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American Mathematical Monthly -December 2006

December 2006

Bernstein Polynomials and Brownian Motion
Emmanuel Kowalski
Brownian motion is one of the most fascinating objects in probability and indeed in mathematics as a whole. It provides a way of speaking of a continuous function "picked at random" and to describe what kind of properties it can have. We present a new and very elementary way to look at Brownian motion by means of the sequence of its Bernstein approximating polynomials. This provides a very intuitive proof of the existence of Brownian motion, and a new very simple proof that a random continuous function is not differentiable at any (given) point.


Asymmetric Rhythms and Tiling Canons
Rachel W. Hall and Paul Klingsberg,
Anyone who listens to rock music is familiar with the repeated drum beat—one, two, three, four—based on a 4/4 measure. Fifteen minutes listening to a Top 40 radio station offers evidence enough that most rock music has this basic beat. But if we tune the radio to different frequencies, we may hear popular music (jazz, Latin, African) with different characteristic rhythms. Although much of this music is also based on the 4/4 measure, some instruments play repeated patterns that are not synchronized with the 4/4 beat, creating syncopation—an exciting tension between different components of the rhythm. This article is concerned with classifying and counting rhythms that are maximally syncopated in the sense that, even when shifted, they cannot be synchronized with the division of a measure into two parts. In addition, we discuss rhythms that cannot be aligned with other even divisions of the measure. Our results have a surprising application to rhythmic canons. Audio recordings of all examples discussed are available at


Some Generalizations of the Notion of Bounded Variation
Pamela B. Pierce and Daniel J. Velleman,
When Jordan introduced the notion of bounded variation in 1881, he was seeking a sufficient condition for a function f to have a Fourier series that sums to f. With this same motivation, Wiener, Young, and Waterman have extended the notion of bounded variation in several directions. We introduce these broader classes of functions, and ask how these spaces are related to one another. For certain spaces, we prove theorems that give a complete description of how the spaces are interleaved.


A Study of Core-Plus Students Attending Michigan State University
Richard O. Hill and Thomas H. Parker,
One important measure of the effectiveness of a high school mathematics program is the success students have in subsequent university mathematics courses. Yet this measure is seldom considered in studies of high school curricula. This article describes a study that we undertook to quantify this measure for one particular curriculum. The study examined the university records of students arriving at Michigan State University from four high schools which began using the Core-Plus Mathematics program. The data show that these students placed into, and enrolled in, increasingly lower level mathematics courses as the implementation progressed. This downward trend is statistically robust (p


Sliding along a Chord through a Rotating Earth
Andrew J. Simoson

Dynamical Systems Method and a Homeomorphism Theorem
A. G. Ramm

A Simple Way of Proving the Jordan-Hölder-Schreier Theorem
Benjamin Baumslag

Complex Numbers and the Ham Sandwich Theorem
Andrew Browder

A New Proof of Euclid’s Theorem
Filip Saidak

Problems and Solutions


Saunders Mac Lane: A Mathematical Autobiography.
By Saunders Mac Lane
Reviewed by Della D. Fenster

Ants, Bikes, and Clocks: Problem Solving for Undergraduates.
By William Briggs
Reviewed by Steven Galovich