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American Mathematical Monthly - June-July 1999

 

JUNE-JULY, 1999

Change of Variables in Multiple Integrals
by Peter D. Lax
[email protected]
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
[email protected], [email protected]
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
[email protected]
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
by Adrian Rice
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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
[email protected]
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
[email protected], [email protected], [email protected]

Hopping Hoops Don't Hop
by James P. Butler
[email protected]

Approximation of Hölder Continuous Functions by Bernstein Polynomials
by Peter Mathé
[email protected]

An Extension of the Wallace-Simson Theorem: Projecting in Arbitrary Directions
by Miguel de Guzmán
[email protected]

Another Short Proof of Ramanujan's Mod 5 Partition Congruence, and More
by Michael D. Hirschhorn
[email protected]

UNSOLVED PROBLEMS

The Cayley Addition Table of Zn
by Hunter S. Snevily
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PROBLEMS AND SOLUTIONS

REVIEWS

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

Reviewed by Ricardo Diaz
[email protected]

Learning Towards Infinity
By Sue Woolfe

Reviewed by John Beebee and Karen Willmore
[email protected], [email protected]

TELEGRAPHIC REVIEWS