AMERICAN MATHEMATICAL MONTHLY - May 1999

MAY, 1999

Conway's ZIP Proof
by George K. Francis and Jeffrey R. Weeks
gfrancis@math.uiuc.edu, weeks@geom.umn.edu
Surfaces arise naturally in many different forms, in branches of mathematics ranging from complex analysis to dynamical systems. The Classification Theorem asserts that all closed surfaces, despite their diverse origins and seemingly diverse forms, are topologically equivalent to spheres with some number of so-called handles or crosscaps. The usual textbook proof, developed in the early 20th century, is satisfyingly constructive, but requires that a given surface be brought into a somewhat artificial standard form. In about 1992, John H. Conway discovered a completely new proof, which retains the constructive nature of its predecessors while eliminating the irrelevancies of the standard form. He calls it his Zero Irrelevancy Proof, or ZIP. Both the proof and this article are easily accessible to undergraduate mathematics students with no prior knowledge of topology.

Chaos, Cantor Sets, and Hyperbolicity for the Logistic Maps
by Roger L. Kraft
roger@calumet.purdue.edu
The family of logistic maps f_r(x) = r x(1 - x) appears in almost every dynamical systems textbook. It is one of the simplest nonlinear systems that one can study, but it is amazingly rich in phenomena. It has a surprising number of connections to other topics in dynamical systems and applied mathematics, e.g., population dynamics, symbolic dynamics, complex analytic dynamics, the Mandelbrot set, period-doubling route to chaos, renormalization, universality, horseshoes, etc. Because of its simplicity, many introductory dynamical systems textbooks use it as a primary example, in particular as the primary example of a chaotic dynamical system. They prove that when r > 2 + 5^1/2 4.236, f_r is chaotic on an invariant Cantor set. Most of these books then state without proof that Ã?r is actually chaotic for all r > 4. The goal of this article is to give a simple proof of this fact.

An Elementary View of Euler's Summation Formula
by Tom M. Apostol
apostol@caltech.edu
Euler's summation formula, which relates integrals and finite sums, is widely used in numerical analysis, analytic number theory, and the theory of asymptotic expansions. It contains Bernoulli numbers and periodic Bernoulli functions and is ordinarily discussed in courses in advanced calculus or real and complex analysis. Starting with a simple diagram, this paper shows how the general formula can be discovered using only elementary calculus, and it also shows how Bernoulli numbers and Bernoulli functions arise naturally along the way. The formula is used to calculate the first 8 digits in Euler's constant.

Marriage, Magic, and Solitaire
by David B. Leep and Gerry Myerson
leep@ms.uky.edu, gerry@mpce.mq.edu.au
Marriage: Hall's Marriage Theorem gives necessary and sufficient conditions for there to
be suitable partners for all the men and women in a given community.

Magic: A semi-magic square is a square matrix with non-negative integer entries, with all
row and column sums equal.

Solitaire: Deal out a deck of cards, face up, into a 4-by-13 array. Try to select
13 cards, one from each column, in such a way as to get one card of each denomination.

Marriage, Magic, and Solitaire: How do these connect?

The Isoperimetric Problem on Surfaces
by Hugh Howards, Michael Hutchings, and Frank Morgan
howards@wfu.edu, hutching@math.stanford.edu, frank.morgan@williams.edu
The authors begin with their three favorite proofs that in the plane the circle minimizes perimeter for given area, including a slight twist on a magical proof of Gromov. In a paraboloid of revolution, the "obvious" solution turns out to be a recent result, though proved here using little more than calculus and the Gauss-Bonnet formula. There are also recent examples in hyperbolic surfaces.

What is a Closed-Form Number?
by Timothy Y. Chow
tchowatalum.mit.edu
Can the root of the equation x = cos(x) be written in closed form? To answer this question we must first define "closed form". While much has been written about closed-form functions, scandalously little has been written about closed-form numbers, even though thousands of students have asked either explicitly or implicitly about them. We propose a definition of the term "closed-form number" that helps answer questions like these; we hope it will become standard. Possibly more interesting than the definition itself are its connections with Schanuel's conjecture in transcendental number theory and Tarski's theorem on the decidability of the first-order theory of the real numbers.

By the way, the x = cos(x) question remains an open problem. Perhaps an enterprising reader will solve it!

NOTES

The Smallest Solution of (30n+1) (30n) Is...
by Greg Martin
gerg@math.toronto.edu

A Matrix Representation for Euler's Constant,
by Frank K. Kenter
frank.kenter@smi.siemens.com

More on a Mean Value Theorem Converse
by H. Fejzíc and D. Rinne
hfejzic@mail.csusb.edu, drinne@mail.csusb.edu

An Elegant Continued Fraction for p
by L. J. Lange
jerry@math.missouri.edu

The Reciprocity Law for Dedekind Sums via the Constant Ehrhart Coefficient
by Matthias Beck
matthias@euclid.math.temple.edu

THE EVOLUTION OF...

Riemann's Dissertation and Its Effect
by Detlef Laugwitz
laugwitz@mathematik.th-darmstadt.de

PROBLEMS AND SOLUTIONS

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cwoodbur@pittstate.edu

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