Contributed Presentations:

6:00 - 6:20 Friday. TCCW 125 C
Buddy Lagani, Western Kentucky University (U)
On the Cardinality of a Collection of Topologies

A topology may be generated on a finite set by manipulating a quasi-order relation on that set.  We will discuss the relationship between these order
relations and the topologies that may be imposed on a finite set.  Finally, we will describe methods for determining the number of topologies on a given set.

6:00 - 6:20 Friday. TCCW 129
Noah Clemons, Vanderbilt University (U)
From Coarse to Finest Level: An Exploration of Uni/Multi Wavelets

In keeping light of the conference theme, I plan to show how elementary wavelet transforms (i.e. Haar and Daubechies) carry analogous properties that are effective for imaging and computer vision. High resolution data sets will be used to show how these transforms become powerful tools for image analysis. I will then show how the basic properties of Haar and Daubechies wavelets are assumed by the GHM family of multiwavelets. No previous knowledge of wavelet theory is needed, as I plan on making this a very visual demonstration. The premise underlying all visual aids will be how Haar's initial device to study the L2 space has become the cornerstone for one of the "hottest fields" in mathematics today.

6:00 - 6:20 Friday. TCCW 116
Christopher Winfield, Western Kentucky University (F)
Quantum Scattering and the Lippmann-Schwinger Equation

Two-particle potential scattering in the 3-dimensional quantum situation will be introduced by a heuristic derivation of the related Lippmann-Schwinger integral equation. Then, more rigorous connections to scattering wave operators and to the spectrum of the Schroedinger operator involving so-called Rollnik potentials will be discussed. Modifications of techniques for Rollnik potentials to study non-Rollnik potentials will also be discussed.

6:25 - 6:45 Friday. TCCW 125 C
Jonny Groves, Western Kentucky University (G)
Absolutely Flat Idempotents

A real n-by-n idempotent matrix A with all entries having the same absolute value is called absolutely flat. The possible ranks of such matrices are considered along with a characterization of the triples: size, constant, and rank for which such a matrix exists. Possible inequivalent examples of such matrices are also discussed.

6:25 - 6:45 Friday. TCCW 129
Bruce Kessler, Western Kentucky University (F)
Undergraduate Mathematics Never Goes Away: A Differentiable, Orthogonal Scaling Vector

The purpose of the talk is two-fold: I would like to illustrate a new scaling vector I recently developed and its application to digital image compression, and, at the same time, I would like to illustrate the importance of the mathematical content in our undergraduate courses in the discovery. The issues in finding the scaling vector, finding a useful prefilter, and applying the basis to digital images will be discussed and explained. In the course of the explanation, I (possibly we) will identify the undergraduate mathematics content that made the discovery possible. Although reporting on a fairly high-level result, the talk will be at an undergraduate level. Specialized definitions and theorems related to wavelet theory will be illustrated visually, so that no special background in wavelets is required.

6:25 - 6:45 Friday. TCCW 116
Jamie Johnson, Western Kentucky University (G)
Representing Numbers with Continued Radicals.

It is well-known that the golden ratio can be represented as the continued radical; other continued radicals of this form are examined. An algorithm for creating a continued radical to represent any given number is explored, and Mathematica code for approximating such representations is discussed. Further results involving continued radicals are also presented.

7:00 - 7:20 Friday. TCCW 125 C
Anne M. Raymond, Bellarmine University (F)
Prime Numbers, Generalization, and Problem Solving in Mathematics Education

In a mathematics course for people preparing to be elementary teachers, my primary objective is to have students revisit topics that they will ultimately teach at the elementary level. However, they are challenged to revisit these elementary topics from a much more advanced perspective. In keeping with the theme of "elementary topics which recur in advanced settings," I plan to share some of my favorite problem-solving activities from this course related to understanding prime numbers. These activities and problems are specifically designed to have students engage in elementary topics from a more adult,
problem-solving, understanding. They focus on generalizations of prime factorizations, classifications of numbers by prime factorization, and proofs related to primes.

7:00 - 7:20 Friday. TCCW 129
Matt Estes, Tennessee Technical University (G)
An Introduction to the Theory of Formal Languages and Parsing

The notion of Formal Languages and their grammars will be introduced and defined and their relevance to mathematics and other fields explained. The topic of parsing will be introduced as an algorithmic method of exploiting the information contained in the grammar to reconstruct a particular sentence from the Language.  This talk will be based on material in the first three chapters of "Parsing Techniques: A Practical Guide" by Grune, et al. freely available online at: http://www.cs.vu.nl/~dick/PTAPG.html.  Material will also be presented from "Grammars for Programming Languages" by Cleaveland as well as "A Decidability Criterion for van Wijngaarden Grammars" by P. Deussen, Acta Informatica, vol. 5, pg. 353-375.

7:00 - 7:20 Friday. TCCW 116
Mark P. Robinson, Western Kentucky University (F)
Taylor's Theorem for Functions of Several Variables

Taylor's Theorem for functions of a single variable is a standard topic in undergraduate calculus. In this presentation we review the single-variable case and then consider the higher-dimensional analogue, Taylor's Theorem for functions of several variables. The basic theory, illustrative examples, and an application of interest will be examined.

7:25 - 7:45 Friday. TCCW 125 C
Amber De More, Austin Peay State University (U)
Mathematics as a Language Course.

In this paper, we discuss mathematics as a language course. The necessity for such a course through examples of discrepancies between mathematics and English is emphasized. Furthermore, it is revealed that the average student on today's college campus misses these discrepancies and an attempt is made to raise the level of awareness of the content of such a course. In particular, we will address specifically the ideas to be taught, methods for teaching such a course and a suggestion for the way it ought to be graded. Finally, we will conclude with statistics on the effectiveness of this course and include a few quotations from students of this course as well as from those who have taught it.

7:25 - 7:45 Friday. TCCW 129
Deane Arganbright, University of Tennessee at Martin (F)
Mathematical Modeling with Excel: Creative Models Linking Fundamental Mathematics and Stimulating Applications

We give a brief presentation of how fundamental mathematical ideas are incorporated into significant modeling procedures through Microsoft Excel. Illustrative models are designed using difference equations and incorporating animated spreadsheet graphics. Illustrations will come from growth and related models with a brief look at chaos examples.

7:25 - 7:45 Friday. TCCW 116
Nick Newman, Troy State University (U)
New Harmonious Labelings of Graphs

Let G = (V,E) be a simple graph in which |V| = n and |E| = m. G is said to be harmonious if there is an injection f:V -> { 0,1, . . . , m-1 } such that for two adjacent vertex-pairs x,y and u,v, [f(x) + f(y)] mod(m) * [f(u) + f(v)] mod(m). When trees or graphs involving trees are under consideration, it is permissible for the function to map exactly one pair of vertices to the same number. In this work, we demonstrate such a function for the disjoint union of an odd cycle and a path on three vertices, thus proving that it is a harmonious graph.

Paul Sally, Jr., University of Chicago
Triangles, Polygons, and Shmuzzles

We discuss the general concept of tessellation and some of the more interesting results.

Contributed Presentations Saturday, November 22

8:30-8:50 Saturday. TCCW 125 C
Chris Christensen, Northern Kentucky University (F)
Blowing-up Points?

We begin with the problem of integrating functions f(z) that have multiple branches--a problem that Riemann solved (1851) by means of the introduction of the Riemann surface. Riemann imagined the surface being a number of sheets lying above the complex plane. On each of the sheets, the function would have only one value. That problem connects with algebraic geometry in 1863 with Cremona's investigation of birational transformations of algebraic curves f(x, y)=0. Algebraic geometers desired to have an algebraic method of separating points. This is often referred to as "blowing-up points."

8:30-8:50 Saturday. TCCW 129
Sheena Richards, Troy State University (U)
Strategies for Another Variation of Nim

In this talk the speaker generalizes a variation of the game called Nim, and then examines winning strategies for each player. Among the topics discussed are:
1. For what number of pebbles can the players be guaranteed a "fair game?"
2. For what number of pebbles are the outcomes of the game predetermined?
3. Upper and lower bounds for the size of the largest pile.
4. Patterns, conjectures, and counter-examples for generating "closed sets."

8:30-8:50 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 December 1999 issue of The American Mathematical Monthly. Such polynomials are called nice polynomials. The earliest paper on this subject was published by Karl Zuser in 1963. 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, where we just need to add the most recent continuation of their work published by Eugene Flynn in 2001, and the paper published by Johann Walter in 1987.

In this talk, we will present a new method to deal with nice polynomials, and we will present new properties and examples obtained with our method. While there is still a long way to go to find a complete description of the set of nice polynomials, we have obtained a lot of new results in a short time, thanks to our new method, and it seems that many more results can be obtained in the near future. Copies of our paper, submitted for publication last July, will be available after the talk.

8:30-8:50 Saturday. TCCW 125 B
Christopher McMahan, Austin Peay State University (U)
A Collocation Method for a Special Class of Higher Order Differential Equations

In this paper, a collocation method constructed using canonical polynomials as basis functions is studied for a special class of higher order differential equations. The method is applied iteratively to solve initial value problems, for instance, the nonlinear pendulum problem that is otherwise solved by linearization. Pertaining to boundary value problems, the method is applied directly using Gaussian elimination with partial pivoting for linear boundary value problems and the modified Newton-Raphson method for nonlinear boundary value problems. The convergence analysis of the method is discussed and numerical examples are given which shows the accuracy of the method.

8:55-9:15 Saturday. TCCW 125 C
Jesse Pratt, Northern Kentucky University (U)
Symmetry Analysis of a Generalized KdV Equation

Symmetry analysis is a very powerful method for solving nonlinear partial differential equations. The KdV equation is one of the more celebrated of these equations. This talk will discuss symmetries of this equation and a reduction for a special case.

8:55-9:15 Saturday. TCCW 129
Kathy Cash, Brescia University (U)
Probabilities in the Game of Monopoly

Using the ideas of steady state probability, I have attempted to calculate the probability of landing on each of the spaces of a Monopoly game board. By using the rules of the game and making a few assumptions, I wrote state equations for each of the spaces and used Mathematica to solve these equations. Finally, I take a look at the assumptions that I made and the problems these may have caused my answer and I also touch on some other ways in which I could extend the problem.

8:55-9:15 Saturday. TCCW 116
Haslinda Ibrahim, Southern Illinois University at Carbondale (G)
Orthogonal Latin Squares

The study of orthogonal latin squares goes back to 1782 when Euler considered the famous Euler Officer Problem. This famous problem has inspired mathematicians to further investigate orthogonal latin squares. Recently Gary L. Mullen suggested that the orthogonal latin squares will be a possible candidate for the "next Fermat problem". Thus, in this paper we will explore diverse aspects of orthogonal latin squares.

Bernd Schröder, Louisiana Tech

Order from 0 to w + 1

Order relations abound in mathematics. Yet order is often taken for granted. The natural, rational and real numbers as well as the integers are linearly ordered. Order properties are often used in passing (such as when proving the squeeze theorem) and when order is mentioned, one usually thinks of a linear order in which all elements are comparable to each other.

This talk shall give a flavor of the theory that arises once incomparability of (pairs of) elements is allowed. (Nonlinear) order relations occur, for example, whenever (set) containment is investigated. Via Zorn's Lemma (and via the continuum hypothesis, which is about linear orders), order-theoretical notions are near the heart of set theory. Simple-sounding tasks, such as counting the number of possible order relations for an n-element set, are actually quite hard (there still is no exact formula). In analysis, the ordering of function spaces allows the proof of existence theorems for solutions of certain types of integral equations. This list of observations about orders can and will be extended, in a coherent fashion, as close to w as the speaker's ability and the audience's patience will allow.

Contributed Presentations Saturday, November 22

10:30 - 10:50 Saturday. TCCW 125 C
Ben Harwood, Northern Kentucky University (U)
Extensions and Enlargements of Groups

Group extensions play a pivotal role in group theory, even though most people don't realize they are working with them. In this talk, we will give a quick intorduction to group extensions, then generalize the concept to a group enlargement, and finally show that extensions and enlargements of quasi p-groups by quasi p-groups are again quasi p-groups.

10:30 - 10:50 Saturday. TCCW 129
Mark Walters, Miami University (U)
Arc Length and Surface Area - Are We on the Same Page?

In calculus textbooks, formulas are developed for the length of a curve in the plane and for the area of a surface in three-space. Many textbooks, including Stewart's calculus book that we use at Miami University, take different approaches to these two very similar mathematical situations. One approach connects the dots along a curve to get a polygonal approximation, while the other approximates via tangential considerations. This raises the question of why we don't take the same approach in both situations. We shall look at these differing approaches, compare them, and prove that each leads to the expected mathematical conclusions. Upon doing so, we remain with the curiosity of this common inconsistency in many calculus textbooks.

10:30 - 10:50 Saturday. TCCW 116
Jeffrey L. Overbey, Southeast Missouri State University (U)
The Order of GL(d, Z/mZ) and Its Involutory Subset

The group GL(d, Z/mZ) arises frequently, particularly in applications such as the Hill cipher. We develop a proof of the order of this group using only undergraduate mathematics, considering first the case of a prime modulus, then the cases of prime power and composite moduli. We briefly review results from the literature on the order of this group's involutory subset. Finally we observe the effects of change in dimension and modulus on the numbers of involutory, invertible and general matrices.

10:55 - 11:15 Saturday. TCCW 125 C
Jason Harrington, Western Kentucky University (U)
Distances in Cantor-Like Sets

It is well-known that any number in [0,1] can be realized as the distance between two points of the Cantor set. We investigate whether every number in [0,1] can be realized as the distance between points of Cantor-like sets formed by removing sections other than middle thirds.

10:55 - 11:15 Saturday. TCCW 129
Claus Ernst, Western Kentucky University (F)
What Kind of Knot Can You Tie with a Rope of Fixed Length?

Given a rope of fixed length and a fixed knot K, a natural question one can ask is: Can I tie the given knot K with the given rope? How do I know whether my rope is too short to tie the knot K? There will be lots of nice pictures and no proofs. The talk is intended to give you an overview. This problem is easy to state but it is actually very difficult to solve for even the simplest knot.

10:55 - 11:15 Saturday. TCCW 116
Jerzy Wojdylo, Southeast Missouri State University (F)
The Density of Invertible Matrices Over Zm

Following the formula for the cardinality of GL(d, Z/mZ), we define a function that expresses the "density" of the invertible d¥d matrices in the ring of all d¥d matrices over Zm. We investigate properties of this function, in particular its behavior when the modulus m is prime or many prime divisors, when dimension d of the matrices is large and suggest some further investigations of the graph of this function.

Paul Sally, Jr., University of Chicago

Problems in Mathematics from Zero to Infinity

We begin with some very elementary problems in mathematics which lead to very sophisticated ideas in a short period.

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

The Department of Mathematics wishes to thank the following companies for their support and participation. Please visit their displays in TCCW 125 A