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        »   vol. 14, issue 1
        »   vol. 14, issue 2


» Vol. 14, Issue 2, 2013 «



Title: On the Coarse Geometry of Lp([a,b])
Author: Phanuel Mariano, University of Connecticut Author Bio    
Abstract: Coarse geometry deals with the large-scale geometry of a space as opposed to its small-scale structure. This paper investigates the concept of coarse geometry and specifically studies the coarse geometry of spaces regularly encountered in real analysis. We construct a non-separable space that is coarse equivalent to the separable space L1([a,b],m).
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Title: Dominance Over ℵ
Author: Tyler Ball, Pacific Lutheran University
Daniel Juda, University of Arkansas
Author Bio    
Author Bio    
Abstract: This paper provides an overview of the b-dominance order over the natural numbers, ℵ, using the base b expansion of natural numbers. The b-dominance order is an accessible partially-ordered set that is less complex than the divisor relation but more complex than ≤; thus, it supplies a good medium through which an undergraduate can be exposed to the subject of order theory. Here we discuss many ideas in order theory, including the Poincare polynomial and the Mobius function.
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Title: The Broncho Tower: A Modification of the Tower of Hanoi Puzzle
Author: Stephen B. Gregg, University of Central Oklahoma
Juan Orozco, University of Central Oklahoma
Author Bio    
Author Bio    
Abstract: We explore ways to find an upper bound on the minimum number of moves to complete a modified version of the Tower of Hanoi puzzle. First, we demonstrate how a simpler modification's minimum number of moves can be obtained using Difference Equations. Next, we show how our modified puzzle differs and how we can apply the same techniques to our puzzle.
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Additional Downloads: Appendix: The Iterative C++ Program
A short animation of this research.

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Title: A Non-geometric Switch Toggling Problem
Author: Megan Duke, Muskingum University Author Bio    
Abstract: Switch toggling games such as Lights Out and the σ+-game are widely studied in mathematics and have been applied to model a variety of situations such as genetic networks and cellular automata. This paper introduces a class of toggling games where at each iteration a fixed number of switches is chosen to be toggled where the only switches changed are the switches chosen to be toggled. The switches all operate independently of each other and do not depend on the proximity or the position relative to any other switch. This paper classifies the conditions necessary and the steps taken to transition from all switches in the on state to all switches in the off state. Further results include the conditions required of the parity between the number of switches in the system and the fixed number of switches toggled at each step in order to transition from a given initial state to a specified terminal state.
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Title: Iterated functions and the Cantor set in one dimension
Author: Benjamin Hoffman, Lewis & Clark College
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Abstract: In this paper we consider the long-term behavior of points in ℜ under iterations of continuous functions. We show that, given any Cantor set Λ* embedded in ℜ, there exists a continuous function F*:ℜ → ℜ such that the points that are bounded under iterations of F* are just those points in Λ* . In the course of this, we find a striking similarity between the way in which we construct the Cantor middle-thirds set, and the way in which we find the points bounded under iterations of certain continuous functions.
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Title: Proof of Euler's φ (Phi) Function Formula
Author: Juan Vargas, UNC Charlotte
Shashank Chorge, Mumbai University
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Author Bio    
Abstract: Euler's φ (phi) Function counts the number of positive integers not exceeding n and relatively prime to n. Traditionally, the proof involves proving the φ function is multiplicative and then proceeding to show how the formula arises from this fact. We ignore this fact, at least directly, and show a practical and sound method to calculate φ. We offer a proof of the closed form formula for this function relying on similar, but subtly different counting techniques.
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Title: The Three-Variable Bracket Polynomial for Reduced, Alternating Links
Author: Kelsey Lafferty, Wheaton College, Wheaton, IL
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Abstract: We first show that the three-variable bracket polynomial is an invariant for reduced, alternating links. We then try to find what the polynomial reveals about knots. We find that the polynomial gives the crossing number, a test for chirality, and in some cases, the twist number of a knot. The extreme degrees of d are also studied.
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Title: Subfield-Compatible Polynomials over Finite Fields
Author: John J. Hull, Georgia State University Author Bio    
Abstract: Polynomial functions over finite fields are important in computer science and electrical engineering in that they present a mathematical representation of arithmetic circuits. This paper establishes necessary and sufficient conditions for polynomial functions with coefficients in a finite field and naturally restricted degrees to be compatible with given subfields. Most importantly, this is done for the case where the domain and codomain fields have differing cardinalities. These conditions, which are presented for polynomial rings in one and several variables, are developed via a universal permutation that depends only on the cardinalities of the given fields.
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Title: Analytic Extension and Conformal Mapping in the Dual and the Double Planes
Author: Conrad Blom, Calvin College
Timothy DeVries, Calvin College
Andrew Hayes, Calvin College
Daiwei Zhang, Calvin College
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Author Bio    
Author Bio    
Author Bio    
Abstract: Many theorems in the complex plane have analogues in the dual (x+jy, j2=0) and the double (x+ky, k2=1) planes. In this paper, we prove that Schwarz reflection principle holds in the dual and the double planes. We also show that in these two planes the domain of an analytic function can usually be extended analytically to a larger region. In addition, we find that a certain class of regions can be mapped conformally to the upper half plane, which is analogous to the Riemann mapping theorem.
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Title: Entanglement-Assisted Quantum Error-Correcting Codes from Generalized Quadrangles
Author: William Thomas, University of Chicago Author Bio    
Abstract: A generalized quadrangle GQ(s,t) is an incidence structure consisting of points and lines in which each line is incident with a fixed number of points, each point is incident with a fixed number of lines, and there is exactly one line connecting any point with a line not incident with the point. Entanglement-assisted quantum error-correcting codes provide a method for correcting data transmission errors in quantum computers. EAQECCs require entangled quantum states, called ebits, and it is desirable to minimize the number of ebits a code uses because ebits are difficult to manufacture. We use a binary incidence matrix N of a generalized quadrangle to create entanglement- assisted quantum error-correcting codes. The rank of NNT gives the number of ebits a code requires. Because incidence matrices of generalized quadrangles are highly structured and reflect the geometric properties of the quadrangles, we can examine the rank of N and NNT and write the parameters of quantum codes in terms of s and t. We identify a class of generalized quadrangles that produce quantum codes that require a low number of ebits, a class that produce quantum codes that require a large number of ebits, and a class that produces quantum codes that are too small to be useful.
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Title: Directed Graphs of Commutative Rings
Author: Seth Hausken, University of St. Thomas
Jared Skinner, University of St. Thomas
Author Bio    
Author Bio    
Abstract: The directed graph of a commutative ring is a graph representation of its additive and multiplicative structure. Using the mapping (a,b) → (a+b,a ⋅ b) one can create a directed graph for every finite, commutative ring. We examine the properties of directed graphs of commutative rings, with emphasis on the information the graph gives about the ring.
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