





» Vol. 10, Issue 2, 2009 «


Title:

Using Biomechanical Optimization to Interpret Dancers' Pose Selection for a Partnered Spin



Author:

Megan E. SelbachAllen, 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.



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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.



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Title:

Weyl group B_{n} as Automorphisms of ncube: 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 B_{n},
and the group of graph
automorphisms of the ncube, Aut(Q_{n}), are known to be isomorphic to
Z_{2}^{n} x S_{n}.
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(Q_{n}). 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.



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Title:

ZeroDivisors 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 zerodivisor 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 zerodivisors
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.



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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.



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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.



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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,
b^{x}≡ 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
b^{1}≡ b (mod m). To uncover additional structure that may be
useful in constructing or refining algorithms, statistical methods are employed to compare
mappings, x ≡ b^{x} (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.



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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. 


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Title:

A padic 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 padic numbers. In this paper, we take some basic
number theory concepts and apply them to rational padic numbers. Using these concepts, a padic division
algorithm is developed along with a padic Euclidean Algorithm. These algorithms produce a generalized
greatest common divisor in the padics along with a padic simple continued fraction. In Section 2,
we describe the padic numbers, and in Section 3 we present our padic Division Algorithm and Euclidean
Algorithm. In Section 4, we show some applications, including a connection to Browkin's padic continued
fractions, which motivatored our investigations in the first place. Finally, in Section 5, we give some
open questions for further study.



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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 ^{k}x^{m} 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.



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Title:

EnvyFree 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 envyfree division involves
each person receiving the same fraction of the object. When money transfers
are allowed, we show that the set of envyfree 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.



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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 Ppositions of some of these variants are
subsets of the list of Ppositions 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.



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Title:

Isoperimetric Regions on a Weighted 2Dimensional Lattice



Author:

Deividas Seferis, Williams College

Author Bio



Abstract:

We investigate isoperimetric regions in the 1^{st} 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.



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Title:

A special case of the YeltonGaines Conjecture on Isomorphic Dessins



Author:

Claudia Raithel, University of Michigan Ann Arbor

Author Bio



Abstract:

Let
(r_{0}, r_{1})
and
(r_{0}^{¢},
r_{1}^{¢})
be two ordered pairs of
permutations in S_{n} and let t be a divisor of n.
The YeltonGaines conjecture states that if at
least one of these four permutations is a product of n/t disjoint tcycles, and
if there is a strong
isomorphism (definition
below)
φ:<
r_{0},r_{1}>
®
<
r_{0}^{¢},
r_{1}^{¢}>
between the two subgroups of S_{n} generated by the elements in each ordered pair, then there is a
fixed permutation t in S_{n} that simultaneously
conjugates
r_{i}
to r_{i}^{¢}
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 fixedpointfree involutions.



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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 edgedcolored graph has an alternating colored even cycle, and determining if a directed graph has an antidirected
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 NPcomplete.



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