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» Vol. 16, Issue 1, 2015 «



Title: Deconstructing the Welch Equation Using p-adic Methods
Author: Abigail Mann, Rose-Hulman Institute of Technology
Adelyn Yeoh, Mount Holyoke College
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Abstract: The Welch map x —> gx-1+c is similar to the discrete exponential map x —> gx, which is used in many cryptographic applications including the ElGamal signature scheme. This paper analyzes the number of solutions to the Welch equation, gx-1+c = x (mod pe), where p is a prime and g is a unit modulo p, and looks at other patterns of the equation that could possibly be exploited in a similar cryptographic system. Since the equation is modulo pe, where p is a prime number, p-adic methods of analysis are used in counting the number of solutions modulo pe. These methods include p-adic interpolation, Hensel's Lemma and the Chinese Remainder Theorem.
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Title: Perimeter-minimizing Tilings by Convex and Non-convex Pentagons
Author: Whan Ghang, Massachusetts Institute of Technology
Zane Martin, Williams College
Steven Waruhiu, University of Chicago
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Abstract: We study the presumably unnecessary convexity hypothesis in a theorem of Chung et al. on perimeter-minimizing planar tilings by convex pentagons. We prove that the theorem holds without the convexity hypothesis in certain special cases, and we offer direction for further research.
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Title: Minimal Number of Steps in the Euclidean Algorithm and its Application to Rational Tangles
Author: M. Syafiq Johar, University of Oxford Author Bio    
Abstract: We define the regular Euclidean algorithm and the general form which leads to the method of least absolute remainders and also the method of negative remainders. We show that if looked from the perspective of subtraction, the method of least absolute remainders and the regular method have the same number of steps which is in fact the minimal number of steps possible. This enables us to determine the most efficient way to untangle a rational tangle.
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Title: E-ergodicity and Speedups
Author: Tyler B. George, Ferris State University Author Bio    
Abstract: We introduce the notion of E-ergodicity of a measure-preserving dynamical system (where E is a subset of the natural numbers). We show that given an E-ergodic system T and aperiodic system S, T can be sped up to obtain an isomorphic copy of S, using a function taking values only in E. We give examples applying this concept to the situation where E is a congruence class, the image of an integer polynomial, or the prime numbers.
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Title: The Multiplicative Structure of the Group of Units of Zp[x]/<f(x)> where f(x) is Reducible
Author: Erika Gerhold, Salisbury University
Jennifer Ferralli, Salisbury University
Jason Jachowski, Salisbury University
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Abstract: Factor rings of the form Zp[x]/<f(x)>, with p prime and f(x) irreducible in Zp[x], form a field, with cyclic multiplicative group structure. When f(x) is reducible in Zp[x] this factor ring is no longer a field, nor even an integral domain, and the structure of its group of units is no longer cyclic. In this paper we develop concise formulas for determining the cyclic group decomposition of the multiplicative group of units for Zp[x]/<f(x)> that is only dependent on the multiplicities and degrees of the irreducible factors of f(x), and p.
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Title: Lie Matrix Groups: The Flip Transpose Group
Author: Madeline Christman, California Lutheran University Author Bio    
Abstract: In this paper we explore the geometry of persymmetric matrices. These matrices are defined in a similar way to orthogonal matrices except the (flip) transpose is taken over the skew diagonal. We give a proof that persymmetric matrices form a Lie group.
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Title: Optimizing the Allocation of Vaccines in the Presence of Multiple Strains of the Influenza Virus
Author: Ana Eveler, Valparaiso University
Tayler Grashel, Valparaiso University
Abby Kenyon, Valparaiso University
Jessica Richardson, Valparaiso University
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Abstract: During the annual flu season, multiple strains of the influenza virus are often present within a population. It is a significant challenge for health care administrators to determine the most effective allocation of multiple different vaccines to combat the various strains when protecting the public. We employ a mathematical model, a system of differential equations, to find a strategy for vaccinating a population to minimize the number of infected individuals. We consider various strengths of transmission of the disease, availability of vaccine doses, vaccination rates, and other model parameters. This research may lead to more effective health care policies for vaccine administration.
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Title: On the Center of Non Relativistic Lie Algebras
Author: Tyler Gorshing, Southwestern Oklahoma State University Author Bio    
Abstract: The center of the Schrodinger Lie algebra is the Lie subalgebra generated by its center of mass. An explicit mathematical proof of this statement doesn't seem to be available in literature. In this paper, we use elementary matrix multiplication to prove it. We also investigate the case of the Galilei Lie algebra, the Harmonic Oscillator Lie algebra and the Heinsenberg-Weyl Lie algebra. We show by calculation that these non-relativistic Lie algebras have no center unless centrally extended.
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Title: Identifying a Coefficient Matrix for Numerical Approximations to the Wave Equation in Two Dimensions
Author: Morgan M. Brown, University of Mary Washington Author Bio    
Abstract: In this paper we develop a coefficient matrix to be used in numerical approximation methods which model the wave equation in two dimensions. In particular, we will briefly introduce the centered difference approximation method. Next, we explain the derivation of the system of equations into which the problem is transformed in order to utilize such a method. Then, we introduce a set of rules to generate the generalized coefficient matrix for use in approximating the wave equation for any number of unknowns per axis. Finally, we write MATLAB code which uses our matrix to solve said approximations, and then use it in a real world application. The desired outcome of this project has been achieved: to generalize the coefficient matrix used in the system of equations which approximates the wave equation in two dimensions so that its algorithm may be used in MATLAB code for any number of unknowns.
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Title: On the Mathematics of Utility Theory
Author: Harshil Sahai, Swarthmore College Author Bio    
Abstract: Utility theory is a field of economics which hopes to model the innate preferences humans have toward different objects. Though it is most obviously economic in spirit and application, the ever-growing discipline finds its theoretical roots in mathematics. This paper will explore the mathematical underpinnings of basic utility theory by following, divulging, and extending the work of Ok [Real Analysis with Economics Applications, 2007]. We will develop necessary analytic and algebraic concepts, and use this mathematical framework to support hypotheses in theoretical economics. Specifically, we establish classical existence theorems (Rader, Debreu, and von-Neumann Morgenstern) in both utility and expected utility contexts. The paper will require only a firm grasp of real analysis in Rn, elementary group theory and linear algebra, and will proceed assuming no prior knowledge of economic theory.
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Title: First Encounters with Option Pricing and Return Simulation
Author: Joshua Cape, Rhodes College
William Dearden, Lehigh University
William Gamber, Pomona College
Linh Nguyen, Lafayette College
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Abstract: We provide a tractable introduction to option pricing models and examine how the complex analysis concept of branch-cutting influences financial mathematics. The Black-Scholes model is introduced to motivate our discussion of the Heston stochastic volatility model, a model which dominates industry and option pricing literature in financial mathematics. We focus on developing mathematical intuition as a tool for stimulating further undergraduate interest and research in financial mathematics. We provide code in R and Mathematica for applications.
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Title: Are circles isoperimetric in the plane with density er?
Author: Ping Ngai Chung, University of Cambridge
Miguel A. Fernandez, Truman State University
Niralee Shah, Williams College
Luis Sordo Vieira, University of Kentucky
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Abstract: We prove that an isoperimetric region in R2 with density er must be convex and contain the origin, and provide numerical evidence that circles about the origin are isoperimetric, as predicted by the Log-Convex Density Conjecture.
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Title: Folding concave polygons into convex polyhedra: The L-Shape
Author: Emily Dinan, University of Washington
Alice Nadeau, University of Minnesota
Isaac Odegard, University of North Dakota
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Abstract: Mathematicians have long been asking the question: Can a given convex polyhedron can be unfolded into a polygon and then refolded into any other convex polyhedron? One facet of this question investigates the space of polyhedra that can be realized from folding a given polygon. While convex polygons are relatively well understood, there are still many open questions regarding the foldings of non-convex polygons. We analyze these folded realizations and their volumes derived from the polygonal family of `L-shapes,' parallelograms with another parallelogram removed from a corner. We investigate questions of maximal volume, diagonal flipping, and topological connectedness and discuss the family of polyhedra that share a L-shape polygonal net.
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Title: Prime Vertex Labelings of Families of Unicyclic Graphs
Author: Nathan Diefenderfer, Northern Arizona University
Michael Hastings, Northern Arizona University
Levi N. Heath, Northern Arizona University
Hannah Prawzinsky, Northern Arizona University
Briahna Preston, Northern Arizona University
Emily White, Northern Arizona University
Alyssa Whittemore, Northern Arizona University
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Abstract: A simple n-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers 1 through n such that adjacent vertices have relatively prime labels. We will present previously unknown prime vertex labelings for new families of graphs, all of which are special cases of Seoud and Youssef's conjecture that all unicyclic graphs have a prime labeling.
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Title: An Introduction to the Birch and Swinnerton-Dyer Conjecture
Author: Brent Johnson, Villanova University Author Bio    
Abstract: This article explores the Birch and Swinnerton-Dyer Conjecture, one of the famous Millennium Prize Problems. In addition to providing the basic theoretic understanding necessary to understand the simplest form of the conjecture, some of the original numerical evidence used to formulate the conjecture is recreated. Recent results and current problems related to the conjecture are given at the end.
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