Abstracts of Student Talks

Abstracts are listed first by day and then time of presentation.
Click the title to go to the abstract.

Speakers:  Here are some  SPEAKER GUIDELINES for this conference.

Index

Friday Afternoon

Saturday Morning

To Be Announced

Abstracts

Simultaneous s-core and t-core Partitions
By David Aukerman of Taylor University

Abstract:
A t-core partition is a special kind of partition of a positive integer whose Ferrers-Young diagram contains no hook numbers that are multiples of t. Some very simple generating functions describe the numbers of t-core partitions for a given integer n. Interesting variations on this topic involve looking at self-conjugate partitions of n as well as those that are simultaneously s-core and t-core partitions of n. This talk will present a graphical method of representing partitions to aid in determining if a partition is self-conjugate. Also, this talk will present the generating function for simultaneous s-core and t-core partitions, with gcd(s,t)>1. Finally, the generating function for self-conjugate simultaneous s-core and t-core partitions will be presented, as well.

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On Knots: An Application to DNA
By Meredith Deters of Greenville College

Abstract:
After basic definitions on knots and the equivalence of two knots, we study properties of a link and the problems that occur in the world of all knots and each given knot. The applications of knots in the sciences (particularly in Chemistry and in Molecular Biology) are growing. So we will discuss the use of tangles to interpret the reaction of an enzyme on DNA.

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A Brief Explanation of the Robinson-Schensted Algorithm
By Randy Pistor of University of Michigan

Abstract:
The Robinson-Schensted Algorithm is a useful combinatorial algorithm that uses a one-to-one correspondence to match a permutation in the symmetric group on n elements with a pair of Standard Young Tableaux. Standard Young Tableaux (SYT) are boxes filled with the numbers one through n in a Ferrers shape, with values increasing along rows and columns.

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Chord Analysis in the Key of Math and Physics
By Jenny Fromme of College of Mount St. Joseph

Abstract:
Musical chords are made up of three or more unique pitches. The musical quality (major, minor, etc.) of these chords is determined by the differences in the frequencies of the individual pitches of the chord. These pitches are formed by acoustical energy, which travels in waves that can be approximated by sinusoidal waves. When two pitches are sounded simultaneously, the pitches created by the sum and difference of each sinusoidal waves are also perceived. This research focuses on the sums and differences of each combination of the pitches in the chord and tries to determine patterns that account for the pleasing quality of some chords and the cacophonous quality of other chords. Also analyzed, are the differences in the patterns created from notes formed using the just intonation system, which utilizes rational frequencies, and the equal temperament system, which utilizes irrational frequencies to establish the musical scale.

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Ion Channels and Spontaneous Oscillations: The Effects of Stochastic Noise on a Subcellular Mathematical Model
By David Maduram of University of Illinois at Chicago

Abstract:
Ever since the early 1920s, mathematical models have been used by scientists to explicitly describe biological phenomena. Quite recently, a mathematical model was created to describe the effects of ion channel behavior found in the auditory inner hair cells of the red-eared turtle (Trachemys scripta elegans). During my talk, I will (1) describe the mechanisms of this model and (2) illustrate how the introduction of a stochastic noise component creates interesting nonlinear behavior within this mathematical model. Additionally, I shall discuss the practical implications of this mathematical model as it applies to the auditory reception of T. s. elegans.

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An Introduction to Swarms and Swarm Algorithms
By Jon Murton of John Carroll University

Abstract:
Some insect behavior exemplifies the ultimate in `distributed programming' with many simply programmed units accomplishing a gigantic task by working together. This talk will include discussion of the undergraduate research on swarm intelligence that is done at John Carroll University using robots and even humans.

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Cut-and-Paste Topology
By William Willis of Southern Illinois University

Abstract:
I start with a brief overview of topology, along with algorithms for cut-and-paste transformations. The Mobius strip and it's connections to the projective plane will be studied and expanded on. Subsequently, I will display a result, followed by method of derivation. The projective plane identifies the boundary of the e-nhd of the point at infinity to itself with 180 degree rotational symmetry. My result is similar, but the boundary is identified by 120 degree rotation. Most work done will be visually oriented, with the intent to be as accessible as possible to the widest audience.

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How to Solve a Rubik's Hypercube
By Nathanael Berglund of Rose Hulman

Abstract:
Since Erno Rubik invented his famous "Rubik's Cube" in 1974, people have invented many systematic ways to solve it. This talk will examine some examples of move sequences given by these systems, why they work, and how they might possibly be generalized to higher dimensional Rubik's Cubes. In particular, we will examine a 4-dimensional generalization of Rubik's Cube (refered to as a Rubik's Hypercube). We will discuss the fact that there are at least two different ways one might generalize the allowable moves on a Rubik's Cube to 4-space, and each of these yields a different method for solving it, making it difficult to talk of a "standard" higher dimensional Rubik's Cube.

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Classifying Alpha-Almost Squares
By Bobbe Cooper of Taylor University

Abstract:
We look at a twist on the basic calculus problem of minimizing the cost of fencing a given area ­ we can only use integer fence lengths. Dr. Greg Martin defined the best solutions to this problem as Œalmost-squares¹ and characterized them completely in his 1999 paper. We have generalized his work by introducing a weighting constant alpha, and defining alpha-almost-squares. We see how these numbers can also be characterized completely.

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Quasi-p or Not Quasi-p? That is the Question.
By Ben Harwood of Northern Kentucky University

Abstract:
In a 1957 paper entitled "Coverings of Algebraic Curves," Abhyankar conjectured that the algebraic fundamental group of the affine line over an algebraically closed field of prime characterstic p in the set of quasi p-groups. In 1995, Harbater and Raynaud shared the AMS's Cole prize in Algebra for proving the Abhyankar Conjecture. Although the conjecture recieved much attention, not much had been by way of studying quasi p-groups in terms of GROUP THEORY. We shall define and discuss these quasi p-groups.

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Algebraic Properties of a Number-Theoretic Permutation
By Lucas R. Wiman of Illinois State University

Abstract:
If x is any real number, let {x} denote its remainder modulo 1, that is, its fractional part. Choose an irrational number alpha, such that 0 < alpha < 1. For each alpha and each positive integer n we can define a permutation pi(alpha, n) which orders the numbers {alpha}, {2 alpha},..., {n alpha}.

In 2001, Kevin O'Bryant (UIUC) proved a number of algebraic results regarding these permutations. He established some theorems regarding their order (which is often surprisingly far from the "average" permutation), and created a somewhat strange representation of the symmetric group on n symbols. This representation agrees with matrices which come up in the study of Sturmian words and Beatty sequences, implying that an algebraic study of these combinatorial objects may be fruitful.

I shall prove that for fixed n, a certain subset of {pi(alpha, n): 0 < alpha < 1} forms a group isomorphic to the multiplicative group modulo n+1. Also, I shall survey some of O'Bryant's algebraic results, focusing mainly on the order results, and list some open questions.

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Gray Code Labeling of Complete Iterated Graphs
By Shawn Alspaugh of Taylor University

Abstract:
We set up codes that have a "word" corresponding to each vertex in the complete graph Knd. We define a perfect one-error correcting code, and we show that the location of the codeword vertices for a perfect one-error correcting code is unique up to isomorphism. However, there are numerous ways to label the vertices. We define a gray code labeling, and we explore one such labeling called the a-method labeling. We devise a finite method of creating the labels for any Knd, and then characterize these codes. Finally, we discuss codeword detection and error correction of the a-method-labeled Knd. This work is a result of research completed at an REU at Oregon State University in 2001.

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Airport Security: A Queuing System Model
By Maureen McMilin & Reijiro Matsuo of University of Evansville

Abstract:
Due to the events of September 11, airport security has been heightened and, as a result, time at security checkpoints has greatly increased. We use queuing systems to model the flow of passengers through one of these checkpoints. We created an initial model of a checkpoint before September 11 to use as a baseline. A second model with increased security was then created. Various combinations of workers and metal detectors were tested with the goal that total time in the queue be no more than 10% of the time established in the baseline. Computational results will be presented.

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The Optimality of Morse Code
By Lucas Beverlin of Rose Hulman

Abstract:
Morse code has been around for over 150 years, and yet no one has been able to improve upon it. By looking at ideas in probabilistic coding theory such as uncertainty and compact codes that have been discovered in the last 75 years, one can analyze Morse code and determine whether or not it can be improved.

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Graph Colorings with Restrictions
By Stephen Young of Rose-Hulman

Abstract:
One important type of restricted graph coloring is a T-coloring. In a T-coloring, the coloring of the graph must be such that if two vertices are adjacent, the absolute value of the difference of thier colors does not exist in the finite set T. In this talk we will consider a stronger restriction, that of being colorable by D. A graph is colorable by D if there exists a coloring such that for all adjacent vertices, the absolute value of the difference of their colors exists in D. In this talk we will answer, for certain D, which graphs are colorable by D. We will also outline some furture areas of research regarding colorings by D.

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Finding the Mathematically Optimal 4-Team Double Elimination Tournament
By Maroof Khan of Wabash College

Abstract:
A problem with the standard 4-team double-elimination tournament is that initial pairings have a deep impact on deciding who wins the tournament. The purpose of this presentation is to take a comprehensive look at the tournaments that are potentially better than the standard double-elimination tournament. In particular, all reasonable tournaments that reduce the pairing effect by having the team that did not play the 0-2 team in rounds one or two sit out for the 3rd round are analyzed. For all tournaments considered, the strongest team has a better chance of becoming the victor than in the standard tournament. Three of these tournaments also give a lower standard deviation of the probability of the best team becoming the champion.

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Using Sets to Count Sets
By Joseph McClain of University of Michigan

Abstract:
A common problem in combinatorics is to find a formula for the number of elements in a given set. Often, it is difficult to find such a formula directly. In this talk, I will illustrate the technique of proving that two sets have the same cardinality without actually counting them. The core of the technique is devising a one-to-one correspondence between the elements of the two sets. Using this technique, I will show that several quite different sets are counted by the same integer sequence, the Catalan numbers.

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Unit Interval Orders and Related Posets
By Eric Hendrickson of University of Michigan

Abstract:
A totally ordered set consists of elements all of which can be compared to any other element in the set...that is, for any two elements x and y, you can say x < y or x > y. An example is the set of integers. A partially ordered set (poset) on the other hand, consists of elements which may be incomparable to other elements in the set...a good example being intervals of real numbers. the interval [1,2] is less than the interval [3,4] but incomparable to [1.5, 2.5].

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