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» Vol. 8, Issue 1, 2007 «



Title: Isoperimetric Regions in Gauss Sectors
Author: Elizabeth Adams, Williams College
Ivan Corwin, Harvard University
Diana Davis, Williams College
Michelle Lee, Williams College
Regina Visocchi, Michigan State Universiy
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Abstract: We consider the free boundary isoperimetric problem in sectors of the Gauss plane. The solution is not always a circular arc as in sectors of the Euclidean plane. We prove that the solution is sometimes a ray and we conjecture that the solution is sometimes a "rounded n-gon" which we discovered computationally using Mathematica.
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Title: Some Geometry of the p-adic Rationals
Author: Catherine Crompton, Agnes Scott College Author Bio    
Abstract: Given a prime p, we introduce the p-adic absolute value on the rational numbers. We define triangles and angles using this absolute value and investigate their behavior in p-adic geometry and how it differs from Euclidean geometry. Finally, we consider the existence of other polygons using the new absolute value.
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Title: A Study of Approximated Solutions of Heat Conduction Problems Using Approximated Eigenfunctions
Authors: Justin Taylor, Southeast Missouri State University Author Bio    
Abstract: Let L be the length of a rod and u(x,t) be its temperature for [0,L]X[0,infinity) and assume the initial and boundary temperatures of the rod are f(x) and 0 respectively. In this paper we explain and demostrate a method for estimating the eigenfucntions appearing in the solution to the corresponding heat conduction problem.
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Title: Monopolist Strategies in a Durable Goods Market
Authors: Shikha Basnet, Simpson College Author Bio    
Abstract: In his classical model for a durable goods monopoly, Ronald Coase conjectured that a monopoly will never be able to charge a price above the equilibrium competitive price and the monopoly will end up forgoing dominant market power. Under certain circumstances, the ideas in the Coase conjecture break down, which we can see in the high-end fashion industry. In this paper, we will analyze some classical and modern contributions in the field of a durable goods monopolist. Based on the ideas of various contributors, we introduce a new model while less formidable vividly presents the complexity in the topic.
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Title: Euclid's Partition Problem and Ceva's Theorem
Authors: Max Nosiglia, Willamette University Author Bio    
Abstract: Euclid's Partition Problem is the problem of constructing one-nth of a given segment using only a compass and straightedge. There are many well-known constructions that solve this problem, including the standard construction involving parallel lines. This new construction uses Ceva's Theorem and is simpler than many of the other constructions. Furthermore, it easily generalizes to construct m-nths of any given segment using only compass and straightedge.
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Title: Markov Chaines and Traffic Analysis
Author: Emanuel Indrei, Georgia Institute of Technology Author Bio    
Abstract: In this paper, we use Markov chains to construct a theoretical traffic system. The paper is organized into three parts: The first two deal with the construction of two spaces in which objects may interact. The third part analyzes the evolution of one particular object. Using bounds given by the law of iterated logarithm and the central limit theorem, we prove that after a large number of time steps, the probability of locating this object in the traffic network diminishes to zero. We conclude with several suggestions on the evolution of multiple objects.
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Title: A Labelling of the Faces in the Shi Arrangement
Authors: Felipe Rincon, Universidad de los Andes, Bogota, Colombia Author Bio    
Abstract: Let Fn be the face poset of the n-dimensional Shi arrangement, and let Pn be the poset of parking functions of length n with the order defined by (a1, a2 , , an ) <= (b1 , b2 , , bn ) if ai <= bi for all i. Pak and Stanley constructed a labelling of the regions in Fn using the elements of Pn. We show that under this labelling, all faces in Fn correspond naturally to closed intervals of Pn, so the labelling of the regions can be extended in a natural way to a labelling of all faces in Fn. We also explore some interesting and unexpected properties of this bijection. We finally give some results that help to characterize the intervals that appear as labels and consequently to obtain a better comprehension of Fn. As an application we are able to count in a bijective way the number of one dimensional faces.
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Title: Design and Optimization of Explicit Runge-Kutta Formulas
Author: Stephen Dupal, Rose-Hulman Institute of Technology
Michael Yoshizawa, Pomona College
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Abstract: A model of the pretzel knot is described. Explicit Runge-Kutta methods have been studied for over a century and have applications in the sciences as well as mathematical software such as Matlab's ode45 solver. We have taken a new look at fourth- and fifth-order Runge-Kutta methods by utilizing techniques based on Gröbner bases to design explicit fourth-order Runge-Kutta formulas with step doubling and a family of (4,5) formula pairs that minimize the higher-order truncation error. Gröbner bases, useful tools for eliminating variables, also helped to reveal patterns among the error terms. A Matlab program based on step doubling was then developed to compare the accuracy and efficiency of fourth-order Runge-Kutta formulas with that of ode45.
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Title: Linear N-Graphs
Author: Demet Taylan, Suleyman Demirel University, Turkey
Gulnur Baser, Suleyman Demirel University, Turkey
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Abstract: We call a simple graph G a linear N-graph if its ordinary (vertex) chromatic number equals to the linear chromatic number of its neighborhood complex N(G). We prove that the linearity is preserved under taking joins and multiplication of vertices, and give a complete characterization of linear N-trees.
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Title: Geometric Models of the Card Game SET
Author: Cherith Tucker, Southern Nazarene Unversity Author Bio    
Abstract: The card game SET can be modeled by four-dimensional vectors over Z3. These vectors correspond to points in the affine four-space of order three (AG(4,3)), where lines correspond to SETs, and in the affine plane of order nine (AG(2,9)). SETless collections and other aspects of the game of SET will be explored through caps in AG(4,3) and conics in AG(2,9).
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Title: Intrinsic Knotting of Partite Graphs
Author: Chloe Collins, Portland State University
Ryan Hake, California State University, Chico
Cara Petonic, Bryn Mawr
Laura Sardagna, Academy of the Pacific in Honolulu
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Abstract: A graph is intrinsically knotted (IK) if for every embedding of the graph there exists a knotted cycle. Let G be a multipartite graph, and form the multipartite graph G by increasing the number of vertices in each of the parts except one and then deleting an edge. We show that if G is IK, then the resulting graph G is also IK. We use this idea to describe large families of IK multipartite graphs. In particular we use the fact that K5,5\2e is IK to show that a bipartite graph with 10 or more vertices (respectively 12 or more vertices) with exactly 5 (resp. 6) in one part and E(G) <= 4V(G)-17 (resp. E(G) <=5V(G)-27) is IK. Our method cant be improved since we also show that K5,5\3e is not IK in general.
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Title: A Multi-Objective Approach to Portfolio Optimization
Author: Yaoyao Clare Duan, Boston College Author Bio
Abstract: Optimization models play a critical role in determining portfolio strategies for investors. The traditional mean variance optimization approach has only one objective, which fails to meet the demand of investors who have multiple investment objectives. This paper presents a multi-objective approach to portfolio optimization problems. The proposed optimization model simultaneously optimizes portfolio risk and returns for investors and integrates various portfolio optimization models. Optimal portfolio strategy is produced for investors of various risk tolerance. Detailed analysis based on conve x optimization and application of the model are provided and compared to the mean variance approach.
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Title: A Review of the Potts Model
Author: Laura Beaudin, Saint Michael's College Author Bio
Abstract: This paper examines a mathematical modeling tool for complex systems with nearest neighbor interactions known as the Potts model. We begin by explaining the structure of the model and defining its Hamiltonian, probability function, and partition function. We then focus on the partition function, giving examples and showing the equivalence of two different formulations. We then introduce the Tutte polynomial a well known graph invariant. We give details of the equivalence of the Tutte polynomial and the Potts model partition function. Since the Tutte polynomial, and hence the Potts model partition function, is computationally intractable, we explore Monte Carlo simulations of the Potts model. Finally, we discuss three applications illustrating how these simulations model real world situations.
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