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American Mathematical Monthly -February 2001

February 2001

Square-Banded Polygons and Affine Regularity
by Duane DeTemple and Matthew Hudelson
Napoleon's Theorem and its generalizations concern regular m-gons constructed on the sides of general m-gons. We explore the related configuration obtained by erecting a sequence of bands of squares on a given base polygon. We prove several properties of the configuration using the geometry of complex arithmetic and matrix theory. For example, we obtain a sharp inequality that relates the areas of the squares in the first two bands of squares; equality holds if and only if the given polygon is affinely regular. Several linear relationships are obtained for the areas of the squares in consecutive bands; in particular, we generalize a classical theorem of Euler from quadrilaterals to general polygons.


Change of Variables in Multiple Integrals II
by Peter D. Lax
In a paper if the same title in the summer of 1999 [Monthly 106 (1999) 497-501] I gave a simple, algebraic derivation of the change-of-variables formula for multiple integrals. But the change-of-variables function had to be the identity function outside the unit ball, and several correspondents have pointed out that this is not typical of functions that come up in real life. The purpose of this note is to sketch a way to extract the garden variety change-of-variable formula from the main theorem in my 1999 article.


Around the Finite-Dimensional Spectral Theorem
by Adam Koranyi
This article contains what the author claims to be the simplest and most natural way to prove the various forms of the spectral theorem and related results such as the polar decomposition and singular value decomposition of linear transformations. The first theorem is the singular value decomposition, for which a direct geometric proof is given. The definitions of the adjoint and of the notion of normality are made with the aid of this theorem. The spectral theorem for self-adjoint linear transformations is then a very easy consequence. Finally the spectral theorem for normal linear transformations is proved, both in the complex and the real case. The method works unchanged for compact operators on Hilbert space.


Reform Now, Before It's Too Late!
by William Mueller

It is to be hoped that the near future will bring reforms in the mathematical teaching in this country. We are in sad need of them. From nearly all of our colleges and universities comes the loud complaint of inefficient preparation on the part of students applying for admission; from the high schools comes the same doleful cry. ...and educators who have studied the work of (foreign) schools declare that our results in elementary instruction are far inferior.

This is the sobering assessment of the state of American mathematics instruction offered by the Department of Education, reporting on a national study of contemporary educational practices. As timely as it may sound, it was made in 1890.

This article offers a small sampling of voices from that era. Through a collage of quotations and citations, it attempts to stir up a little of the dust that was being kicked around 100 years ago. The dust has never really settled. Indeed, I suspect that these voices will elicit an acute sense of retrospective déjà vu among modern readers. This being the case, I propose a simple question: To what extent can the current state of mathematics education be in such a state of "crisis", if the terms of the debate have changed so little in 100 years?


by H. Turgay Kaptanoglu
We describe several counterexamples in analysis exhibited by functions involving the expression . Our selection ranges from elementary results in limits through classes of functions related to differentiability to the more topological questions.



The Multi-Dimensional Version of
by Jean B. Lasserre and Konstantin E. Avrachenkov

Almost Sure Escape from the Unit Interval Under the Logistic Map
by Maximilian Thaler and Sibylle Zeller

Generalized Rearrangement Inequalities
by Robert Geretschläger and Walther Janous and

A Weighted Erdös-Mordell Inequality
by Shay Gueron and Seannie Dar

A Four-Point Set That Cannot Be Split
by Jan J. Dijkstra

Unsolved Problems

Problems and Solutions


Chapter Zero-Fundamental Notions of Abstract
By Carol Schumacher

Reviewed by Robert A. Fontenot

Geometry Civilized. History, Culture, and Technique.
By J. L. Heilbron

Reviewed by Doris Schattschneider

Two- and Three-dimensional Patterns of the Face.
By Peter L. Hallinan, Gaile G. Gordon, A.L. Yullle, Peter Giblin, and David Mumford

Reviewed by Ben M. Herbst