American Mathematical Monthly - March 2003

MARCH 2003

Yuen-Gin Gung and Dr. Charles Y. Hu Award for
Distinguished Service to Clarence F. Stephens
by Robert E. Megginson
meggin@msri.org

Quatum Computation
by Stan Gudder
sgudder@math.du.edu

Quantum computers may be the next revolution in the computer industry. Primitive quantum computers have already been constructed and fast quantum algorithms for factorization and data searches have been derived. Employing mathematical methods from linear algebra, this paper surveys some of the latest developments in quantum computation. This is an exciting new field that is only about ten years old. The topics covered included quantum circuits, superdense coding, quantum teleportation, Grover's search algorithm, quantum Fourier transforms, and Shor’s factorzation algorithm.

The Crystallographic Restriction, Permutations, and Goldbach’s Conjecture
by John Bamberg, Grant Cairns, and Devin Kilminster
john.bam@maths.uwa.edu.au, g.cairns@latrobe.edu.au, devin@maths.uwa.edu.au

We examine the connection between the crystallographic restriction, the orders of the elements of the symmetric group, and Goldbach’s conjecture. In particular, we observe that Goldbach’s conjecture, in the strong sense that every even integer greater than six can be written as the sum of two distinct odd primes, is equivalent to the following condition: for every even integer n greater than six there is an integer matrix of order pq for distinct odd primes p and q, and there is no smaller integer matrix of this order.

Generalizing the Petr-Douglas-Neumann Theorem on n-gons
by Stephen B. Gray
stevebg@adelphia.net

In 1908, Karel Petr published an unusual theorem about triangle constructions on n-gons. It was rediscovered twice in the early 1940s by Jesse Douglas and B. H. Neumann. The author rediscovered it again in 1961, and has recently generalized it. In both the special and general versions, the integer n can range from three upwards without limit, and the number of lines in a full illustration makes it impossible to visualize for all but very small n. The generalization also stands out because of the large number of defining parameters that can be varied independently. Its new proof needs only the most elementary methods in complex variables and matrices. Several avenues are open for even further extension of this result.

Problems and Solutions

Notes

Can Kirkman’s Schoolgirls Walk Abreast by Forming Arithmetic Progressions?
by Lorenz Halbeisen
halbeis@qub.ac.uk

Newton’s Identities Once Again!
by Ján Minác
minac@uwo.ca

An Illuminating Counterexample
by Michael Hardy
hardy@math.mit.edu

A Norm Inequality for Hermitian Operators
by Ritsuo Nakamoto
nakamoto@base.ibaraki.ac.jp

Reviews

Fragments of Infinity, A Kaleidoscope of Math and Art
by Ivars Peterson
Reviewed by William Mueller
wmueller@alum.mit.edu

Mathematical Models in Population Biology and Epidemiology
by Fred Brauer and Carlso Castillo-Chávez
Reviewed by Shandelle M. Henson
henson@andrews.edu

Telegraphic Reviews