Given a positive integer \(m\), the authors exhibit a group with the...

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**Solving Equations, An Elegant Legacy **

by Jerry L. Kazdan

kazdan@math.upenn.edu

Solving equations--one of the primary themes in mathematics. In discussing this, my secondary theme is that classical and modern mathematics are tightly intertwined, that contemporary mathematics contributes real insight and techniques to understand traditional problems. One way to summarize this article is "Elementary Mathematics from an Advanced Point of View."

In the long second section I discuss procedures that help to solve equations. Applications are given to problems in number theory, Markov chains, and differential equations. There the discussion of symmetry is extensive because in courses it is usually treated only as a special aspect of group theory rather than as a fundamental thread that runs throughout mathematics and its applications.

The third section shows how to use (i) critical points of functions and (ii) fixed point theorems to prove that equations have solutions.

**Separation of Subspaces by Volume **

by Donald W. Robinson

robinson@math.byu.edu

A measure of the "angle" between two complementary subspaces of an inner product space is given in terms of the volume of the projection onto one subspace along the other.

**The Geometric Series in Calculus **

by George E. Andrews

andrews@math.psu.edu

The object of this paper is to show that the Finite Geometric Series and its limiting case, the Infinite Geometric Series, are helpful for explaining several important but complicated results in calculus.

The Infinite Geometric Series is most familiar in calculus for establishing the Ratio and Root Tests. We show that the Finite Geometric Series allows an easy evaluation of both the derivative and the integral of polynomials. Also we explain its use in motivating the fundamental theorem of calculus.

**The Surprise Examination or Unexpected Hanging Paradox **

by Timothy Y. Chow

tchow@umich.edu

A teacher announces that there will be a surprise exam next week; the students argue by backwards induction that this is impossible, and yet when the exam occurs the students are totally surprised, thus vindicating the announcement. What is wrong with the students' argument? This "surprise exam paradox" is the subject of nearly a hundred papers in prestigious philosophy journals. Many of these papers connect the paradox with interesting and nontrivial mathematics, including Gödel's incompleteness theorem and game theory. This article describes these connections and dispels much of the confusion that surrounds the paradox.

**NOTES**

**Two or Three Identities of Ramanujan **

by Michael David Hirschhorn

mikeh@maths.unsw.EDU.AU

**The Bull and the Silo: An Application of Curvature **

by Michael E. Hoffman

meh@sma.usna.navy.mil

**THE EVOLUTION OF . . . **

Function: Part I by N. Luzin

**PROBLEMS AND SOLUTIONS **

**REVIEWS **

Analysis by its History. By E. Hairer and G. Wanner

Reviewed by John L. Troutman

Principles and Practices of Mathematics. By COMAP

Reviewed by Anthony Ralston ar9@doc.ic.ac.uk

A Mathematical Mosaic: Patterns & Problem Solving. By Ravi Vakil

Reviewed by Andre Toom toom@ime.usp.br