##
AUGUST/SEPTEMBER 2001

**The Twin Paradox in a Closed Universe**

by Jeff Weeks

weeks@northnet.org

In the classical twin paradox, one twin, call him Albert, stays at home, while his sister Betty travels at relativistic speed to a nearby star and back. According to special relativity, traveler Betty measures less time than stay-at-home Albert, and is therefore younger upon their reunion. But from Betty's point of view, she's at rest, and it's Albert who makes the high speed journey and should return younger. The resolution of this classical twin paradox is that Betty experiences an acceleration and a change of inertial frame at the turnaround point, while Albert stays in a single inertial frame throughout. The symmetry is broken, and Betty is truly younger at the reunion.

The twin paradox hits harder in a closed universe. Albert stays home, while Betty takes a trip around the universe! The situation is completely symmetrical: Albert sees Betty moving in a straight line at constant velocity from the moment of departure until their reunion, while Betty sees Albert moving in the opposite direction at constant velocity for the same period. Using special relativity, each calculates that the other should be younger at the reunion. Who is right?

**Linear Preserver Problems**

by Chi-Kwong Li and Stephen Pierce

ckli@math.wm.edu

Linear preserver problems is an active research area in matrix and operator theory. These problems involve certain linear operators on spaces of matrices or operators. In this article, we give a general introduction to the subject. In the first three sections, we discuss motivation, results, and problems. In the last three sections, we describe some techniques, outline a few proofs, and discuss some exceptional result.

**Towering Figures in American Mathematics, 1890-1950**

by David E. Zitarelli

zit@vm.temple.edu

How did the state of mathematics in America in 1950 differ from what it had been in 1890? Who were the major players in the dramatic transformation that ensued? This article recounts the main features of this development using the lives of six mathematicians to illustrate the overarching themes. These six figuresÂ–E.H. Moore, O.Veblen, G.D. Brikhoff, R.L. Moore, N. Wiener, and M. StoneÂ–towered over the mathematical landscape and ushered in a completely new generation of American mathematicians. This account describes their achievements in a attempt to provide us with a clearer picture of our heritage.

**Rapidly Growing Fourier Integrals**

by Erik Talvila

etalvila@math.ualbeta.ca

If a function is absolutely integrable on the real line then its Fourier transform exists and tends to zero as the transform parameter tends to infinity. This is the Riemann-Lebesgue Lemma. When the integral is allowed to converge conditionally then the Fourier transform can have arbitrarily large pointwise growth. We prove this by constructing an example. We also look at the growth properties of various trigonometric integrals.

**NOTES**

A Useful Strengthening of Stone-Weierstrass Theorem

by Soren Boel and Toke Meier Carlsen

TOKE@MATH.KU.DK

**Linear Preservers that Permute the Entries of a Matrix**

by Xingzhi Zhan

zhan@math.is.tohoku.ac.jp

**A Real Antiderivative by Contour Integration**

by Darrell Desbrow

dd@maths.ed.ac.uk

**Mortality in Matrix Semigroups**

by Ves Halava and Tero Harju

vehalava@utu.fi

**EVOLUTION**

**Mathematics in the 20th Century**

by Abe Shenitzer and Michael Atiyah

shenitze@mathstat.yorku.ca

**REVIEWS**

**The Nine Chapters on the Mathematical Art: Companion and Commentary**

By Shen Kangshen

Reviewed by Frank Swetz

jswetz3@yahoo.comb

**Number: From Ahmes to Cantor**

By Nidhat Gazalé

Reviewed by Eli Maor

emaor@suba.com