April 2005

Spaces of Polygons in the Plane and Morse Theory
by Don Shimamoto and Catherine Vanderwaart
dshimam1@swarthmore.edu, cvanderwaart@alum.swarthmore.edu
This article studies the topology of spaces of polygons in the plane using advanced calculus and elementary notions from Morse theory. The basic approach and main results are not new, but, for the most part the arguments have been simplified to follow from standard parts of the upper-level undergraduate mathematics curriculum. The article concludes by applying the techniques to the particular case of spaces of pentagons, where, for example, we recover the fact that the space of equilateral pentagons is topologically an orientable surface of genus four.

A Tail of Two Palindromes
by Edward B. Burger
eburger@williams.edu
In this paper, written in a lively “literary” style, the question of determining when a real quadratic irrational is a linear fractional transformation of its conjugate is addressed. The surprising answer involves pairs of palindromes and continued fractions. A necessary and sufficient condition for a quadratic to have integral trace is also presented.

Remarks on the Gravity Equation
by Sherman Stein
stein@math.ucdavis.edu
The equation f(x+h) - f(x-h) = 2hf' (x) arises in the study of central forces. Klamkin and Newman showed that if a function satisfies this equation for all real x and for two fixed values of h, then it must be a quadratic polynomial. Their proof is fairly complicated. The present paper simplifies that proof and offers two different proofs, one mainly algebraic, the other based on a theorem of Titchmarsh concerning solutions of the equation for a single value of h.

An Elementary Approach to the Monster
by Christopher S. Simons
simons@rowan.edu
Using little more than undergraduate abstract algebra, we reach the celebrated finite simple Monster group. We do so by describing a construction of the smallest non-Abelian finite simple group, the alternating group on five letters, and showing that it very closely parallels a construction of the square of the Monster.

The Generalized Simpson’s Rule
by Daniel J. Velleman
djvelleman@amherst.edu
In this paper, we introduce a generalization of Simpson's rule for estimating a definite integral. Like Simpson's rule, our generalization involves estimating the integral by using quadratic polynomials to approximate the integrand on a sequence of intervals. However, some instances of our generalized Simpson's rule are significantly more accurate than Simpson's rule.

Notes

Sufficient Conditions for Poncelet Polygons Not to Close
by Boris Mirman
bmirman@rcn.com

A Simple Proof of Sion’s Minimax Theorem
by Jürgen Kindler

A Constructive Approach to Singular Value Decomposition and Symmetric Schur Factorization
by John Clifford, David James, Michael Lachance, and Joan Remski
jcliff@umd.umich.edu, dajames@umd.umich.edu, malach@umich.edu, remski@umd.umich.edu