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» Vol. 5, Issue 2, 2004 «



Title: Another Proof of the Steiner Result for Three Equidistant Points in Euclidean Space and an Analogous Proof in Hyperbolic Space
Author: Diana Dimond, Brigham Young University Author Bio    
Abstract: In the early 19th century, Jacob Steiner wanted to find the shortest path to connect three villages. He concluded that the shortest path depended on the angles of the triangle created. If all the angles were less than 120° then the shortest path involved a fourth interior point, a Steiner point, at which the segments from the vertices all meet at 120°.

Our research group developed a new "slicing" method that can be used to prove which paths are minimal. To demonstrate this new method, we first give a new proof of a particular Steiner result. Then we use it to prove an analogous result in hyperbolic space; that is, the shortest path between three equidistant points in hyperbolic space is formed by hyperbolic geodesics that meet at 120°.

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Title: Population Models with Diffusion and Constant Yield Harvesting
Authors: A. Collins, Oklahoma Baptist University
M. Gilliland, Murray State University
C. Henderson, Jarvis Christian College
S. Koone, Northwestern State University
L. McFerrin, Virginia Polytechnic Institute and State University
E. K. Wampler, Rose-Hulman Institute of Technology
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Abstract: In this paper we discuss reaction-diffusion equations arising in population dynamics with constant yield harvesting in one dimension. We focus on the mathematical models of the logistic growth, the strong Allee effect, and the weak Allee effect and their influence on the existence of positive steady states as well as global bifurcation diagrams. We analyze the equations using the quadrature method and the method of sub-super solutions.
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Title: Extensions of Turán’s Theorem for Multigraphs
Authors: Cristian Aldea, Seton Hall University
Brandon Cruz, Seton Hall University
Michael Gaccione, Seton Hall University
Oksana Jablonski, Seton Hall University
Danielle Shelton, Seton Hall University
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Abstract: Some results in graph theory state: if a graph G is of fixed order n and has at least size f(n) ( # of edges), then G contains a particular subgraph , or G has some kind of property. Determining the sharp bounds and resulting extremal graphs is an area of graph theory called extremal graph theory.

We extend some existing extremal results for simple graphs ( On the extensions of Turan's theorem ) to multigraphs. Also, Ramsey Theory in the context of multigraphs is introduced.

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Title: A Procedure for Determining the Exact Solution to a 2x2 First Order Homogeneous Nonautonomous System
Author: Alexander K. Shveyd, University of California, Riverside Author Bio    
Abstract: First order linear nonautonomous homogenous systems of differential equations are characterized by a matrix differential equation where the matrix is a function of the independent variable. These nonautonomous systems are used extensively in the study of Floquet and Lyapunov theories, and the applications of such systems reaches into fields such as physics, biology, and engineering. The following paper develops a technique for finding the closed form solution to a 2×2 nonautonomous system. The paper shows that the solution to such a system is directly related to the solution of a Riccati differential equation constructed from the coefficients of the system's matrix. The primary findings also demonstrate that the system can be solved exactly if a solution to the corresponding Riccati equation can be determined.
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Title: The Effect of a `Smart' Predator in a One Predator, Two Prey System
Author: Elizabeth Green, University of Nebraska-Lincoln Author Bio    
Abstract: This paper analyzes a food web with a predator and two non-competing preys where the predation follows the density gradient of the prey. The long-term dynamics of the food web and short-term population crashes and outbreaks are analyzed using singular perturbation analysis.
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Title: On codes generated from quadratic surfaces in PG(3,q)
Authors: Mandy Passmore, University of Mary Washington
Jenny Stovall, University of Mary Washington
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Abstract: We construct two families of low-density parity-check codes using point-line incidence structures in PG (3 ,q ). The selection of lines for each structure relies on the geometry of the two classical quadratic surfaces in PG (3, q ), the hyperbolic quadric and the elliptic quadric.

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Title: Small-worlds: Beyond Social Networking
Author: Andrew R. Curtis, University of Wyoming Author Bio    
Abstract: Small-world phenomena were initially studied in the 1960s through a series of social network experiments, and are, as evidenced by the game "The six degrees of Kevin Bacon", even part of our pop-culture. Recently, mathematicians and physicists have shown that most small-world phenomena are expected consequences of the mathematical properties of certain networks -- known as {\em small-world networks}. In this paper, we survey some recent mathematical developments dealing with small-world networks, as well as present a new small-world network model and discuss some new ideas for decentralized searching. The goal is to give the reader a sense of the importance of small-world networks, and some of the useful applications dealing with these networks.
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Title: Developing And Comparing Numerical Methods For Computing The Inverse Fourier Transform
Author: Edgar J. Lobaton, Seattle University Author Bio    
Abstract: Computing the Fourier transform and its inverse is important in many applications of mathematics, such as frequency analysis, signal modulation, and filtering. Two methods will be derived for numerically computing the inverse Fourier transforms, and they will be compared to the standard inverse discrete Fourier transform (IDFT) method. The first computes the inverse Fourier transform through direct use of the Laguerre expansion of a function. The second employs the Riesz projections, also known as Hilbert projections, to numerically compute the inverse Fourier transform. For some smooth functions with slow decay in the frequency domain, the Laguerre and Hilbert methods will work better than the standard IDFT. Applications of the Hilbert transform method are related to the numerical solutions of nonlinear inverse scattering problems and may have implications for the associated reconstruction algorithms.
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Title: Tight Subdesigns of the Higman-Sims Design
Authors: Steven Klee, Valparaiso University
Leah Yates, East Carolina University
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Abstract: The Higman-Sims design is an incidence structure of 176 points and 176 blocks of cardinality 50 with every two blocks meeting in 14 points. The automorphism group of this design is the Higman-Sims simple group. We demonstrate that the point set and the block set of the Higman-Sims design can be partitioned into subsets X1, X2,...,X11 and B1, B2,...,B11, respectively, so that the substructures (Xi, Bi), i = 1, 2,...,11, are isomorphic symmetric (16, 6, 2)-designs.
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Title: A Search for Patterns in a Sequence of Integers: Proving Trends in a Recurrence Relation
Author: Qaiser Saify, Brandeis University Author Bio    
Abstract: Let {f(n} be the sequence of integers determined by the recurrence relation f(n+2) equivalent to 2f(n+1) - f(n) (mod(n+2)) with f(n) greater than or equal to 0 and less than n for every integer n greater than or equal to m, where m is a certain positive integer with initially assigned arbitrary integers f(m) and f(m+1). We investigate how the sequence {f(n} increases or decreases. We call a maximal increasing or decreasing subsequence of consecutive elements of the sequence a run. We show that each run is an arithmetic progression and that the common difference in an increasing run is one more than the common difference in the previous increasing run, and similary for decreasing runs.
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Title: Symbolic Powers of Edge Ideals
Authors: Scarlet Worthen Ellis, University of Texas at Tyler
Lesley Wilson, University of Texas at Tyler
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Abstract: In this paper we discuss a connection between graph theory and ring theory. Given a graph G there exists a corresponding edge ideal I generated by xi xj where xi and xj are vertices in G connected by an edge. Simis, Vasconcelos, and Villarreal show that a graph G is bipartite (contains only even cycles) if and only if its corresponding edge ideal I satisfies I(n)=In for all n greater than or equal to 1. We explore what happens when G is not bipartite - in particular, when G is an odd sided polygon.
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