# American Mathematical Monthly - June-July 1999 ## JUNE-JULY, 1999

Change of Variables in Multiple Integrals
by Peter D. Lax
lax@cims.nyu.edu
The usual proof of the change of variables formula for multiple integrals is a mess; this paper contains a simple algebraic proof that avoids approximations. It is suitable for presentation in a course on advanced calculus. The mappings considered need not be one-to-one. The intermediate value theorem and Brouwer's fixed point theorem are simple corollaries. Random Walks and Plane Arrangements in Three Dimensions
by Louis J. Billera, Kenneth S. Brown, and Persi Diaconis
billera@math.cornell.edu, kbrown@math.cornell.edu
This paper explains some modern geometry and probability in the course of solving a random walk problem. Consider n planes through the origin in three dimensional Euclidean space. Assume, for simplicity, that they are in "general position". They then divide space into n(n -1) + 2 regions. We study a random walk on these regions. Suppose the walk is in region C. Pick a pair of the planes at random. These determine a line through the origin. Pick one of the two halves of the line with equal probability. The walk now moves to the region adjacent to the chosen half-line that is closest to C. We determine the long-term stationary distribution: All regions of i sides have stationary probability proportional to i - 2. We further show that the walk is close to its stationary distribution after two steps if n is large. A Counting Formula for Primitive Tetrahedra in Z3
by Mizan R. Khan
khanm@ecsuc.ctstateu.edu
A primitive tetrahedron is a tetrahedron whose vertices are (integer) lattice points but does not contain any other lattice points. The standard example of such a tetrahedron is the tetrahedron whose vertices are (0,0,0), (1,0,0), (0,1,0), and (1,1,n) where n is any non-zero integer. We give an exposition of primitive tetrahedra, and describe an elegant, but little known, characterization of such a tetrahedra that was discovered over 30 years ago. We then present a formula that counts the number of equivalence classes of primitive tetrahedra of a given volume. The proof is an application of Burnside's lemma. What Makes a Great Mathematics Teacher? The Case of Augustus De Morgan
ar6n@virginia.edu
To shed some light on what goes into making a great mathematics teacher, we investigate the teaching of Augustus De Morgan (1806-1871) in the middle of the 19th century. We use some valuable and hitherto untapped sources: over 300 of his own manuscript notebooks; and a single book, from 1847, containing the lecture notes of one of his students. The TEAM Approach to Investing
by Frank Gerth III
gerth@math.utexas.edu
Consider an investor with a portfolio of "stock-like" investments (e.g., an S&P 500 Index fund) and "cash-like" investments (e.g., Treasury bills, CDs, and high-grade commercial paper). This paper compares the results of two strategies. The first is a buy-and-hold strategy with no exchange between the stock fund and cash fund. The second (the "TEAM strategy") periodically reallocates money between the stock fund and cash fund in a certain way. The main result is that the TEAM strategy produces a higher expected total portfolio value than does the buy-and-hold strategy for the same level of risk. NOTES

A Physically Motivated Further Note on the Mean Value Theorem for Integrals
by William J. Schwind, Jun Ji, and Daniel E. Koditschek
schwindw@eecs.umich.edu, kod@eecs.umich.edu, junji@valdosta.edu

Hopping Hoops Don't Hop
by James P. Butler
jbutler@hsph.harvard.edu

Approximation of Hölder Continuous Functions by Bernstein Polynomials
by Peter Mathé
mathe@wias-berlin.de

An Extension of the Wallace-Simson Theorem: Projecting in Arbitrary Directions
by Miguel de Guzmán
mdeguzman@bitmailer.net

Another Short Proof of Ramanujan's Mod 5 Partition Congruence, and More
by Michael D. Hirschhorn
m.hirschhorn@unsw.edu.au

UNSOLVED PROBLEMS

The Cayley Addition Table of Zn
by Hunter S. Snevily
snevily@uidaho.edu

PROBLEMS AND SOLUTIONS

REVIEWS

An Imaginary Tale: The Story of (-1)1/2
. By Paul J. Nahin

Reviewed by Ricardo Diaz
rdiaz@bently.unco.edu

Learning Towards Infinity
By Sue Woolfe

Reviewed by John Beebee and Karen Willmore
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