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        »   vol. 14, issue 1


» Vol. 14, Issue 1, 2013 «



Title: Existence of the limit at infinity for a function that is integrable on the half line
Author: James Patrick Dix, San Marcos High School, Texas Author Bio    
Abstract: It is well known that for a function that is integrable on [0, ), its limit at infinity may not exist. First we illustrated this statement with an example. Then, we present conditions that guarantee the existence of the limit in the following two cases: When the integrable function is non-negative, if the first, second, third, or fourth, derivative is bounded in a neighborhood of each local maximum of f, then the limit exists. Without the non-negative constraint, if an integrable function has a bounded derivative on the entire interval [0, ), then the limit exists.
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Title: Tangent circles in the hyperbolic disk
Author: Megan Ternes, Aquinas College Author Bio    
Abstract: Constructions of tangent circles in the hyperbolic disk, interpreted in Euclidean geometry, give us examples of four mutually tangent circles. These are shown to satisfy Descartes's Theorem for tangent circles. We also show that the Archimedes twin circles in the hyperbolic arbelos are usually not hyperbolic congruent, even though they are Euclidean congruent. We include a few construction instructions because all items under consideration require surprisingly few steps.
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Title: Rotationally Symmetric Rose Links
Author: Amelia Brown, Simpson College Author Bio    
Abstract: This paper is an introduction to rose links and some of their properties. We used a series of invariants to distinguish some rose links that are rotationally symmetric. We were able to distinguish all 3-component rose links and narrow the bounds on possible distinct 4 and 5-component rose links to between 2 and 8, and 2 and 16, respectively. An algorithm for drawing rose links and a table of rose links with up to five components are included.
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Title: The Weierstrass representation always gives a minimal surface
Author: Roshan Sharma, Williams College Author Bio    
Abstract: We give a simple, direct proof of the easy fact about the Weierstrass Representation, namely, that it always gives a minimal surface. Most presentations include the much harder converse that every simply connected minimal surface is given by the Weierstrass Representation.
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Title: Trees of Irreducible Numerical Semigroups
Author: Taryn M. Laird, Northern Arizona University
Jose E. Martinez, Northern Arizona University
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Author Bio    
Abstract: A 2011 paper by Blanco and Rosales describes an algorithm for constructing a directed tree graph of irreducible numerical semigroups of fixed Frobenius numbers. This paper will provide an overview of irreducible numerical semigroups and the directed tree graphs. We will also present new findings and conjectures concerning the structure of these trees.
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Title: Klein Links and Braids
Author: David Freund, The College of Wooster
Sarah Smith-Polderman, The College of Wooster
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Author Bio    
Abstract: We introduce the construction of Klein links through an alteration to the orientation on the rectangular representation of a torus knot. We relate the resulting Klein links to their corresponding braid representations, and use these representations to understand the relationship between Klein links and torus knots as well as to prove relationships between several different Klein links.
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Title: Directed Graphs of Commutative Rings with Identity
Author: Christopher Ang, Michigan State University
Alex Schulte, University of St. Thomas
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Author Bio    
Abstract: The directed graph of a ring is a graphical representation of its additive and multiplicative structure. Using the directed edge relationship (a,b) → (a+b,a ⋅ b), one can create a directed graph for every ring. This paper focuses on the structure of the sources in directed graphs of commutative rings with identity, with special concentration in the finite and reduced cases.
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Title: Equilateral Dimension of Riemannian Manifolds with Bounded Curvature
Author: Jeremy Mann, Johns Hopkins University Author Bio    
Abstract: The equilateral Dimension of a riemannian manifold is the maximum number of distinct equidistant points. In the first half of this paper we will give upper bounds for the equilateral dimension of certain Riemannian Manifolds. In the second half of the paper we will introduce a new metric invariant, called the equilateral length, which measures size of the equilateral dimension. This will then be used in the recognition program in Riemannian geometry, which seeks to identify certain Riemannian manifolds by way of metric invariants such as diameter, extent, or packing radius.
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Title: Generic Polynomials for Transitive Permutation Groups of Degree 8 and 9
Author: Bradley Lewis Burdick, The Ohio State University
Jonathan Jonker, Michigan State University
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Author Bio    
Abstract: We compute generic polynomials for certain transitive permutation groups of degree 8 and 9, namely SL(2,3), the generalized dihedral group: C2 \ltimes (C3 x C3), and the Iwasawa group of order 16: M16. Rikuna proves the existence of a generic polynomial for SL(2,3) in four parameters; we extend a computation of Grobner to give an alternative proof of existence for this group's generic polynomial. We establish that the generic dimension and essential dimension of the generalized dihedral group are two. We establish over the rationals that the generic dimension and essential dimension of SL(2,3) and M16 are four.
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Title: Category Theory and Galois Theory
Author: Amanda Bower, University of Michigan--Dearborn Author Bio    
Abstract: Galois theory translates questions about fields into questions about groups. The fundamental theorem of Galois theory states that there is a bijection between the intermediate fields of a field extension and the subgroups of the corresponding Galois group. After a basic introduction to category and Galois theory, this project recasts the fundamental theorem of Galois theory using categorical language and illustrates this theorem and the structure it preserves through an example.
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Title: Bounds for elements of the degree sequence of an unknown vertex set in a balanced bipartite graph
Author: Sam Pine, Elmira College Author Bio    
Abstract: Consider the set of all balanced bipartite graphs. Given the degree sequence of one vertex set in one of these graphs, we find bounds for any given position in the degree sequence of the unknown vertex set. Additionally, we establish bounds for the median of the unknown degree sequence, as well as bounds for any given percentile. We discuss the connection between this paper and the High School Prom Theorem.
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Title: Counting Maximal Chains in Weighted Voting Posets
Author: George Story, Wake Forest University Author Bio    
Abstract: Weighted voting is built around the idea that voters have differing amounts of influence in elections, with familiar examples ranging from company shareholder meetings to the United States Electoral College. We examine the idea that each voter has a uniquely determined weight, paying particular attention to how voters leverage this weight to get their way on a specific yes/no motion (for example, by forming coalitions). After some more background on weighted voting, we describe a natural partial order relation between these coalitions of voters. This ordering can be modeled by a partially ordered set (poset), which we call a coalitions poset. Using this poset, we derive another important poset via a natural ordering on collections of coalitions. Our results begin by detailing a method for counting the number of maximal chains in the derived poset. After employing this method to find the number of maximal chains in the derived poset with 5 voters, we extend our method for use in the coalitions poset. Finally, we conjecture a formula for the number of maximal chains in the coalitions poset with n voters.
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Title: Sectioning Angles Using Hyperbolic Curves
Author: Julie Fink, Tulane University
Nicholas Molbert, University of Louisiana at Lafayette
Author Bio    
Author Bio    
Abstract: In this paper, we construct a single hyperbola G that, along with a straight edge and compass, allow for the trisection of any angle. Descartes constructed a parabola with this property in his original treatment of analytic geometry. Unlike Descartes's proof, the proof that all angles can be trisected with the hyperbola G is a geometric rather than an algebraic argument.
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