long standing conjecture is that the Besicovitch triangle, i.e., an
equilateral triangle with side SQRT(28/27) is a worm-cover. We will
indicate the proof of the conjecture and time permitting we will show
that there is a class of isosceles triangles which contain the
Becicovitch triangle, are worm-covers, and are of minimum size for
their similarity class. (Work to appear in Geom. Dedicata)
Constance C. Edwards (WKU) - Modeling the Effects
of Drugs
What happens if you become addicted to nonprescription cough
medicine? How many beers can you drink an hour and still be able to
legally drive? How long does it take you to sober up after consuming
a six pack? In this talk we find out the answers to these and other
interesting questions.
Claus Ernst (WKU) - Unknotting of Knots Through Diagrams
Knots can be represented through drawings, i.e. diagrams. A crossing
in a diagram happens when one string lies on top of another. A knot L
is called a predecessor of a knot or link K if a diagram of L can be
obtained from a diagram of K by changing the over and under at a
single crossing change. Changing a knot in this way to simpler and
simpler knots is one way of how to unknot a knot. There will be lots
of pictures and your calculus knowledge will not be needed.
Jean-Claude Evard (WKU) - A Tangent Line Method
for Fast Factorization of Large Integers
This is a joint work with Professor Keith Wolcott, who is at Eastern
Illinois University. We first found considerable improvements to the
method published in the following paper: Chun-Xuan Jiang, Factoring
large numbers (Algebras Groups Geom. 19, no. 1, 85--95, 2002). We
have also improved several additional ideas that have been posted by
different mathematicians on the internet. We have completely tested
the first versions of the improved method, by factoring all integers
from 4 to 2^60, and by checking that all of the results agree with
the factorizations done with Maple and Mathematica. Our work is still
very unstable in the positive sense that we are still finding so many
ideas for improvements that we have not had time to write a new
version and to test it. But the method has the best chances to be at
least the fastest and the simplest of all elementary methods. I will
present the part that is in good form, and I will present the new
idea for a tangent line method. This tangent line method has not been
tested yet.
Jonny Groves (University of Kentucky) - D-Nice
Polynomials with Three or Four Roots
Let D be an integral domain of any characteristic. We say that a
polynomial p(x) in D[x] is D-nice if p(x) and its algebraic
derivative p'(x) split over D. If D=Z, we say nice rather than
Z-nice. Mathematicians originally began working on the problem of
constructing nice polynomials because these are “nice” for
their calculus students to sketch. Later, other mathematicians, like
Dr. Jean-Claude Evard here at Western Kentucky University, have
considered the problem of constructing and classifying nice
polynomials worthy of further research. He has considered arbitrary
integral domains of characteristic 0, especially the integers and
Gaussian integers. I began researching this problem with him in the
academic year of 2003-2004 on a research assistantship and have
written a successful Master’s thesis on my results. I am now
preparing several papers on my results. Dr. Evard has found a new
approach to this problem, and we both have used it successfully. My
thesis focused on nice polynomials; however, earlier this year, I
have generalized many of my earlier results to arbitrary integral
domains, including integral domains of nonzero characteristic. I
present this new approach to investigating D-nice polynomials, and I
give explicit formulas for D-nice polynomials with three roots and
D-nice symmetric polynomials with four roots. I conclude by counting
the number of equivalence classes of such polynomials and by giving
examples I have found with these formulas.
Bruce Kessler (WKU) - A "Sound" Approach
to Fourier Transforms
While wavelet decompositions can sometimes provide a better analysis
of non-periodic data (like image data and network usage levels),
Fourier series are ideally suited for the analysis of sound. Fourier
series are composed of sine and cosine waves of various frequencies,
and the amplitudes of the waves that coincide with each frequency
tell us how much of that particular pitch is in our signal. This talk
will give a brief introduction to the ideas of sound generation and
the discrete Fourier transform. We will conduct live experiments
during the talk to demonstrate how this mathematical concept can be
used to analyze samples of sound.
Buddy Lagani (WKU) - Some Implications of the
Weierstrass Approximation Theorem
The Weierstrass Approximation Theorem states that any continuous
function over a compact domain may be expressed as a uniform limit of
polynomials. We will discuss this theorem and its generalizations to
continuous functions of a complex variable. We might also discuss how
this theorem may be applied to the set of continuous functions over a
compact domain to show that it is separable as a metric space.
Dominic Lanphier (WKU) - Integrity of Manifolds
The Cheeger constant of a graph measures how easy (or hard) it is to
tear it into two pieces. Another measure of the connectedness of
graphs is edge-integrity, which measures how hard it is to shatter a
graph into small pieces. Both the Cheeger constant and the
edge-integrity of graphs can be used to measure the efficiency of
communication networks. Cheeger constants can also be defined for
2-dimensional objects. We give a definition of integrity for such objects.
Yale Madden (WKU) - Counting Special Prefix Vectors
A prefix vector is a string of one's and zero's, such that when read
from left to right there are never more zero's than one's. We want to
count the number of such vectors, of length 2n, with k "1 0"
pairs. These vectors arise in the generation process of random
4-regular graphs.
David K. Neal (WKU) - Generating Fibonacci
Fractions from Heron's Method
Heron's method is used to approximate the square root of 5 in order
to find successive rational approximations of the Golden Ratio, and a
characterization is given for when the results always will be ratios
of successive Fibonacci numbers.
Lan Nguyen (WKU) - Generalization of the
Exponential Function
Given a real-valued exponential function f(t)=exp(at), then we think
about its characteristics like f'(t)=af(t) or f(t+s)=f(t)f(s) and the
Taylor expansion of f(t). Using these characteristics, we generalize
the concept of exponential functions, when their range is in other
spaces like complex plane, n-dimensional, or even infinitely
dimensional spaces. Many results from calculus can be obtained from
properties of generalized exponential functions.
Mark P. Robinson (WKU) - Fixed-Point Methods for
Solving Nonlinear Equations
One of the problems encountered frequently in mathematics is that of
solving a nonlinear equation of the form F(x) = 0,
where F goes from Rn to Rn. Such
a problem may be reformulated as one of finding solutions of an
equation G(x) = x, that is, of seeking fixed points of
some function G. This gives rise to the consideration of the
fixed-point iteration method x(k) = G(x(k-1))
for the approximation of such solutions. An example of a fixed-point
method that is quite well known is Newton’s method. This
presentation will focus on the speed of convergence of fixed-point
methods, starting with the one-dimensional case and then considering
fixed-point methods for functions of several variables.
Jason Rosenhouse (James Madison University) - Monty
Hall vs. Howie Mandel: Bayes Theorem Hits Prime Time
On the game show "Deal or No Deal," the contestant is
presented with 26 briefcases, which contain amounts of money ranging
from one penny to one million dollars. One of these briefcases is
chosen at random. After making this choice, some of the remaining
twenty-five briefcases are randomly opened and their contents
revealed. If the game proceeds far enough, a situation arises in
which only two cases remain: the one originally chosen and one other.
The contestant is given the option either of sticking with the
original briefcase, or switching to the other one. Though this
situation is superficially similar to the Monty Hall problem, in this
case there is no advantage to be obtained by switching briefcases.
Explaining the difference between the two situations involves some
interesting probability theory, particularly Bayes' Theorem.
Andrew T. Wilson (Austin Peay State University) - Involving
Pre-service Teachers in Standards-based Mathematics Instruction: A
Case Study
Attendees will gain understanding of change in children's thinking by
examining pre-tests and post-tests related to proper and improper
fractions and mixed numbers. Attendees will gain appreciation for the
importance of building connections among real life, concrete,
pictorial, spoken, and written representations of fractions by
examining activities that occurred between the pre- and post-tests.
Attendees will learn about a model for involving pre-service teachers
in standards-based instruction.
Di Wu (WKU) - Distance Geometry in Protein
Structural Modeling
The coordinates of the atoms and hence the protein structure can be
determined by using the known distances for certain pairs of atoms,
which can often be obtained based on our knowledge on various types
of bond-lengths and bond-angles or from physical experiments such as
nuclear magnetic resonance (NMR). However, it requires the solution
of a mathematical problem called the distance geometry problem, which
is proved to be computationally intractable in general. In this talk,
we will review some mathematical problems and their applications in
protein structural modeling.
Uta Ziegler (WKU) - Simplifying Knots
This presentation describes an algorithm that can be used to simplify
a given diagram of a 'knot'. The algorithm reduces the problem to a
problem in graph theory using the language of cut-vertices and
spanning trees in a graph.
We gratefully
acknowledge the funds which were provided for student travel by MAA
NSF-RUMC (NSF Grant DMS-0241090, via the MAA) and Ogden College of
Science and Engineering, WKU.
 
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