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ABSTRACTS  for the 20th Annual Rose-Hulman Undergraduate Mathematics Conference

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


Title Speaker Institution Room/Time
Generalizing Cwatsets Daniel Smith Wabash College G219/3:00

Simulation of Communicable Diseases

J.B. Brown University of Evansville G222/3:00
A Cost-Benefit Analysis of International Cigarette Smuggling Using Computationally-Intensive Mathematical Models David Maduram University of Illinois at Chicago G219/3:30

Analysis of Disease Mutations By Evolutionary Comparison and Codon Position

Jacob Reidhead Arizona State University G222/3:30

Three-Minute Dating and Other Social Mixers

Tyrel Fisher Valparaiso University G219/4:00

On Binary Quadratic Forms

Andrew Clausen Greenville College G222/4:00


Saturday Morning

Title Speaker Institution Room/Time

Simple and Multiple Circular Arc Graphs

Lucas Wiman Illinois State University G219/10:20

Approximations Using Two Almost-Binomial Random Variables

Tony Smith Wabash College G222/10:20

On the Bookthickness of Graphs

Noah Prince University of Illinois at Urbana-Champaign G219/10:50

Quasi [p,q]-groups

Ben Harwood Northern Kentucky University G222/10:50

On the Edge Set of Graphs of Essentially Equivalent Lattice Paths

Steven Klee Valparaiso University G219/11:20

War: the Vicious Cycle

Kellan Wampler Rose-Hulman G222/11:20


Abstracts

Generalizing CWATsets
Daniel Smith, Wabash College

Abstract:  A CWATset is a subset of n copies of Z2 that is closed with a twist under componentwise addition.  We will define a gc-set, which generalizes the cwatset structure, and determine some of its basic properties. Some of the group theoretic properties gc-sets will then be explored.

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Simulation of Communicable Diseases
J.  B. Brown, University of Evansville

 Abstract:  Direct contact communicable diseases continue to exist.  The threat of bio-terrorism could create additional problems with the introduction of smallpox or microbial pathogens.  We simulate the spread of direct contact viruses in two-dimensional space, having initially considered the case where all people are in fixed locations and, more recently, removing the restriction of fixed positions, allowing people to interact.  We will examine the results and look for patterns explaining how diseases spread so that protocols of effective quarantine and cure can be developed.

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A Cost-Benefit Analysis of International Cigarette Smuggling Using Computationally-Intensive Mathematical Models
David Maduram, University of Illinois at Chicago

Abstract:  This study uses a stochastic Markov model to investigate the economics of tobacco smuggling.  By using Norton's model of international smuggling risks as well as Chaloupka's cigarette demand equations, this study attempts to assess the profit gained by smuggling tobacco products between countries and the effects of localized tax reforms on the smuggling economy.

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Analysis of Disease Mutations By Evolutionary Comparison and Codon Position
Jacob Reidhead, Arizona State University

Abstract:  Recent accumulation of genome sequence data of humans and other species has opened the door for interesting interspecies comparisons and phylogenetic analysis.  The availability of data about mutations that cause disease in humans allows us to characterize genetic disease in humans with respect to evolutionary history.  Previous work has compared the severity, location, and type frequency of disease mutations and interspecies changes, and has pointed out distinct differences between the two.  Disease causing mutations were of greater severity (in terms of the biochemical differences) than those observed in the long-term evolutionary history, occurred with higher frequency at evolutionarily conserved sites, and comprised a unique type frequency signature.  We have conducted similar comparisons within putative protein domains.  We find that similar types of domains exhibit similar ranges of tolerated mutation severity, propensity for mutations at corresponding key sites, and similar types of frequency signatures.  These computational analyses help in uncovering the nature of disease causing mutations and provide insights into the contrast between the genome diversity between species and that in disease mutations.  Information of this kind may ultimately be useful in the future to better understand the function of different domains of a gene and be useful in the genetic disease diagnosis.

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Three-Minute Dating and Other Social Mixers
Tyrel Fisher, Valparaiso University

 Abstract:  Suppose you have k people sitting at each of n tables.  They are allowed to get up and rearrange themselves under the following conditions: 1) no person may sit at the same table more than once, and 2) no two people may sit together more than once.  How many times x can a new seating be generated?  Using properties of families of orthogonal Latin Squares, we obtain solutions to this problem when k = 1, k = 2, kn, and n = pt, where p is some prime number.

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On Binary Quadratic Forms
Andrew Clausen, Greenville College

 Abstract:  We will offer basic definitions concerning binary quadratic forms and discuss properties of an equivalence relation on binary quadratic forms.  Examples of equivalence classes of definite quadratic forms will be considered, and a program in Maple that determines the class number of definite quadratic forms will be demonstrated.

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Simple and Multiple Circular Arc Graphs
Lucas Wiman, Illinois State University

Abstract:  Given k circles in the Euclidean plane, and a set S of n arcs of these circles, define the circular arc graph G(S) as a graph on n vertices such that each vertex represents an arc in S, and two vertices are adjacent precisely when the corresponding arcs have a nonempty intersection.  Which graphs can arise in this way?  The question for k = 1 has been well studied.  In 2001, Ross M. McConnell published a linear-time algorithm for recognizing simple circular arc graphs.  Some aspects of simple circular arc graphs (k = 1) will be summarized, after which some original results on double circular arc graphs (k = 2) will be presented, including a number of examples of nonrepresentable graphs.

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Approximations Using Two Almost-Binomial Random Variables
Tony Smith, Wabash College

Abstract:  The distribution of a sum of independent Bernoulli random variables is, in general, complex.  We can approximate this distribution with that of a sum of two independent almost-binomial variables whose parameters are chosen so that the first four moments of the approximating distribution agree with the first four moments of the sum of Bernoulli random variables.  In this talk, we model the errors we get using this approximation method.  They are shown to be typically quite small, and our model of the errors is shown to be very accurate.

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On the Bookthickness of Graphs
Noah Prince, University of Illinois at Urbana-Champaign

Abstract:  Let the vertices of a graph G be ordered along a line L.  A book embedding of G is an assignment of each edge of G to exactly one half-plane, called a page, with L as its boundary, such that the edges on any page do not intersect.  Define the bookthickness of G to be the least number of pages into which G has a proper book embedding, taken over all vertex orderings.  In this talk, I will discuss results on the bookthickness of special classes of graphs.

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Quasi [p,q]-groups
Ben Harwood, Northern Kentucky University

Abstract:  A quasi p-group is any finite group generated by all of its Sylow p-subgroups.  At last year's conference, I presented a classification of the quasi p-groups of order less than 24.  During that presentation, a member of the audience asked the question, "Can a group be a quasi p-group for more than one prime p?"  My answer at the time was somewhat limited.  In this talk we will discuss the properties of a quasi [p,q]-group.

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On the Edge Set of Graphs of Essentially Equivalent Lattice Paths
Steven Klee, Valparaiso University

 Abstract:  This study is a continuation of work done last year investigating a family of graphs created from essentially equivalent lattice paths.  To obtain these graphs, each path in an m n lattice corresponds to a vertex in the graph; two paths are considered equivalent if they share more than k edges and therefore their corresponding vertices in the graph are connected.  We denote such a graph by G(m, n, k).  This year we identified some interesting complete sub-graphs and determined the number of edges in G(m, 2, 0), G(m, 2, 1), and G(m, 2, m - 1).  We also generated data sets giving the number of edges to use in further investigations of this type.

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 War: the Vicious Cycle
Kellan Wampler, Rose-Hulman Institute of Technology

Abstract: This talk is a follow-up to an article written by Dr. Gary J. Sherman on the card-game war.  The motivating question is whether or not it is necessary for a game of war to end.  In order to obtain a more manageable problem, the game was abstracted by removing suits and considering a deck with fewer cards.  Non-terminating, or cyclic, games were found in this abstracted version of war, and the purpose of this talk is to characterize these games.  Terminating games will be discussed briefly along with possibilities for further study.

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