» Vol. 10, Issue 2, 2009 «

Title: Using Biomechanical Optimization to Interpret Dancers' Pose Selection for a Partnered Spin Author: Megan E. Selbach-Allen, U.S. Naval Academy Author Bio     Abstract: Swing dancing is a high tempo, athletic form of dancing. In performing their physically rigorous jumps, lifts, and spins, dancers often talk about using the laws of physics. However, they do not have mathematical evidence to support these claims. Our goal was to determine whether expert swing dancers physically optimize their pose for a partnered spin. In a partnered spin, two dancers connect hands and spin around a single vertical axis. A biomechanical model built in Mathematica allowed comparisons of mathematically produced optimal poses to live dancers with the use of a motion capture system. We hypothesized that expert swing dancers would achieve a higher fraction of their optimal acceleration than beginners. We were unable to determine a statistically significant difference between the posses of expert and beginner dancers. However, the optimal pose predicted by the model was intuitively reasonable. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: A Brief History of Elliptic Integral Addition Theorems Author: Jose Barrios, Montclair State University Author Bio     Abstract: The discovery of elliptic functions emerged from investigations of integral addition theorems. An addition theorem for a function f is a formula expressing f(u+v) in terms of f(u) and f(v). For a function defined as a definite integral with a variable upper limit, an addition theorem takes the form of an equation between the sum of two such integrals, with upper limits u and v, and an integral whose upper limit is a certain function of u and v.In this paper, we briefly sketch the role which the investigation of such addition theorems has played in the development of the theory of elliptic intgrals and elliptic functions. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Weyl group Bn as Automorphisms of n-cube: Isomorphism and Conjugacy Authors: David Chen, University of California, Los Angeles Author Bio     Abstract: The Weyl groups are important for Lie algebras. Lie algebras arise in the study of Lie groups, coming from symmetries of differential equations, and of differentiable manifolds. The Weyl groups have been used to classify Lie algebras up to isomorphism. The Weyl group associated to a Lie algebra of type Bn, and the group of graph automorphisms of the n-cube, Aut(Qn), are known to be isomorphic to Z2n x Sn. We provide a direct isomorphism between them via correspondence of generators. Geck and Pfeiffer have provided a parametrization of conjugacy classes and an algorithm to compute standard representatives. We believe we have a more transparent account of conjugacy in the Weyl group by looking at Aut(Qn). We give a complete description of conjugacy in the automorphism group. We also give an algorithm to recover a canonical minimal length (in the Weyl group sense) representative from each conjugacy class, and an algorithm to recover that same representative from any other in the same conjugacy class. Under the correspondence with the Weyl group, this representative coincides precisely with the minimal length representative given by Geck and Pfeiffer, leading to an easier derivation of their result. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Zero-Divisors and Their Graph Languages Authors: Harley D. Eades III, University of Iowa Author Bio     Abstract: We introduce the use of formal languages in place of zero-divisor graphs used to study theoretic properties of commutative rings. We show that a regular language called a graph language can be constructed from the set of zero-divisors of a commutative ring. We then prove that graph languages are equivalent to their associated graphs. We go on to define several properties of graph languages. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Virtual Mosaic Knots Authors: Irina T. Garduno, University of Illinois at Chicago Author Bio     Abstract: We extend mosaic knot theory to virtual knots and define a new type of knot: virtual mosaic knot. As in classical knots, Reidemeister moves are applied to a virtual mosaic knot to transform one knot diagram into another. Additionally, given the mosaic number of a virtual knot, we find an upper bound on the sum of the classical and virtual crossing numbers. Furthermore, given the classical and virtual crossing numbers of a knot, we find a lower bound on the virtual mosaic number of a knot. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: The Most General Planar Transformations that Map Hyperbolas to Hyperbolas Author: James Hays, Calvin College Todd Mitchell, Calvin College Author Bio     Author Bio     Abstract: The space of vertical and horizontal right hyperbolas and the lines tangent to these hyperbolas is considered in the double plane. It is proved that an injective map from the middle region of a considered hyperbola that takes hyperbolas and lines in this space to other hyperbolas and lines in this space must be a direct or indirect linear fractional transformation. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Statistical Investigation of Structure in the Discrete Logarithm Authors: Andrew Hoffman, Wabash College Author Bio     Abstract: The absence of an efficient algorithm to solve the Discrete Logarithm Problem is often exploited in cryptography. While exponentiation with a modulus, bx≡ a (mod m), is extremely fast with a modern computer, the inverse is decidedly not. At the present time, the best algorithms assume that the inverse mapping is completely random. Yet there is at least some structure, such as the fact that b1≡ b (mod m). To uncover additional structure that may be useful in constructing or refining algorithms, statistical methods are employed to compare mappings, x ≡ bx (mod m), to random mappings. More concretely, structure will be defined by representing the mappings as functional graphs and using parameters from graph theory such as cycle length. Since the literature for random permutations is more extensive than other types of functional graphs, only permutations produced from the experimental mappings are considered. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: A Stochastic Shell Model of Turbulence: Numerical and Analytical Results Author: Kristen Campilonga, University of Maryland Dennis Gucker, University of Northern Colorado Joshua Keller, Emory University Author Bio     Author Bio     Author Bio     Abstract: We consider an inviscid shell model of turbulence with the addition of Itô and Stratonovich multiplicative stochastic forcing. Numerical simulations are performed for both models that show dissipation of energy at an algebraic rate. A comparison between the Itô and Stratonovich effects is examined. Positivity of solutions is discussed and demonstrated numerically. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: A p-adic Euclidean Algorithm Author: Cortney Lager , Winona State University Author Bio     Abstract: The rational numbers can be completed with respect to the standard absolute value and this produces the real numbers. However, there are other absolute values on the rationals besides the standard one. Completing the rationals with respect to one of these produces the p-adic numbers. In this paper, we take some basic number theory concepts and apply them to rational p-adic numbers. Using these concepts, a p-adic division algorithm is developed along with a p-adic Euclidean Algorithm. These algorithms produce a generalized greatest common divisor in the p-adics along with a p-adic simple continued fraction. In Section 2, we describe the p-adic numbers, and in Section 3 we present our p-adic Division Algorithm and Euclidean Algorithm. In Section 4, we show some applications, including a connection to Browkin's p-adic continued fractions, which motivatored our investigations in the first place. Finally, in Section 5, we give some open questions for further study. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Extensions of Extremal Graph Theory to Grids Author: Bret Thacher, Williams College Author Bio     Abstract: We consider extensions of Turán's original theorem of 1941 to planar grids. For a complete kxm array of vertices, we establish in Proposition 4.3 an exact formula for the maximal number of edges possible without any square regions. We establish with Theorem 4.12 an upper bound and with Theorem 4.15 an asymptotic lower bound for the maximal number of edges on a general grid graph with n vertices and no rectangles. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Envy-Free Divisions Author: Seth Unruh, Goshen College Author Bio     Abstract: We consider the division of a single homogeneous object and transfers of money among several people who may have different valuations for the object. A division is envy free if every person believes the division he or she received is at least as valuable as the division received by each other person. If no money is transferred, the only envy-free division involves each person receiving the same fraction of the object. When money transfers are allowed, we show that the set of envy-free divisions is a simplex whose vertices involve giving the same fraction of the object to each person in a set of persons who most value the object, and in turn those people pay the same amount of money to the other people who receive none of the object. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Variants of the Game of Nim that have Inequalities as Conditions Author: Toshiyuki Yamauchi, Kwansei Gakuin University, Japan Taishi Inoue, Kwansei Gakuin University, Japan Yuuki Tomari, Kwansei Gakuin University, Japan Author Bio     Author Bio     Author Bio     Abstract: In this article the authors are going to present several combinatorial games that are variants of the game of Nim. They are very different from the traditional game of Nim, since the coordinates of positions of the game satisfy inequalities. These games have very interesting mathematical structures. For example, the lists of P-positions of some of these variants are subsets of the list of P-positions of the traditional game of Nim. The authors are sure that they were the first people who treated variants of the game of Nim conditioned by inequalities. Some of these games will produce beautiful 3D graphics (indeed, you will see the Sierpinski gasket when you look from a certain view point). We will also present some new results for the chocolate problem, a problem which was studied in a previous paper and related to Nim. The authors make substantial use of Mathematica in their research of combinatorial games. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: Isoperimetric Regions on a Weighted 2-Dimensional Lattice Author: Deividas Seferis, Williams College Author Bio     Abstract: We investigate isoperimetric regions in the 1st quadrant of the 2- dimensional lattice, where each point is weighted by the sum of its coordinates. We analyze the isoperimetric properties of six types of regions located in the first quadrant of the Cartesian plane: squares, rectangles, quarter circles, diamonds, crosses and triangles. To compute volume and perimeter of each region we use summation and integration methods which give comparable but not identical results. Among our candidates the diamond has the least perimeter for given volume. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: A special case of the Yelton-Gaines Conjecture on Isomorphic Dessins Author: Claudia Raithel, University of Michigan- Ann Arbor Author Bio     Abstract: Let (r0, r1) and (r0¢, r1¢) be two ordered pairs of permutations in Sn and let t be a divisor of n. The Yelton-Gaines conjecture states that if at least one of these four permutations is a product of n/t disjoint t-cycles, and if there is a strong isomorphism (definition below) φ:< r0,r1> ® < r0¢, r1¢> between the two subgroups of Sn generated by the elements in each ordered pair, then there is a fixed permutation t in Sn that simultaneously conjugates ri to ri¢ for i=0,1. The conclusion of this conjecture can be restated to say that the two dessins d'enfants corresponding to the two ordered pairs are isomorphic. In this paper a proof of this conjecture is given in the case in which all of the initial four permutations are fixed-point-free involutions. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP Title: General Models for Variations of the Even Cycle Problem Author: Rana Kunkar , Illinois State Don Andrew Macatangay, Illinois State Rachel Pepich, Illinois State Author Bio     Author Bio     Author Bio     Abstract: We consider three related problems in graph theory: determining if a directed graph has a directed even cycle, determining if a two edged-colored graph has an alternating colored even cycle, and determining if a directed graph has an anti-directed even cycle. We show that each of these, and two other variations, are special cases of a more general graph problem. We also show that one of these variations is NP-complete. Article: Downloadable PDF     Additional Downloads: » BACK TO TOP