Abstracts of Contributed Papers

s of Contributed Papers Listed Alphabetically by Author ! Friday 4:15 pm Room 1 One Semester Calculus with the TI-92 Barbara Ashton Faculty: Wittenberg Univ Abstract: In this presentation, I will talk about my experiences teaching a one semester calculus course for manangement and biology majors. The course takes full advantage of the power of the TI-92 calculator. The discussion will include the topics covered, materials used, student assessment and students evaluations of the course and technology. ! Saturday 10:40 am Room 2 The Application of Group Theory to Chemistry Isabel Averill Student: The Univ of Akron Abstract: In Abstract Algebra, it is possible to define a group based on the rotations of a polygon called a Dihedral Group. Applying the same principle to the rotation of a molecule, we can find the point group of the molecule, and know a little bit more about its chemical properties. In this talk, we will show several examples of such groups and briefly discuss their applications. ! Saturday 10:20 am Room 5 Optimization in Precalculus Floyd Barger Faculty: Youngstown State Univ Abstract: Standard precalculcus syllabi include graphing functions with local extrema, but do not include methods for locating these points. Two methods appropriate for precalculus are included. This material provides reinforcement of other topics in the course. One method uses graphing transformations and the arithmetic-geometric mean inequality, focusing on the x-coordinates. The other method finds the y-coordinates first. ! Friday 4:55 pm Room 2 1/19th of a generating function Curtis Bennett Faculty: Bowling Green St Univ Abstract: When students were asked to find the 18th digit of 1/19th without doing pencil and paper work, they discovered two interesting patterns in the decimal expansion. In this talk we will discuss these patterns in the and their relationships to generating functions. ! Friday 4:35 pm Room 5 The Analysis of Truth Tables to Help Ellen get Home Beka Black Student: Ashland Univ Abstract: Truth tables describe a logic function by listing all possible combinations of input values and indicating, for each combination, the output value. They also help to exercise critical reasoning along with mathematical thinking. In the article, A Truth Table on the Island of Truth-tellers and Liars, I will analyze four different situations and show how truth tables can provide us with the correct conclusion. ! Friday 5:55 pm Room 3 Weak Associative Laws in Quasigroups Christopher Bomba Student: Univ of Dayton Abstract: Quasigroups are algebraic structures related to Latin Squares. It is well known that every quasigroup satisfying the full associative law is a group. Quasigroups satisfying certain weak associative laws are loops. (A loop is a quasigroup that contains an identity element.) The question of which weak associative laws of size four imply a quasigroup is a loop was settled by Prof. Kenneth Kunen of the University of Wisconsin—Madison, in his paper “Quasigroups, Loops, and Associative Laws”, [J. Algebra 185 (1996)] . I have extended his results in an examination of weak associative laws of size five. ! Friday 4:15 pm Room 5 Sines and Chords Justus Brake Student: Cleveland St Univ Abstract: How sinusoidal functions combine in music to form chords. Harmony and dissonance are products of simple, or complex ratios of periods. ! Friday 4:55 pm Room 4 Thermal Damage Model to Predict Cell Death for MRI Guided Surgery Miyuki Breen Streetsboro, OH 44241 Student: The Univ of Akron Abstract: Certain cancerous tumors do not respond to traditional cancer treatments. One of the alternative treatments is to destroy the tumor with heat. By using MRI image guided surgery, the treatment can be minimally invasive. However, the problem arises when MRI data are used to predict cell death. I developed a probability model to determine cell death from the temperature history. I will investigate maximizing the region of cell death by analyzing the simulation result for different temperatures. ! Saturday 10:20 am Room 2 A new way of looking at primitive roots Mihai Caragiu Faculty: Ohio Northern Univ Abstract: We will present a new result about primitive roots, based on a deep result from mathematical logic (modeltheoretic algebra). Namely, we will search for a first-order formula F(x) written in the language of rings, with the following property: for any prime p and any element g of the Galois field GF(p), g satisfies F(x) if and only if g is a primitive root in GF(p). We will explain why such a search cannot be successful. Thus, one may say that the primitive roots elude a unified first-order definition. ! Saturday 11:40 am Room 3 Four Equivalent Forms of Cauchy-Riemann Equations Rebekah Carter Student: Otterbein College Abstract: Cauchy-Riemann equations play an important role in complex analysis. These equations are named in honor of A.L. Cauchy who discovered and used them, and in honor of G.F.B. Riemann who used them in the development of his theory of functions of a complex variable. These equations are used today to determine if a function of a complex variable is differentiable. However, most text books, such as Complex Variables and Applications by James Ward Brown and Ruel V. Churchill, used at Otterbein College only provide the student with two equivalent equations. The purpose of this presentation is to provide students and professors with four equivalent forms of the Cauchy-Riemann equations. ! Friday 5:35 pm Room 5 How a couple of crazies tried to outfox the Great Whoosie Thomas Dence Faculty: Ashland University Abstract This talk reflects a true life, yet humorous and off-beat relationship between the Fibonacci numbers 1,2,3,5,,8,... and the sequence of squares 1,4,9,16,... .In addition to being entertaining and mathematically correct, the talk will be appropriate for you to take home and share with your classes. ! Friday 4:15 pm Room 2 A Last Digit Effort Ian Deters Student: Malone Coll Abstract: When one raises positive integers to positive integer powers, a pattern occurs among the last digits of the resultant numbers. In this talk, I will define what is meant by a pattern, give some examples, and show that if a base n number system satisfies certain properties, then such patterns will always occur among the last digits of positive integers raised to positive integer powers. ! Friday 4:35 pm Room 1 Dragons and other Creatures on a TI-89 Bob Dieffenbach Faculty: Miami Univ-Middletown Abstract: Iterations of the Dragon Design are used to separate chapters of the original Jurassic Park novel. An program to recreate these designs on a TI-89 will be described. The limit of the Dragon Designs appears to be a fractal. Mandelbrot's definition of fractal dimension will be used to decide. Stage 5 of the Dragon Design ! Saturday 11:00 am Room 1 Magic in Robertson Squares Christine Faller Student: Ashland Univ Abstract: Magic squares have many interesting properties that date back for centuries. Come join me as the magic of Robertson squares is unleashed and discover what Pythagorean triples have to offer. ! Saturday 10:20 am Room 1 Divisibility of Generalized Fibonacci Sequences by 2n-1 Leah Frazee Graduate Student: Miami Univ-Oxford Abstract: This talk presents results concerning Fibonacci-like sequences of the form a_n = b*a_n-1 + c*a_n-2, n > 1, where a_0 and a_1 are given and b and c are chosen as integers. We will examine the divisibility of the nth term of such sequences by 2n-1. Specifically, we will consider the two cases where b=c=1 and where b=1, c=3 and determine what constraints must be placed on 2n-1 in order for 2n-1 to divide an. ! Saturday 11:20 am Room 1 Scheduling Tournaments Using Combinatorial Designs David Gerberry Student: Youngstown St Univ Abstract: At first glance, the process of scheduling a season or tournament does not seem to be a mathematical one. However, when facing a list of conditions a schedule must satisfy, one quickly realizes that a systematic approach is needed to attack the problem. I will discuss how combinatorial designs can be used to create tournaments that satisfy various criteria. ! Saturday 11:40 am Room 4 Save a lot or a little! Sarah Grove Student: Youngstown St Univ Abstract: When you go out into the world and start to make pricier purchases, is it worth your while to pay in full or to make payments? Of course we all know that with interest you will end up paying more if you take out a loan. However, is that extra money worth the hassle of saving up until you can make a purchase in full? How does this decision affect your financial status for the rest of your life? ! Saturday 11:40 am Room 2 Automated Generation of Twoand Three-Dimensional Fractal Tilings Jonathan Hafner Student: The Univ of Akron Abstract: We will open with a basic introduction to the generation of "rep tiles," fractal tilings of the plane. We will then introduce a simple algorithm for automating the often-tedious task of determining the principle residue vectors and will demonstrate some of its results. Finally, we will extend this method to generating "tilings" in threedimensional space. ! Friday 4:55 pm Room 5 Kepler's Conjecture Kelli Hall Student: Marshall Univ Abstract: What is the densest possible packing in the second, third and other dimensions? Johannes Kepler's conjecture about the densest possible packing of spheres has mystified mathematicians for hundreds of years. I will briefly explain the problem and it's relevance in mutiple dimensions, talk about past strategies and attempts at legitimate proofs, and give examples of practical applications. ! Friday 6:15 pm Room 5 An Introduction to Elliptic Curve Cryptography Amy J. Herron Student: Miami University Abstract: This presentation will explain what an elliptic curve is, show that an elliptic curve over a field forms an Abelian group, and provide a concrete example of how elliptic curves are used to encrypt messages. ! Friday 4:35 pm Room 3 Singular Value Decomposition Visualization: a New Web-based Version Thomas Hern Faculty: Bowling Green St Univ Abstract: We will demonstrate an applet version of an SVD visualization previously written in Matlab. This was developed using David Meel's method of writing web applets using Geometer's Sketchpad. The advantage is obvious: anyone with a Java enabled web browser can use it. ! Friday 4:55 pm Room 1 The classroom setting does it make a difference? Aparna Higgins Faculty: Univ of Dayton Abstract: In the last two years, I have taught mathematics in classrooms with very different set-ups. I will discuss two of those situations -the typical classroom at the United States Military Academy, and the Studio at the University of Dayton. I will comment on how I felt compelled to change my teaching to best use the room I was in, and I will reflect on whether I believe this improved my teaching or my students' learning. ! Friday 5:35 pm Room 2 On a Theorem of Touchard and the Form of Odd Perfect Numbers Judy Holdener Faculty: Kenyon Coll Abstract: Perfect numbers have been studied for well over two millennia, yet the nature of their very existence remains a mystery. Are there infinitely many perfect numbers? Are there infinitely many even perfect numbers? Are there any odd perfect numbers? These questions have stood the test of time and remain open today. In this talk, I will present a surprisingly elementary proof of a well-known theorem of Jacques Touchard regarding the form of odd perfect numbers. ! Saturday 11:00 am Room 3 A square root algorithm Thomas Jonell Student: Ohio Northern Univ Abstract: We present a new algorithm (implemented by a C++ program) which uses a class of recurrent sequences to calculate square-root approximations of integers. ! Saturday 10:40 am Room 5 The Parity Theorem & The Magic 15 Puzzle Christopher Jones Student: Youngstown State Univ In this presentation, we discuss the Magic 15 Puzzle created in 1878 by Sam Loyd, the famous American puzzlist. We will give a solution to the puzzle based on elementary group theory and the Parity Theorem. An elegant proof of the theorem will also be given. ! Friday 5:15 pm Room 3 The Roots of Unity Katie Jones Student: The Univ of Akron Abstract: The Fundamental Theorem of Algebra guarantees that any polynomial with complex coeffiecients and degree greater than or equal to one can be factored completely into linear terms over the complex numbers, and thus give its roots. For this talk, I will deal with the special case when this polynomial takes the form of some complex number to an integer power n, minus one. I will define the roots of unity for the general nth case and prove that the nth roots of unity form a cyclic group under multiplication with order n. I will solve an equation finding its roots of unity and diagram them on the unit circle. There will also be some application shown between the roots of unity and polar coordinates. ! Friday 5:55 pm Room 5 Orthogonality in Function space Aditya Joshi Student: Cleveland St Univ Abstract: This talk explores the concept of Orthogonality as applied to functions of real variables. We begin by introducing the concept in the framework of basis vectors in position space. We then extend the idea to Real functions and polynomials and discuss its meaning in this context. We conclude by discussing some applications of a few orthogonal functions and polynomials to Theoretical Physics. ! Saturday 11:00 am Room 5 A Model for Biochemical Switches Jacob Land and Amanda Marple Students: Cleveland St Univ Abstract: In this session the mathematics behind the stripes on a zebra and the wing patterns of butterflies will be examined. It will be determined using a model for a biochemical switch, a gene which produces a pigment or other gene product when the concentration of chemical signal substance reaches a threshold. We use Maple to analyze this model ! Saturday 10:40 am Room 4 An application of a Markov chain. Megan Lipiec Student: Cleveland St Univ Abstract: A Markov chain is a sequence of probability experiments. One of the main conditions of a Markov chain is that the outcome of any one experiment only depends on the present state. In this discussion, an application of a discrete-time Markov chain will be presented. ! Friday 5:15 pm Room 2 Slick Cyclic Numbers Erika Loomis Student: Ashland Univ Abstract: There are many natural phenomena that occur in mathematics. Cyclic numbers are one of these phenomena. A cyclic number is an integer of n digits with the following characteristic: when multiplied by a number from 1 to n, the product has the same digits as the original number and in the same cycle. Not only are these numbers slick but so is the method to find them. ! Friday 4:15 pm Room 4 Identification Numbers, Check Digits, and Error Correction Lew Ludwig Gambier, OH 43022 Faculty: Kenyon Coll Abstract: We will discuss various types of identification numbers (ISBN, UPC, etc.) and how their respective check digits are computed. Specifically, we will discuss a method involving a modified ISBN scheme used by my computer science students. Given a ten-digit ISBN, they were able to detect a single digit error and correct it. This technique is a nice application of some basic number theory and can be used in classes ranging from computer science to liberal arts. ! Saturday 11:40 am Room 1 Pythagorean Triples and Plouffe's constant Barbara Margolius Faculty: Cleveland St Univ Abstract: A primitive Pythagorean triple is a Pythagorean triangle with sides whose lengths have no common factors. (7,24,25) is a primitive Pythagorean triple, but (14,48,50) is not. We use a sequence of these triples to prove that Plouffe's constant, arctan(1/pi) is irrational. ! Friday 5:35 pm Room 3 Is it Possible to Square this Circle? Benjamin Marko Student: The Univ of Akron Abstract: My talk is on the classical math problem of Squaring the Circle, that is, if one is given a line of length one, can a square with the exact same area as the unit circle be constructed with only a straightedge and compass. I will start by giving a very brief history of this problem, including a few facts about the intense obsession that the Greeks had with this problem. I will then explain the idea of constructible and algebraic numbers by relating them to the roots of polynomials with rational coefficients. I will go into the idea of Euler's Formula, and then state Lindemann's famous theorem which put all of the pieces of this problem together once and for all. ! Saturday 11:20 am Room 2 Publishing Maple Animations on the Web Felipe Martins Faculty: Cleveland St Univ Abstract: Everybody is familiar with the "unending annoying animation" phenomenon from banner adds in web pages. The culprits are animated GIFs, which contain a series of images that browsers cycle through. Maple is capable of exporting animations as GIFs, and in this talk we present a Java applet that allows control the animation with VCR buttons, much in the way it is done inside Maple. As an added benefit, animations are compacted for faster download times. ! Friday 5:55 pm Room 4 In the Ballpark of Perfection Traci McLaughlin Student: Ashland Univ Abstract: In 1998 and 1999 two New York Yankee players pitched perfect games. Along with player batting average and on-base average, probability theory is used to explore the chance that a perfect game will be pitched on any given day. Through the events in baseball history the relationship between theoretical and experimental probability will be illustrated. ! Friday 4:15 pm Room 3 Linear Algebra Web Modules: The future is now David Meel Faculty: Bowling Green St Univ Abstract: This talk will present several new linear algebra web modules useful in the examination of concepts such as linear transformations, change of bases, and coordinate systems. By utilizing Geometer's Sketchpad 4.01's ability to produce applets, web-based modules can be developed to help students explore problematic concepts of linear algebra. ! Saturday 11:20 am Room 5 PascGalois Triangles and Visualization Techniques of Abstract Algebra Joel Rabe Student: The University of Akron Abstract: PascGalois Triangles provide a unique method of visualizing the complexities of group theory and an effective teaching tool. The basic idea behind PascGalois Triangles is to apply the methodology of Pascal’s Triangle to finite groups. By indexing the elements of groups by specific colors, a visual picture of the group structure can be created. Applications of the triangles include identifying group generators, finding homomorphic and isomorphic images of a group, identifying whether or not a group is abelian, and determining all possible subgroups of a group.. ! Friday 5:35 pm Room 4 The Luck Of The Draw: More than Kicking and Punching Lorenzo Rashid Student: Cleveland St Univ Abstract: When faced with conflict, should you rely on your skills of kicking and punching or the luck of the draw. This presentation examines the luck of the draw in a Martial Arts tournament. ! Friday 4:35 pm Room 4 My summer experience in bioinformatics Lindsay Sabik Student: Kenyon Coll Abstract: In this talk, we will discuss my summer internship at Texas Children's Hospital sponsored by the Baylor College of Medicine Summer Medical and Research Training (SMART)Program. While there I worked with Dr. Chris Man, on a bioinformatics project involving genomic data from pediatric cancer patients. He and I attempted to identify patterns in the false positives generated when RNA from certain samples underwent a particular amplification process. I will discuss my original interest in this field, how I became involved in this program and the mathematical background needed in such an area. ! Saturday 11:00 am Room 2 Self-complimentary graphs and Ramsey numbers Eric Schulte Student: Kenyon Coll Abstract: In this talk we will examine the connection between self-complimentary graphs and Ramsey numbers. After a brief introduction to the basics of graph theory, we will look at the properties and symmetries of selfcomplimentary graphs. Then we will explore how self-complimentary graphs are used to solve Ramsey number problems. No previous knowledge of graph theory is assumed in this talk. ! Saturday 11:00 am Room 4 How the ENIAC took a Square Root Brian Shelburne Faculty: Wittenberg Univ Abstract: The ENIAC (Electronic Numerical Integrator and Computer) is considered to be the world's first electronic computer. However it could only store twenty 10-digit decimal numbers and was programmed by wiring the computational units together. These limitations made it very unlike today's stored program computers. The ENIAC had hardware to add, subtract, multiply, divide and take a square root. This last operation is interesting since computers normally don't do square roots in hardware. So given the "limited" capability of the ENIAC, how did it take a square root? ! Saturday 11:40 am Room 5 Mathematical Pointilism James Shuster Student: Youngstown State University Abstract: In this paper we will address the problem of what would happen if there existed an infinite set of points, S, such that the distance between any two points is a whole number. In considering two possible cases in which the set of points could be arranged, we will show that if such a property held on S, then the points must lie in a straight line. ! Friday 4:55 pm Room 3 Quaternions Robert Shuttleworth Student: Youngstown St Univ Abstract: William Hamilton invented quaternions as an algebra in which quotients of vectors are well defined. Today, computer graphicists use quaternions, represented by orthonormal matrices, to rotate objects and spline camera paths. Examples of real-time interactive quaternionic animation will be presented. This research was developed at illiMath2001, an NSF-VIGRE supported REU program at UIUC. ! Friday 5:15 pm Room 1 The Absolute Value of Teaching Math: Turning Negatives into Positives Dave Sobecki Faculty: Miami Univ Abstract: We're all very familiar with a handful of questions that students ask, and most of us like to avoid: "Is this going to be on the test?"; "Why do we have to do story problems?"; and the crown jewel, "Why do I have to know this stuff? I'm never going to use it". I've devised some answers to these and other questions that turn them into positive learning and motivational opportunities. ! Saturday 10:20 am Room 3 Lights Out and its variants Jon Stadler Faculty: Capital Univ Abstract: Lights Out and Lights Out 2000 are handheld electronic games played on a 5x5 array of lights. Each time a light is pressed, its state and the states of its neighbors cylce through to the next state red or off for Lights Out; red, green or off for Lights Out 2000. To win, one must turn off all of the lights. We will use elementary linear algebra to solve each game and will discuss a generalization in which each button takes on an integer value. ! Saturday 11:00 am Room 5 Interesting Problems on the 2002 AMC-12 Exams David Stenson Faculty: John Carroll Univ Abstract: For the first time there were two versions of the AMC-12 exam given two weeks part in February. A summary of the results will be presented and the problems that challenged the top students will be discussed. ! Friday 5:15 pm Room 4 An Analysis of Bertrand's Paradox in Geometric Probability Hilary Stork Student: Otterbein College Abstract: Bertrand Russell has presented a mathematical problem titled "Bertrand's Paradox": Given a circle. Find the probability that a chord chosen at random be longer than the side of an inscribed equilateral triangle. Thia problem has more than one solution. I would like to present an analysis of this problem. ! Friday 5:15 pm Room 5 Sequences, differentials, divergence, and convergence David Stroup Student: Cleveland St Univ Abstract: In this talk we look at sequences from the perspective of difference and differential equations. ! Saturday 11:20 am Room 4 Fractal Exploration: A Look at Mandelbrot and Julia sets in 2 and 3 dimensions Matt Valerio Student: Ohio Northern Univ Abstract: Fractals are fascinating designs of infinite complexity and structure. Two of the most popular designs, the Mandelbrot Set and the Julia Set, will be explored. Example images will be presented and discussed, as well as the algorithms used to generate each. The Mandelbrot set will be generalized to 3 dimensions and exploration techniques will be discussed. ! Saturday 10:40 am Room 1 On a Fibonacci identity Nicholas Vidovich Student: Ohio Northern Univ Abstract: We discuss a Fibonacci identity for which we provide both an obvious direct proof based on Binet's formula, as well as a combinatorial proof based on the interpretation of Fibonacci numbers in terms of 0-1 strings with no consecutive ones. ! Saturday 10:20 am Room 4 Archways into Mathematics: Teaching a new Carpenter an old Trick Sarah Wetzel Student: Ashland Univ Abstract: When a carpenter attempts to create a wooden archway, a mathematical trick is required. This trick, or technique, and its accuracy will be tested. ! Friday 5:55 pm Room 2 The Complexonacci Numbers, a New Type of Fibonacci Sequence Coral Wheeler Student: The Univ of Akron Abstract: Let Fn denote the Fibonacci sequence, and Ln denote the Lucas sequence. The Complexonacci numbers, a sequence of complex numbers Fn + iLn, are defined and several identities are stated and proven. It is shown that the gcd of two consecutive Complexonacci numbers can be found, and that the result is the same for all natural numbers. It is also shown that the ratio of two consecutive Complexonacci numbers approaches the golden ratio. Finally, Complexonacci Polynomials are formed and patterns in the coefficients are discussed. ! Saturday 11:20 am Room 3 A real world application of the catenary function. Aric Whitington Student: Kenyon Coll Abstract: We will explore our attempts to solve the following real world problem: Find the area beneath a chain hanging from two fixed points. We used three different approaches in solving this problem. Our first, Riemann sums, gave a good estimate, but we wanted something more accurate for our purposes. Next, we decided to treat the chain like a parabola and apply the Fundamental Theorem of Calculus (FTC). This approximation was much closer, but we discovered that the shape of a hanging chain is actually represented by the catenary function, y=a*cosh(x/a)+b. However, before we could apply the FTC, we needed to find the parameters a and b. This turned out to be nontrivial. We will discuss our solution to finding a and b and ultimately the area under the chain. ! Saturday 10:40 am Room 3 The Restricted Three-Body Problem Lisa Zimmerman Student: Capital Univ Abstract: The three-body problem is a fundamental problem of astrophysics that has been the focus of much research for hundreds of years. The problem deals with the motions and positions of three masses in space, interacting only through their mutual gravitational attraction. For any three masses, an infinite number of starting conditions are possible that lead to many diverse and interesting interactions between the bodies. This presentation will introduce Kepler’s Laws of Planetary Motion and Newton’s Law of Universal Gravitation, which will be used in the formation of a system of differential equations for a particular subset of starting conditions. The restrictions on this system, which include limiting the motion of the masses to one plane and considering one body to be negligable in mass, lead to what is known as the restricted three-body problem. A Matlab solution for this restricted problem will follow. Using this solution, the interesting case of the LaGrange equilibrium points will be explored. ! Friday 4:35 pm Room 2 Is This The Right Number? Erin Zuercher Student: John Carroll Univ Abstract: Identification numbers are used everywhere in our world today, from UPC codes on products at the grocery store to social security numbers to checking account numbers and credit card numbers. It is possible to make different kinds of errors when dealing with these numbers and thus many identification numbers, which have become more complex over the years, involve a built-in means for verifying whether certain numbers are valid. Certainly, different types of identification numbers have different check schemes and each method may detect a different kind of error. Currently no such scheme exists for Social Security Numbers, which are widely used in our society. This research has led to the development of a check digit scheme which could be implemented for Social Security Numbers in the future.