Department of Mathematics
§MUURMaC§
Fourth Biennial Mercer University Undergraduate Research in MAthematics Conference
February 12, 2011
The period to submit abstracts has passed. A list of abstracts for the talks that will be given is available below.
Invited Addresses by Dr. Robert Bosch
MAA State Lunch Address: Opt Art: An Introduction
Abstract: Optimization is the branch of mathematics concerned with optimal performance---finding the best way to complete a task. It has been put to good use in a great number of diverse disciplines: advertising, agriculture, biology, business, economics, engineering, manufacturing, medicine, telecommunications, and transportation (to name but a few). In this lecture, we will showcase its amazing utility by demonstrating its applicability in the area of visual art, which at first glance would seem to have no use for it whatsoever! We will begin by describing how to use integer programming to construct a portrait out of complete sets of double nine dominoes. We will then describe how high quality solutions to certain large-scale traveling salesman problems can lead to beautiful continuous line drawings. We will conclude by presenting other examples of Opt Art---art constructed with the assistance of mathematical optimization techniques.
Second Address: Opt Art: Special Cases and New Directions
Abstract: In this lecture we will present additional examples of Opt Art, including edge-matched mosaics and map-colored mosaics. We will also introduce special cases that can be solved very quickly. Finally, we will talk about directions for future research.
Presenter: Bambo Adedire
Affiliation: Emory University
Title: Model of Fast Inactivation for Kv4.3
Abstract: Potassium ions regulate the membrane potential across the surface of muscle. Voltage gated channels, such as Kv4.3 channels, are imbedded within the surface of a membrane. The opening and closing of such channels has been mapped and modeled. A working model for potassium channels has been the Markova model, which takes into account the stochastic nature of these voltage gated potassium channels. We develop a mathematical model that illustrates probability of conformation done through computational experimentation, and the model developed fully illustrates fast inactivation of Kv4.3 channels.
Presenter: Chan Jin
Affiliation: Agnes Scott College
Title: A Class of Commuting Integer Matrices Modulo a Prime
Abstract: For n > 1 we examine certain collections of n strictly upper triangular commuting n x n matrices mod p, and obtain a description of these collections in terms of a system of quadratic equations. In the case of 2 x 2 matrices there are precisely p such collections. When n = 3 we give a formula, as a function of p, for the number of such collections. For n = 4 we count the number of collections for each prime less than 31, and we form conjectures as to the behavior for arbitrary p.
Presenter: Chris Kirkland
Affiliation: Mercer University
Title: Use-It-or-Lose-It Trees and Graph Dynamics
Abstract: In this talk, we explore properties of random, labeled trees which arise from the "Use-It-or-Lose-It" algorithm through particular dynamical "frameworks", which we refer to as Multiplicative Decremental Tag Systems.
Presenter: Adam Lewis
Affiliation: Mercer University
Title: A set of points equidistant from a line in the Beltrami-Klein Disk
Abstract: I will be discussing the shape of the set of points equidistant from a hyperbolic line in different models of hyperbolic geometries, in particular the Beltrami-Klein and Poincaré Disks. The shape formed by this set of points is quite unique and different from Euclidean geometry in the fact that it is not linear and does not visually "look
equidistant." In order to fully explain the the shape of this set, I will utilize the relationships between the Poincaré Disk, the Beltrami-Klein disk, and a sphere in 3-space.
Presenter: Tim Michaels Canceled
Affiliation: University of Tenessee
Title: TBA
Abstract: How many ways are there to partition a rectangle into n sub-rectangles? We derive a recurrence relation for generating the number of tilings of n sub-rectangles by counting the number of ways to "slide" in lines from a certain side and create higher order tilings from lower ones. We can think of the set of all rectangles tiled with n sub-rectangles as a topological space, so we conclude with a study of their Euler Characteristics and underlying patterns as well as possible homotopy types.
Presenter: Aaron Ostrander
Affiliation: Berry College
Title: Studying Topologies on Directed Graphs Using Algebraic Graph Theory
Abstract: We look at certain classes of directed graphs and interior operators associated with them. We show that due to the definitions of these directed graphs and their associated interior operators we are able to define equivalence relations between the classes such that equivalent graphs have identical topologies. We examine the topological aspects of these graphs in the context of linear algebra and matrices associated with graphs. Ultimately we conjecture, and provide numerical evidence, that the calculation of the topologies induced by our interior operators on directed graphs can in fact be viewed as an eigenvector problem.
Presenter: Rose Psalmond, Hilary Tobiasz, Christine Franzel
Affiliation: Agnes Scott College
Title: On the Period of a Linear Recursive Sequence Modulo a Prime
Abstract: The Fibonacci sequence is periodic when examined modulo a prime. This property is shared by any integer sequence defined by a linear recurrence relation. Here we consider the case of second order linear recurrence relations. We show that any such sequence modulo p has period length dividing either p^2 – 1 or p(p – 1). Conversely, any divisor of these two numbers can be realized as a period length for some such sequence.
Presenter: Tyler Steele
Affiliation: Georgia Southwestern State University
Title:Hausdorff Lines in the Second Symmetric Product
Abstract: This presentation will begin with a description of the Hausdorff metric and the symmetric product, followed by some interesting aspects of the geometry of the second symmetric product $F_2(\mathbb{R})$. More specifically, the presentation will provide a comparison of Euclidean lines in $\mathbb{R}^2$ and Hausdorff lines on $F_2(\mathbb{R})$.
Presenter: Cecilia Villagomez, Mercedes Mixon, Gloria Ananing, Melanie King
Affiliation: Mercer University
Title: Opportunities for Women in Mathematics: The Importance of Student Participation
Abstract: A group of five female Math Majors from Mercer University's Math Department recently traveled to Lincoln, Nebraska for the Thirteenth Annual Nebraska Conference for Undergraduate Women in Mathematics (NCUWM). This presentation will provide valuable insight from a student's perspective on what the conference entailed. The hope is to encourage other undergraduates in Mathematics, especially women, to become aware of various conference and research opportunities available.
Other Information
If you have questions not answered on this site, please contact Dr. Julie Beier at Mercer University.
Partial funding from this conference is provided by NSF grant DMS-0846477 through the MAA Regional Undergraduate Mathematics Conference program, http://www.maa.org/RUMC/.