Western Kentucky University

24th Annual Mathematics Symposium

November 19-20, 2004

 

Elegant Mathematics:

Discrete, Continuous, and Everything in Between

 

 

 

ABSTRACTS

 

The code (U), (G), or (F) after the speaker’s affiliation indicates whether the speaker is an Undergraduate, Graduate student, or Faculty member.

 

 

Contributed Presentations               Friday, November 19

 

6:00 - 6:20 Friday.  TCCW 125 C

 

David Benko, Western Kentucky University (F)

 

The hidden geometry of the "hyper convex" functions

The sign of the first and second derivative of a function gives us important information about the geometry of the graph of the function. Ever wondered if the third or other higher order derivatives carry any information about the function? To demonstrate the affirmative answer we will solve the following problem. Let n and k be fixed numbers. Given any k points on the plane (with different x-coordinates) can we always find a function f(x) so that f(x) is going through the given points and its n-th derivative is a non-negative function?

 

 

 

6:00 - 6:20 Friday.  TCCW 129

 

Mark Robinson, Western Kentucky University (F)

 

Topics in Numerical Methods for Initial-Value Problems

 

In this talk we examine several difference methods which are used for the numerical solution of initial-value problems in ordinary differential equations. Various topics of

interest will be investigated, including the natural connection between certain difference schemes and well-known methods for approximating integrals, difficulties encountered by difference schemes, and appropriate selection of methods.  Treatment of systems of ordinary differential equations by difference methods  will also be discussed.

 

6:25 - 6:45 Friday.  TCCW 125 C

 

Chris Christensen, Northern Kentucky University (F)

 

Group theory helped win World War II

 

In the 1920’s, Poland felt threatened by its neighbor Germany.  In order to determine Germany’s intent, the Poles monitored German radio transmissions.  Beginning in 1928, the Poles ran into a roadblock decrypting the German messages and began to suspect that the Germans were using a machine cipher. 

The Germans had begun using Enigma.  In 1932, Polish Intelligence recruited three mathematicians to attack the Enigma cipher.  One of these mathematicians Marian Rejewski was able to use group theory to exploit patterns in German messages; he was able to determine the wiring of the rotors and the rotor settings.  Until 1938 … .  We will look at how Rejewski used group theory to determine the settings of the Enigma rotors.

 

 

6:25 - 6:45 Friday.  TCCW 129

 

Julian A. Allagan, Troy University (U)

 

Chromatic Polynomial of a Cycle of sizes n

 

The chromatic polynomial of a graph G is denoted by P(G, λ) and is defined as the number of proper colorings of the graph G using at most λ colors. 

Here, we prove that for the cycle Cn of order n, the chromatic polynomial is P(Cn, λ) =(λ-1)n + (-1)n(λ-1).

 

 

6:50 - 7:10 Friday.  TCCW 125 C

 

Jessa Pratt, Northern Kentucky University (U)

 

What's a quasi $\pi$-group?

 

Based upon work that led to his thesis under Professor Oscar Zariski, Abhyankar defined a group to be a quasi p-group if the group is generated by the union of its p-Sylow subgroups.  In 1957 Abhyankar conjectured that the set of quasi p-groups is exactly the set of groups that should occur as Galois groups of unramified covers of affine curves over an algebraically closed field of characteristic p.  The truth of this conjecture was proved by Raynaud and Harbater in independent papers in 1993 - 94.  For their work, Harbater and Raynaud received the Cole Prize in Algebra.  Ben Harwood in undergraduate research leading to his Honors Thesis at Northern Kentucky University examined the elementary group theory properties of quasi p-groups and determined for each group of order less than 64 for which primes p it is a quasi p-group.  He also posed the question "what does it mean for a group to be a quasi p-group for all primes p dividing its order?"  We will examine that question and a related question.

 

 

6:50 - 7:10 Friday.  TCCW 129

 

Jason Bell, Troy University (U)

 

The Use of Linear Algebra in Balancing Chemical Equations

 

We describe an application of linear algebra to a chemistry problem.  In particular,we show how the problem of balancing equations in chemistry can be modeled and solved using matrices and row reduction.

 

 

 

Welcoming Adress

 

7:15 - 7:45 Friday. TCCW 129

 

James Barksdale, Western Kentucky University (F)

 

Esoteric Renditions of Celebrated Theorem

 

The Pythagorean theorem, the Law of Cosines, and the Parallelogram Law all epitomize the notion of a celebrated theorem. This presentation demonstrates how such celebrated propositions can be implemented to create esoteric renditions such fundamental laws.

 

 

 

Invited Lecture

 

8:05 - 8:55  TCCW 12

 

Jack Robertson, Washington State University

 

Elegant Elementary Mathematical Potpourri

 

We will survey some of my favorite accessible results from such diverse areas as number theory, geometry, calculus, geometric probability and mathematical billiards. Can you form a circular disc using similar copies of two subsets which partition a square? What is the probability that a line in the plane which strikes a bounded convex set A also strikes a disjoint second bounded convex set B? The answer can be elegantly given using rubber bands.

 

 

 

 

 

 

Contributed Presentations     Saturday, November 20

 

8:30-8:50 Saturday.  TCCW 125 C

 

Amber De Moore, Austin Peay (U)

 

Female Contributions to Scandal in Mathematics

 

This presentation is primarily historical in nature. The history of mathematics is infused with scandal. Although not widely known, it is none the less very true. Throughout history it has been deemed inappropriate for a woman to study mathematics until recent times, and even now in many cultures it is frowned upon. Although culturally it was unacceptable to study mathematics, some women found a way to make their contributions anyway. This presentation will include various women’s contributions to mathematics and the scandal that surrounded these contributions. The women of primary focus will be Ada Byron Lovelace, Sophie Germain, and Mileva Maric, each of whom made great strides in the field of mathematics.

 

 

 

8:30-8:50 Saturday.  TCCW 129

 

Nick Wintz, University of Missouri-Rolla (G)

 

Eigenvalue Comparisons For An Impulsive Boundary Value Problem With Sturm-Liouville Boundary Conditions

 

We will consider the second order impulse equation

 

subject to the impulse effects

satisfying the Sturm-Liouville boundary conditions

                                                 

We look at a system of second order differential equations

  

We will use the theory of -positive operators to establish the existence of the smallest eigenvalues as well as compare the eigenvalues  and .

 

 

 

 

 

 

 

8:30-8:50 Saturday.  TCCW 116

 

Justin Grieves, Western Kentucky University (U)

 

Average Extrema of a Geometric Random Walk

 

In this talk, we discuss the average maximum and average minimum values obtained by one-dimensional random walks that are stopped upon a first “occurrence”. For example go up and stop the first time we move downward or stay constant and find the average maximum and average minimum.  Results are applied to a player’s expected earnings (or losses) on a casino game when using a geometric strategy.

 

 

8:55-9:15 Saturday.  TCCW 125 C

 

Asli Guldurdek, Western Kentucky University (G)

 

On γ-Semi Open Sets

 

The idea of examining generalized open sets in generalized topological spaces was given by À. Csàszàr. Generalized  –sets and generalized -sets were introduced by Miguel Caldas and Julian Dontchev in general topology. Maheswari and Prasad in 1975 introduced the two new classes called semi-T1 spaces and semi-R0 spaces. We give the definitions of γ-semi open set, γs-set and γs-set by using γ-open sets. Also we aimed to show that the concepts of g.  –set, g. -set, semi-T1 space and semi-R0 space can be generalized by replacing semi-open sets with γ-semi open sets for any arbitrary γ(X). These concepts should be considered in generalized topological spaces instead of general topology.

 

 

 

8:55-9:15 Saturday.  TCCW 129

 

Michael Chmutov, Ohio State University (G)

 

The Chromatic Polynomial and Cohomology of Graphs

 

The chromatic polynomial of a graph is a polynomial in n which gives the number of ways of coloring the vertices of a graph in n colors so that no two adjacent vertices have the same color. Recently, L. Helme-Guizon and Y. Rong from George Washington University suggested a definition of a chain complex of a graph so that the graded Euler characteristic of this complex is the chromatic polynomial. In this talk, I will describe this chain complex and some interesting results about the calculations of the corresponding homology groups.

 

 

 

8:55-9:15 Saturday.  TCCW 116

 

Matt Dawson, Western Kentucky University (U)

 

Random Walks Motivated by Unit-Circle

 

We will present a two-dimensional circular random walk that is constructed as follows:  Starting at the origin, a random angle between 0 and 2pi is chosen. One step of length r is then taken in the direction of this angle. This process is then repeated for additional steps, starting with the ending points of the previous step. The distribution for this process will be developed for the first step and will be approximated for a large number of steps.  Then the probability that the random walk is within a certain radius after a large number of steps is given.  From this result, we can analyze the limiting behavior as the number of steps increases. This basic idea for this problem was originally by Dr. Benko.

 

 

Invited Lecture

 

9:30 - 10:20 Saturday.  TCCW 129

 

Martin Bohner, University of Missouri-Rolla

 

Dynamic Equations on Time Scales

 

Time scales have been introduced in order to unify continuous and discrete analysis and in order to extend those theories to cases "in between". We will offer a brief introduction into the calculus involved, including the so-called delta derivative of a function on a time scale. This delta derivative is equal to the usual derivative if the time scale is the set of all real numbers, and it is equal to the usual forward difference operator if the time scale is the set of all integers. However, in general, a time scale may be any closed subset of the reals.

We present some basic facts concerning dynamic equations on time scales (those are differential and difference equations, resp., in the above two mentioned cases) and initial value problems involving them. We introduce the exponential function on a general time scale and use it to solve initial value problems involving first order linear dynamic equation. We also present a unification of the Laplace and Z-transform, which serves to solve any higher order linear dynamic equations with constant coefficients.

Throughout the talk, many examples of time scales will be offered. Among others, we will discuss the following examples:

  1. The two standard examples (the reals and the integers).
  2. The set of all integer multiples of a positive number (this time scale is interesting for numerical purposes).
  3. The set of all integer powers of a number bigger than one (this time scale gives rise to so-called q-difference equations).
  4. The union of closed intervals (this time scale is interesting in population dynamics; for example, it can model insect populations that are continuous while in season, die out in say winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population).

 

 

Contributed Presentations     Saturday, November 22

 

10:30 - 10:50 Saturday.  TCCW 125 C

 

James R. Stapleton, Clemson University (U)

 

 FLT: Fermat’s Last Triangle

 

The unweighted total distance location problem, as stated by Fermat, is to find a point X in the plane that minimizes the sum of the Euclidean distances from X to three given points P1, P2, P3.  The weighted total distance problem assumes a positive weight wi associated with each given point and seeks the point X that minimizes the total weighted distance to the three points.  A geometrical solution and geometrical dual was discovered by Toricelli and Simpon for the unweighted problem.  This paper extends the geometrical solution and dual to the weighted problem.

 

 

10:30 - 10:50 Saturday.  TCCW 129

 

Serap Topal, Ege University –Turkey (F)

 

Calculus of Variations on Time Scales

 

In this study, the concept of variational problems on the theory of differential equations and difference equations is motivated on time scales and we examine necessary conditions for the simplest variational problems. Also, some examples are given

on this subject.

 

 

10:30 - 10:50 Saturday. TCCW 116

 

Jean-Claude Evard, Western Kentucky University (F)


Polynomials whose roots and critical points are integers

 

The problem of finding properties, characterizations, and methods of construction of polynomials whose coefficients, roots, and critical points are integers is on the list of unsolved problems published in the issue of December 1999 of The American Mathematical Monthly. Such polynomials are called nice polynomials. To our knowledge, the earliest paper on this subject was published in 1960. The most important paper was published by Ralph Buchholz and James MacDougall in the Journal of Number Theory in January 2000. Their paper contains a comprehensive bibliography on the subject. I am preparing a paper on this subject. The first version of my paper is posted on the internet at the address:

 

http://front.math.ucdavis.edu/math.NT/0407256

 

I am currently working on the revision of this first version. After the first version of my paper was finished, an important work on nice polynomials was achieved by Jonathan Groves in his Master's thesis at Western Kentucky University during the academic year 2003--2004. He is preparing several papers on the results of his thesis, and one of them is already submitted. He finished his Master's thesis last July, and has started the Ph. D. program of the University of Kentucky. He will also give a talk on his work at this Symposium. We have obtained a lot of new results, and opened roads in several directions. Our work has raised many exciting problems at all levels. Many of these problems are likely to be solved in a short time, while many other problems will require the creation of new methods that may interest mathematicians with very different background.

 

10:55 - 11:15 Saturday.  TCCW 125 C

 

Mustafa Atici, Western Kentucky University (F)

 

Searching or "googling"

 

Sometimes referred to as seek, search is the process of locating a letter(s), word(s), file(s), website(s), etc. Many operating systems, software programs, websites contain some kind of search or find utility to locate data within the file being worked on or data within other files. Suppose we have large number of distinct integer numbers stored in some structure such as array. If you are looking an integer key in array, then  how can you determine whether key is in array or not?

 

 

10:55 - 11:15 Saturday.   TCCW 129

 

Raman Arora, Western Kentucky University (G)

 

Calculus of Variations in Economy

 

Calculus of variation is one of the oldest methods in the field of applied mathematics. My aim has been to use this method in solving complex economic issues. I have tried to create a model that gives us an indication of the effect of education on the growth rate of the country by using the Euler-Lagrange equations.

 

 

 

 

 

10:55 - 11:15 Saturday.   TCCW 116

 

Jonny Groves, University of Kentucky (G)

 

 Nice Polynomials with Four Roots

Nice polynomials are polynomials whose coefficients, roots, and critical points are integers.  If the coefficients, roots, and critical  points of p(x) are rational numbers, then we call p(x) Q-nice.  To begin, we give the relations between the roots and critical points for all polynomials with four roots.  We then give the relations between the roots and critical points for all symmetric polynomials with four roots.  Using these relations, we derive a formula for all Q-nice symmetric polynomials with four roots.  The existence and number of equivalence classes of such polynomials are also discussed.  We conclude by giving several examples that illustrate our results.

 

 

Invited Lecture

 

11:25 - 12:15  TCCW 129

 

Jack Robertson, Washington State University

 

The Cake-Cutting Problem:

Be Fair If You Can As quickly As You Can

 

Since the problem was formally introduced to the mathematical community in 1947 by Professor Hugo Steinhaus, it has blossomed and has an extensive literature. We will survey what is known, using different definitions of “fair” and using different classes of algorithms. (Come learn how to make a risk-free bet!)}

 

 

 

 

 

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The Department of Mathematics gratefully acknowledges funds from the MAA NSF-RUMC (NSF Grant DMS-0241090) for support of student speakers.

 

 

 

 

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The Department of Mathematics wishes to thank the following publishing companies and representatives for their support and participation. 

Please visit their displays in TCCW 125 B.

 
 Brooks/Cole
 Kathleen Fitzgerald

                                           

  W. H. Freeman
  Melissa Valentine



  Houghton Mifflin Company
  Mike Schenk


   Pearson Learning Group
   Roz Paul


 
    A. K. Peters
    Susannah Sieper