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» Vol. 4, Issue 2, 2003 «



Title: A Survey of Relative Difference Sets
Authors: Christine Berkesch, Butler University
Jeff Ginn, Central Michigan University
Erin Haller, University of Missouri - Rolla
Erin Militzer, Central Michigan University
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Abstract: A (v,k,lambda) difference set D in a group G is a subset of G such that every nonidentity element of G is covered exactly lambda times by quotients d1d2-1 where d1 and d2 are in D. In the group ring, this means that D obeys the equation DD(-1) = k1 + lambda(G - 1). An (m,n,k,lambda) relative difference set R is a difference set relative to a normal subgroup N of G satisfying the similar equation RR(-1) = k1 + lambda(G - N).

We will describe various search techniques for relative difference sets (RDS), including the exhaustive search method for small groups using the computer program GAP, as well as the multiplier theorem and group representations methods used for larger groups. We will provide a catalog of RDS found, as well as those eliminated, using these methods. Next, a proof is presented of the non-existence of (2m,2,2m,m) relative difference sets in quaternion groups of order 4m where m is odd. In conclusion, we will state several interesting results found for specific parameters, including (12,2,12,6) and (12,3,12,4).

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Title: Knots in Four Dimensions and the Fundamental Group
Authors: Jeff Boersema, Seattle University
Erica Whitaker, Ohio State University
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Abstract: This paper is an introduction to knotted spheres in four dimensions (analogous to knotted circles in three dimensions). We define what a knotted sphere is and describe a to visually represent them, via movies. The fundamental group of the complement of a knot is a powerful invariant and we describe this invariant in detail giving a convenient algorithm for computing it. Lots of examples are given, including the simplest non-trivial locally flat knot.
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Title: Take Me Out to/of the Ball Game
Authors: Brandon Alleman, Hope College
Michael Cortez, Hope College
Katie McKinnon, Lenoir-Rhyne College
Ann Monville, California Lutheran University
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Author Bio    
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Abstract: When should a person leave a baseball game in order to maximize his/her enjoyment? How does this decision depend on the score of the game? Assuming a modified logistic rate of departure from the stadium, and a constant maximum exit rate from the parking lot, we find optimal strategies for leaving games of various scores.
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Title: Properties of Divisor Graphs
Author: Christopher Frayer, Grand Valley State University Author Bio    
Abstract: For a finite and nonempty set S of positive integers, the divisor graph G(S) of S has vertex set S, and two distinct vertices i and j are adjacent if and only if i|j or j|i, while the divisor digraph D(S) of S has vertex set S and (i, j) is an arc of D(S) if and only if i|j. A graph G is a divisor graph if there exists a set S of positive integers such that G is isomorphic to G(S). It is shown that for a divisor graph G with a transitive vertex, G x H is a divisor graph if and only if H has no edges. For m, n in N, with m greater than or equal to five there exists a non-divisor graph G, of order m + n, that has m neighborhoods that are divisor graphs and n neighborhoods that are not divisor graphs.
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Title: A Characterization of Tree Type
Author: Lon H. Mitchell, University of Kansas Author Bio    
Abstract: Let L(G) be the Laplacian matrix of a simple graph G. The characteristic valuation associated with the algebraic connectivity a(G) is used in classifying trees as Type I and Type II. We show a tree T is Type I if and only if its algebraic connectivity a(T) belongs to the spectrum of some branch B of T.
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Title: Long Term Dynamics for Two Three-Species Food Webs
Authors: Brian Bockelman, University of Nebraska-Lincoln
Elizabeth Green, University of Nebraska-Lincoln
Leslie Lippitt, Iowa State University
Jason Sherman, Kent State University
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Abstract: In this paper, we analyze two possible scenarios for food webs with two prey and one predator (a food web is similar to a food chain except that in a web we have more than one species at some levels). In neither scenario do the prey compete, rather the scenarios differ in the selection method used by the predator. We determine how the dynamics depend on various parameter values. For some parameter values, one or more species dies out. For other parameter values, all species co-exist at equilibrium. For still other parameter values, the populations behave cyclically. We have even discovered parameter values for which the system exhibits chaos and has a positive Lyapunov exponent. Our analysis relies on common techniques such as nullcline analysis, equilibrium analysis and singular perturbation analysis.
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Title: Generalized Cwatsets
Author: Daniel Smith, Wabash College Author Bio    
Abstract: We define the gc-set, a generalization of the cwatset, and find some of its basic properties. Then some of the gc-set's group theoretic results are shown.
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