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NOVEMBER 2001

**Mathematical Wunderkammern**

by William Mueller

wmueller@alum.mit.edu

We investigate the historical currents leading up to and away from the proliferation of geometric models in 19th century mathematics education. A continuum of intellectual history from early 17th century "wonder cabinets" to modern computer visualizations is evoked.

**The Geometric Mean, Matrices, Metrics, and More**

by Jimmie D. Lawson and Yongdo Lim

lawson@math.lsu.edu

Beginning with the modest concept of the geometric mean of two positive numbers (the square root of their product), we give eight alternative characterizations, all of which have direct generalizations to positive definite matrices. These generalizations, though elementary in nature, touch on an interesting variety of modern mathematics: matrix and operator equations and inequalities, Bruhat-Tits metrics, weakly symmetric Riemannian spaces, Cartan-Hadamard manifolds, symplectic geometry, Riccati differential equations and related dynamical systems, convex programming, and symmetric cones and Jordan algebras.

**Kepler’s Laws, Newton’s Laws, and the Search for New Planets**

by Robert Osserman

ro@msri.org

One of the major astronomical stories of the past decade has been the discovery of a series of planets orbiting other stars. None of these planets has actually been seen, but their presence is detected by the "wobble" they induce on the accompanying star. Even more impressive is the amount of detailed information about the unseen planets that can be derived, simply by using Newton’s Laws applied to the motion of the star. This article explains how it is done, with particular emphasis on the minor changes that would have to be made to the standard treatment of Newton’s Laws in calculus courses in order to give students access to this exciting new area of astronomical research.

**Polynomial Equations and Circulant Matrices**

by Dan Kalman and James E. White

kalman@american.edu and mathwrig@gte.net

Eigenvalues of circulant matrices provide a route to the zeroes of quartic polynomials. In fact, circulant matrices provide a unified approach to quadratic, cubic, and quartic equations. This article shows how special matrix algebras can provide information about polynomial equations and their roots. In addition to circulant matrices, which lead to Cardano’s solution of the cubic, read about Klein matrices and Cartesian matrices. Klein matrices lead to Euler’s solution of the quartic, while Cartesian matrices lead to a solution nearly identical to that of Descartes. An interactive graphical exploration of some related ideas is available on-line at the New Mathwright Library (www.mathwright.com). Look for the Mathwright workbook Cubic Equations (www.mathwright.com/book_pgs/book241.html), or for the Lava workbook Cardano (www.mathwright.com/lavapage.htm) (viewable on a webpage).

**The Evolution of Â…**

**Weierstrass, Luzin, and Intuition**

by Jeremy Gray

j.j.gray@open.ac.uk

**Notes**

**The Sum of Some Convergent Series **

by Ákos László

lakos@math.ttk.pte.hu

**A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators.**

by A.G. Ramm

Ramm@math.ksu.edu

**An Elementary Product Identity in Polynomial Dynamics**

by Robert L. Benedetto

mailto:bene@bu.edu

**Reviews**

**The Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics (1869-1926)**

By Thomas Hawkins

Reviewed by Armand Borel

borel@maths.hku.hk

**Linear Algebra** By Peter D. Lax

Reviewed by Mark A. Kon

mkon@bu.edu