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American Mathematical Monthly -May 2000


May 2000

Topology in the Complex Plane
by Andrew Browder
In this article we use the complex exponential function to study topological questions in the plane R2, which we identify with the complex plane C. We obtain rapidly the Brouwer fixed point theorem, the fundamental theorem of algebra, and (with a little more difficulty) the invariance of domain. We obtain the Jordan curve theorem as a special case of the Alexander duality theorem. No use us made of the notion of winding number, also a concept derived from the exponential function, which Ahlfors (following E. Artin) used as the main weapon in dealing with topological issues in complex analysis. The reader should know some point set topology, and the definition of the exponential function, but no knowledge of complex analysis is required (the necessary tools are developed as needed).


Is a 2000-Year-Old Formula Still Keeping Some Secrets?
by Keith M. Kending
We introduce the reader to heretofore untold secrets of Heron's ancient formula giving the area of a triangle in terms of its three sides. You will discover that it works even for "impossible" triangles, where one side is longer than the sum of the other two. In fact, the formula turns out to be an excellent tour guide, leading us to triangles in space-time (as in relativity) and beyond, into "anti-Euclidean" space. You will learn that one normally sees less than 10% of all triangles, like the tip of an iceberg. We will "lift the iceberg out of the water", exposing all those triangles. For any of them, we can know the full suite of vital statistics--side lengths, angles, area, . . .


Optimal Running Strategy to Escape from Pursuers
by Joseph B. Keller
When a band of Indians pursued an escapee, one or two would run out in front of the band to force him to run fast and get tired. They would tire, drop back, and be replaced by one or two others, etc. Is this the best way to pursue an escapee, and if not, what is? To answer this question, a model of running is used to formulate the following problem: What is the smallest head start needed by the escapee to reach a fort at a given distance without being caught? To solve this problem , we find the optimal way for the escapee and each pursuer to vary his running velocity as a function of time.


Fixed Points and Fermat: A Dynamical Systems Approach to Number Theory
by Michael Frame, Brenda Johnson, and Jim Sauerberg
Results from number theory are often used in dynamical systems, but the process can be reversed. That is, one can use ideas from dynamical systems to prove number theoretic facts. This has been done on a sophisticated level by people such as Furstenberg, who has shown that number theoretic results of van der Waerden and of Szemeredi can be derived from dynamical systems results of Birkhoff and of Poincare. In this paper we work at a more elementary level, showing how fundamental concepts from discrete dynamics can be used to prove some standard elementary number theory results, including Fermat's Little Theorem.


Lamps, Factorizations, and Finite Fields
by Laurent Bartholdi
The 1993 International Mathematical Olympiad held in Istanbul contained the following problem: "Given n lamps placed around a table, repeatedly perform the following: if the previous lamp in lit, switch the current lamp; move to the next lamp. Assuming that the lamps are initially all lit, prove that after some time t(n) they will again all be lit. Give explicit values of t(n) if n is a power of 2 or one more than a power of 2." We construct an equivalent problem on polynomials over a finite field that answers the Olympiad question, and discover a surprising phenomenon occurring when n is one less than a power of 2.



Zeroless Positional Number Representation and String Ordering
by Raymond T. Boute

An Integer Programming Problem with a Linear Programming Solution
by Kevin Broughan and Nan Zhu

A Short Proof That Every Prime p = 3 (mod 8) is of the Form x2 + 2y2
by Terence Jackson

The Komornik-Loretti Constant is Transcendental
by Jean-Paul Allouche and Michel Cosnard

A Nowhere Differentiable Continuous Function
by Liu Wen

Evolution of. . .

By Jaques Tits
translated by John Stillwell

Problems and Solutions


Multivariable Calculus and Mathematica
By Kevin R. Coombes, Ronald L. Lipsman, and Johnathan M. Rosenburg

Reviewed by Allen C. Hibbard

Mathematical Modeling in the Environment
By Charles R. Hadlock

Reviewed by Mic Jackson