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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 Z**^{3}

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**

by Adrian Rice

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 Z**_{n}

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

afjcb@uaa.alaska.edu, afkew@uaa.alaska.edu

**TELEGRAPHIC REVIEWS**