» Vol. 11, Issue 2, 2010 «

Title: Minimal connected partitions of the sphere Author: Edward Newkirk, Williams College Author Bio     Abstract: We consider the soap bubble problem on the sphere S2, which seeks a perimeter-minimizing partition into n regions of given areas. For n = 4, it is conjectured that a tetrahedral partition is minimizing. We prove that there exists a unique tetrahedral partition into given areas, and that this partition has less perimeter than any other partition dividing the sphere into the same four connected areas. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: A Metacalibration Proof of the Isoperimetric Problem Author: James Dilts, Brigham Young University Author Bio     Abstract: The isoperimetric problem asks, among all figures with the same perimeter (iso-perimetric means ``same perimeter''), which has the greatest area. This paper proves the classic isoperimetric problem using a generalization of calibration techniques which we call metacalibration. We then generalize to arbitrary dimensions and to spherical spaces. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: On the Fixed Points of Abelian Group Automorphisms Author: James Checco, St. Olaf College Rachel Darling, St. Olaf College Stephen Longfield, St. Olaf College Katherine Wisdom, St. Olaf College Author Bio     Author Bio     Author Bio     Author Bio     Abstract: In this article, we present general properties of fixed-point groups of the automorphisms of finite groups. Specifically, we determine the form of fixed-point groups and partition $\aut(G)$ according to the number of fixed points possessed by each automorphism. A function $\theta$ records the size of each partitioning set; we find properties for $\theta$ in general and develop formulae for $\theta$ with respect to certain classes of finite abelian groups. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Counting Modular Tableaux Authors: Nathan Meyer, St. Olaf College Daniel Mork, St. Olaf College Benjamin Simmons, St. Olaf College Bjorn Wastvedt, St. Olaf College Author Bio     Author Bio     Author Bio     Author Bio     Abstract: In this paper we provide a bijection between all modular tableaux of size $kn$ and all partitions of $n$ labeled with $k$ colors. This bijection consists of a new function proven in this paper composed with mappings given by Garrett and Killpatrick in \cite{An1} and Stanton and White in \cite{An2}. We also demonstrate the novel construction and proof of a mapping essentially equivalent to Stanton and White's, but more useful for the purposes of the bijection mentioned above. By using the generating function for the number of $k$-colored partitions of $n$ in conjunction with our bijection, we can count the number of modular tableaux of size $kn$. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Graphs and principal ideals of finite commutative rings Authors: J. Cain, University of Dayton, Dayton OH L. Mathewson, Carroll University, Waukesha, WI A. Wilkens, Beloit College, Beloit, WI Author Bio     Author Bio     Author Bio     Abstract: In \cite{ABM}, Afkhami and Khashyarmanesh introduced the cozero-divisor graph of a ring, $\Gamma'(R)$, which examines relationships between principal ideals. We continue investigating the algebraic implications of the graph by developing the reduced cozero-divisor graph, which is a simpler analog. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Structure and Statistics of the Self-Power Map Author: Matthew Friedrichsen, St. Olaf College Brian Larson, Wheaton College Emily McDowell, University of Pennsylvania Author Bio     Author Bio     Author Bio     Abstract: We investigate the structure of a function relevant to cryptography, given by $f: x \mapsto x^x \bmod{p}$, for $p$ a prime. We call $f$ the \textit{self-power map}. Given $x$, it is easy to calculate $f(x) \equiv x^x \pmod{p}.$ However, it is thought to be difficult to quickly calculate $f^{-1}(x^x)$. That is, given $x^x \equiv c \pmod{p}$, for a fixed $c$, it is difficult to quickly solve for $x$. We call the problem of finding the inverse of the self-power map the \textit{Self-Power Problem}. As a variation of the Discrete Logarithm Problem, the Self-Power Problem is thought to be difficult to solve and therefore considered safe for use in some versions of the ElGamal Digital Signature Algorithm. Nonetheless, utilizing functional graphs to represent the map has revealed non-random structural properties, which we describe primarily through number theory and statistics. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Spectra of Semidirect Products of Cyclic Groups Author: Nathan Fox, University of Minnesota-Twin Cities Author Bio     Abstract: The spectrum of a graph is the set of eigenvalues of its adjacency matrix. A group, together with a multiset of elements of the group, gives a Cayley graph, and a semidirect product provides a method of producing new groups. This paper compares the spectra of cyclic groups to those of their semidirect products, when the products exist. It was found that many of the interesting identities that result can be described through number theory, field theory, and representation theory. The main result of this paper gives a formula that can be used to find the spectrum of semidirect products of cyclic groups. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Calibrating the Complexity of Ternary Propositional Connectives Author: William Bradley, Appalachian State University Alex Dunn, Appalachian State University Steve Harenberg, University of North Carolina at Chapel Hill Matthew Owen, Appalachian State University Matthew Roberts, North Carolina State University Author Bio     Author Bio     Author Bio     Author Bio     Author Bio     Abstract: For each ternary propositional connective, we determine the minimum number of binary connectives needed to construct a logically equivalent formula. In order to reduce this problem to a computably feasible one, we prove a number of lemmas showing that every element of a large set of formulas is logically equivalent to a formula in a much smaller associated set. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Computing Fixed Point Floer Homology Author: Jin Woo Jang, Columbia University Rachel Vishnepolsky, Columbia University Xuran Wang, Columbia University Author Bio     Author Bio     Author Bio     Abstract: In the summer of 2009, our group developed a computer program that computes Hochschild Homology, a topological invariant. While we must assume that the reader has at least encountered algebraic topology, in this paper we provide the mathematical background and motivation for our algorithm. After presenting a number of definitions, we will explain how the algorithm works. Specifically, we first define the Floer complex of two curves on surface; the resulting homology is invariant under isotopies. Then, we introduce the Fukaya category associated to a sequence of curves. Next, we define the Hochschild complex of the Fukaya category. And finally, we describe an algorithm for computing Hochschild Homology and provide some examples. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Phase Transitions in the Ising Model Author: Eva Ellis-Monaghan, Villanova University Author Bio     Abstract: This paper investigates the Ising model, a model conceived by Ernst Ising to model ferromagnetism. This paper presents a historical analysis of a model which brings together aspects of graph theory, statistical mechanics, and linear algebra. We will illustrate the model and calculate the probability of individual states in the one dimensional case. We will investigate the mathematical relationship between the energy and temperature of the model, and, using the partition function of the probability equation, show that there are no phase transitions in the one dimensional case. We endeavor to restate these proofs with greater clarity and explanation in order for them to be more accessible to other undergraduates. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Hyperbolic polygonal spirals Author: Jillian Russo, Aquinas College Author Bio     Abstract: This article is based on the construction of Nested Hyperbolic Polygonal Spirals. The construction uses constructible Euclidean angles to create hyperbolic polygons of five or more sides. The nested polygons are formed by connecting the midpoints of the sides of the original polygon, thus creating a spiral. The construction is included for the readers to be able to construct one for themselves as they read along. This construction, along with hyperbolic trigonometric formulas, led to the results: measures of the angles, side lengths and areas of all the parts of the spiral. Furthermore, the construction is used to prove the constructible hyperbolic regular polygons have the same number of sides as the constructible Euclidean polygons. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP