##
December 2006

**Bernstein Polynomials and Brownian Motion**

Emmanuel Kowalski

emmanuel.kowalski@math.u-bordeaux1.fr

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

rhall@sju.edu, pklingsb@sju.edu

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 http://www.sju.edu/~rhall/Rhythms.

**Some Generalizations of the Notion of Bounded Variation**

Pamela B. Pierce and Daniel J. Velleman

ppierce@wooster.edu, djvelleman@amherst.edu

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

hill@math.msu.edu, parker@math.msu.edu

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

**Notes**

**Sliding along a Chord through a Rotating Earth**

Andrew J. Simoson

ajsimoso@king.edu

**Dynamical Systems Method and a Homeomorphism Theorem**

A. G. Ramm

ramm@math.ksu.edu

**A Simple Way of Proving the Jordan-Hölder-Schreier Theorem**

Benjamin Baumslag

gilbert@sci.ccny.cuny.edu

**Complex Numbers and the Ham Sandwich Theorem**

Andrew Browder

abrowder@math.brown.edu

**A New Proof of Euclid’s Theorem**

Filip Saidak

filip@math.missouri.edu

**Problems and Solutions**

**Reviews**

*Saunders Mac Lane: A Mathematical Autobiography. *

By Saunders Mac Lane

Reviewed by Della D. Fenster

dfenster@richmond.edu

*Ants, Bikes, and Clocks: Problem Solving for Undergraduates. *

By William Briggs

Reviewed by Steven Galovich

galovich@lakeforest.edu