American Mathematical Monthly - December 2001

DECEMBER 2001

Visible Structures in Number Theory
by Peter Borwein and Loki Jorgenson
pborwein@cecm.sfu.ca
Until very recently, mathematical pictures consisted almost entirely of black and white graphs in two or three dimensions. There is the potential, in modern computational environments, to completely change this. A visualization is most interesting when it reveals new and unexpected structure; when it makes complicated phenomena transparent; or when it suggests the right method of proof. This paper, primarily through examples of phenomena chosen from some fairly elementary problems in number theory, aims to support these claims.

How to Differentiate and Integrate Sequences
by Donald R. Chalice
chalice@cc.wwu.edu
The main purpose of this article is to show how to use one’s calculus/differential equations intuition to solve certain discrete differential equations. We do this by first developing "discrete calculus" and along the way, we easily obtain closed formulas for sums such as k+k2+Â•Â•Â•+km, and discrete analogues of Taylor’s formula and the Chain Rule (involving moving averages). Discrete "integrating factors" allow us to express the solutions of certain discrete differential equations naturally as convolutions. We can interpret these solutions as the convolution of the "impulse response" with the "driving sequence". This has an analog in differential equations: The driven harmonic oscillator whose solution is the convolution of the "impulse response" with the driver. Our main point is to show how to use discrete calculus to solve certain discrete differential equations, whose solutions are usually found by guessing and induction.

Dr. Veblen Takes a Uniform: Mathematics in the First World War
by David Alan Grier
grier@gwu.edu
In the spring of 1918, Oswald Veblen, a young mathematics professor from Princeton University, created a mathematics research laboratory with the army’s department of ordnance. This laboratory developed new theories of mathematical ballistics for long range guns and antiaircraft fire. The laboratory flourished for about six months. It employed over sixty mathematicians and helped to advance the careers of Norbert Wiener, A. A. Bennett, Gilbert Bliss, and others. The office also employed eight female mathematicians as human computers in a Washington office located near the Lincoln Memorial. Had not the war intervened, these women would have become high school teachers. At least one, Elizabeth Webb Wilson, was able to capitalize on her war experience and was able to take at least a minor role in the American mathematical community.

Quantum Error Correction: Classic Group Theory Meets a Quantum Challenge
by Harriet Pollatsek
hpollats@mtholyoke.edu
Peter Shor’s discovery of an algorithm that enables a quantum computer to factor large integers in polynomial time attracted the attention of computer scientists, physicists, and mathematicians. Experts still debate when and if this algorithm might threaten the security of encryption schemes based on the difficulty of factoring. One of the challenges of quantum computation is error correction. The limitations of current technology and some of the underlying principles of quantum mechanics make quantum computers much more error prone than classical ones, so it is necessary to detect and correct errors in mid-computation. But quantum measurement is inherently destructive. Moreover, the "no-cloning principle" of quantum mechanics seems to preclude the benefits of redundancy for error correction. Shor made another breakthrough when he showed the quantum error correction is in fact possible. We describe how to use stabilizer codes to correct errors in a exposition assumes only a knowledge of linear algebra and a little bit of group theory. We begin with a brief primer of quantum theory and then describe qubit errors and the error group they generate. The error group has an associated geometry via the theory of extra special p-groups, developed in the 1950s by Philip Hall and Graham Higman. The error group and its geometry lead to the construction of stabilizer codes. We work out several examples and conclude with comments about some other approaches, suggestions for further reading, and final words of caution and optimism.

Notes

What is the Inverse Function of y=x 1/x
by Yunhi Cho and Kyunghwan Park
yhcho@uoscc.uos.ac.kr

On the Total Edge-Length of a Tetrahedron
by Hiroshi Maehara
hmaehara@edu.u-ryukyu.ac.jp

Evaluation of Dirichlet Series
by Eugenio P. Balanzario
ebg@matmor.unam.mx

Affine Transformations and Some Problems of Proportionality
by Timothy G. Feeman and Osvaldo Marrero
timothy.feeman@villanova.edu

Problems and Solutions

Reviews

Introduction to Cryptography
By Johannes A. Buchmann
Reviewed by Neal Koblitz
koblitz@math.washington.edu

Knowing and Teaching Elementary Mathematics
By Liping Ma
Reviewed by Anthony Ralston
ar9@doc.ic.ac.uk