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» Vol. 4, Issue 1, 2003 «



Title: A Predator Prey Model with Disease Dynamics
Authors: Chris Flake, North Carolina State University
Tram Hoang, California State University at Fullerton
Elizabeth Perrigo, Midland Lutheran College
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Abstract: We propose a model to describe the interaction between a diseased fish population and their predators. Analysis of the system is performed to determine the stability of equilibrium points for a large range of parameter values. The existence and uniqueness of solutions is established and solutions are shown to be uniformly bounded for all nonnegative initial conditions. The model predicts that a deadly disease and a predator population cannot co-exist. Numerical simulations illustrate a variety of dynamical behaviors that can be obtained by varying the problem data.
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Title: The Old Hats Problem
Author: Heba Hathout, Westridge School for Girls, Pasadena, CA Author Bio    
Abstract: Derangements, a favorite topic in combinatorics, are usually studied using the inclusion-exclusion principle, to calculate the number of derangements of n objects, as well as the probability of a derangement occurring. This paper briefly presents this solution, as well as a second fairly standard solution using a recursion method, and then proceeds to solve for the probability of a derangement using the binomial inversion formula, which is derived in the final section of the paper. To show the utility and elegance of this approach, the expected value of correct assignments is also calculated if n objects are arranged at random.
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Title: Properties of Magic Squares of Squares
Author: Landon W. Rabern, Washington University Author Bio    
Abstract: A problem due to Martin Labar is to find a 3x3 magic square with 9 distinct perfect square entries or prove that such a magic square cannot exist. In this paper, I assume that such a magic square exists and show that the entries must have certain properties. This is accomplished using unique factorization in two different finite extensions of Z. One property that is proven is: no prime congruent to 3 modulo 8 can divide any entry.
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Title: Classifying and Using Polynomials as Maps of the Field F_{p^d}s
Authors: Dylan Cutler, Middlebury College
Jesse Johnson, Middlebury College
Ben Rosenfield, Middlebury College
Kudzai Zvoma, Middlebury College
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Abstract: Every function from a finite field to itself can be represented by a polynomial. The functions which are also permutations give rise to "permutation polynomials," which have potential applications in cryptology. We will introduce a generalization of permutation polynomials called ``degree-preserving polynomials" and show a classification scheme of the latter. The criteria for a polynomial to qualify as degree preserving are certainly less stringent than those for the permuting qualification. Thus the idea to study degree-preserving polynomials allows more opportunity to maneuver and gain intuition about the occurrence of such polynomials.
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Title: Characterizing a Defect in a One-Dimensional Bar
Authors: Cynthia Gangi, Eckerd College
Sameer Shah, MIT
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Abstract: We examine the inverse problem of locating and describing an internal point defect in a one-dimensional rod $\Omega$ by controlling the heat inputs and measuring the subsequent temperatures at the boundary of $\Omega$. We use a variation of the forward heat equation to model heat flow through $\Omega$, then propose algorithms for locating an internal defect and quantifying the effect the defect has on the heat flow. We implement these algorithms, analyze the stability of the procedures, and provide several computational examples.
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Title: Hamilton Cycles in Addition Graphs
Authors: Brian Cheyne, Western Michigan University
Vishal Gupta, Yale University
Coral Wheeler, University of Akron
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Abstract: If A is a square-free subset of an abelian group G, then the addition graph of A on G is the graph with vertex set G and distinct vertices x and y forming an edge if and only if x+y is in A. We prove that every connected cubic addition graph on an abelian group G whose order is divisible by 8 is Hamiltonian as well as every connected bipartite cubic addition graph on an abelian group G whose order is divisible by 4. We show that connected bipartite addition graphs are Cayley graphs and prove that every connected cubic Cayley graph on a group of dihedral type whose order is divisible by 4 is Hamiltonian.
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Title: Quadratic Assignment Problems (QAP) and Its Size Reduction Method
Author: Darwin Choi, University of Pennsylvania Author Bio    
Abstract: The Quadratic Assignment Problem (QAP) is a discrete optimization problem which can be found in economics, operations research, and engineering. It seeks to locate N facilities among N fixed locations in the most economical way. This paper gives a brief introduction to QAP and discusses how to reduce the problem size to N-1 if the original problem satisfies certain conditions.
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