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American Mathematical Monthly - January 1998

January 1998

Solving Equations, An Elegant Legacy
by Jerry L. Kazdan

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

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

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

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.


Two or Three Identities of Ramanujan
by Michael David Hirschhorn

The Bull and the Silo: An Application of Curvature
by Michael E. Hoffman


Function: Part I by N. Luzin



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

A Mathematical Mosaic: Patterns & Problem Solving. By Ravi Vakil
Reviewed by Andre Toom