American Mathematical Monthly -October 2006

October 2006

Almost Alternating Sums
Kevin O'Bryant, Bruce Reznick, and Monika Serbinowska
kevin@member.ams.org, reznick@math.uiuc.edu, mserbinowska@weber.edu
The drunkard who, upon leaving a bar, takes N steps (each step independent of the previous ones, and heading downtown and uptown with equal likelihood), is likely about the square root of N steps from the bar. But what if the drunkard has a plan? What if the drunkard heads downtown on the nth step if the greatest integer of na is even, and otherwise heads uptown? For some irrational real numbers a this keeps her appreciably closer to the bar, but for others this plan is "worse than random.''

Exponential vs. Factorial
Daniel J. Velleman
djvelleman@amherst.edu
What is the largest number you can write with five characters, using ordinary mathematical notation? This simple question leads us into an investigation of the rates of growth of some fast-growing sequences. We discover some surprising cases of different fast-growing sequences that grow at nearly the same rate.

Lagrange's Proof of the Fundamental Theorem of Algebra
Jeff Suzuki
jeff_suzuki@yahoo.com
As every student of the history of mathematics knows, Gauss's doctoral dissertation was a proof of the fundamental theorem of algebra (FTA). But there were no fewer than six attempts to prove the FTA before Gauss. We'll take a look at some of these pre-Gaussian proofs and show that Lagrange was the first to provide a complete and rigorous proof of the Fundamental Theorem of Algebra.

The Prehistory of the Hardy Inequality
Alois Kufner, Lech Maligranda, and Lars-Erik Persson
kufner@math.cas.cz, lech@sm.luth.se , larserik@sm.luth.se
The development of the famous Hardy inequality (in both discrete and continuous forms) during the period 1906Â—1928 has its own history or, as we have called it, prehistory. Contributions of mathematicians other than G. H. Hardy, such as E. Landau, G. Pólya, I. Schur, and M. Riesz are important here. Several facts and early proofs that are not available in books on this subject are included and discussed.

The Sixty-Sixth William Lowell Putnam Mathematical Competition
by Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson

Notes

Variations on a Theorem of Korovkin
Héctor E. Lomelí and César L. García
lomeli@itam.mx, clgarcia@itam.mx

A Statistical Characterization of Regular Simplices
Ian Abramson and Larry Goldstein
iabramson@ucsd.edu, larry@math.usc.edu

How Terminal is Terminal Velocity?
Lyle N. Long and Howard Weiss
lnl@psu.edu, weiss@math.psu.edu

The Spectrum in a Banach Algebra
Dinesh Singh
dinesh_singh@hotmail.com

Problems and Solutions

Reviews

Dark Hero of the Information Age: In Search of Norbert Wiener, the Father of Cybernetics.
By Flo Conway and Jim Siegelman
Reviewed by Ramesh Gangolli
gangolli@math.washington.edu