American Mathematical Monthly -December 1999

DECEMBER, 1999

Emerging Tools for Experimental Mathematics
by Jonathan M. Borwein and Robert M. Corless
jborwein@cecm.sfu.ca, Rob.Corless@uwo.ca
Using mostly elementary examples, we discuss the use of some recent and emerging tools for experimental mathematics. The tools discussed include so-called "inverse symbolic computation", using lattice reduction algorithms such as "LLL" and "PSLQ", and Sloane and PlouffeÕs integer sequence lookup program. We concentrate on computer-assisted discovery of mathematical results, but a little computer-assisted proof creeps in as well. We use MAPLE throughout the paper, but any other good computer algebra system would be as effective.

Reform, Tradition, and Synthesis
by Thomas W. Tucker
ttucker@mail.colgate.edu
Over the last decade, mathematicians have been debating how to teach calculus (and other mathematics) with a vehemence that surprises observers outside the discipline. This article attempts to find some middle ground in "reform" and "traditional" views of some fundamental issues: technology, lecturing, drill, algebra, diversity of textbooks, and assessment.

You Don't Need a Weatherman to Know Which Way the Wind Blows
by Steven G. Krantz
sk@math.wustl.edu
We consider the merits of the teaching reform movementÑparticularly the calculus reform movementÑfrom the point of view of a traditionalist instructor. Special attention is given to the role of lectures, to discovery learning, to group work, and to the purpose of drill in mathematics teaching.

The Weyr Characteristic
by Helene Shapiro
hshapir1@swarthmore.edu
This article describes and derives Eduard Weyr's canonical form for matrices. Weyr's approach is particularly useful for describing algorithms for computing the Jordan form in a stable manner and for developing canonical forms for matrices under unitary similarity. Although Weyr published his work in 1885, the Weyr form rarely appears in linear algebra texts and Weyr's ideas have been rediscovered by various authors. We hope this article will make Weyr's work better known to a wider audience.

Rental Harmony: Sperner's Lemma in Fair Division
by Francis Edward Su
su@math.hmc.edu
Suppose that you and your friends rent a house and are having trouble deciding who should get which room and for what part of the total rent. Is there always a way to price the rooms so that each person prefers a different room?

As we shall see, with mild assumptions, the answer is yes. This is an example of a fair division question and generalizes the age-old cake-cutting problem as well as the dual problem of dividing chores. We show how a simple combinatorial lemma due to Sperner can be used to prove the existence of "envy-free" divisions. Better yet, its proof yields constructive methods for finding approximate solutions to fair division questions.

A java applet based on this article may be found at http://www.math.hmc.edu/~su/fairdivision/

NOTES

Which Tanks Empty Faster?
by Leonid G. Hanin
hanin@isu.edu

Two Uniformly Distributed Parameters Defining Catalan Numbers
by David Callan
callan@stat.wisc.edu

Determinants of Commuting-Block Matrices
by Istvan Kovacs, Daniel S. Silver, and Susan G. Williams
silver@mathstat.usouthal.edu, williams@mathstat.usouthal.edu

Mixtilinear Incircles
by Paul Yiu
yiu@fau.edu

The Area of the Medial Parallelogram of a Tetrahedron
by David N. Yetter
dyetter@math.ksu.edu

UNSOLVED PROBLEMS

MONTHLY Unsolved Problems, 1969-1999
by Richard Nowakowski
rjn@mscs.dal.ca

PROBLEMS AND SOLUTIONS

REVIEWS

Wavelets: A Primer.
By Christian Blatter

Wavelets in a Box.
By Charles K. Chui, Andrew K. Chan, and C. Steve Liu

Reviewed by Edward Aboufadel, Matthew Boelkins, and Steven Schlicker
Aboufade@gvsu.edu, boelkinm@gvsu.edu, schlicks@gvsu.edu

Poincaré and the Three Body Problem.
By June Barrow-Green

Reviewed by Daniel Henry Gottlieb
gottlieb@math.purdue.edu

TELEGRAPHIC REVIEWS