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

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**Rethinking the Elementary Real Analysis Course**

By Brian S. Thomson

thomson@cs.sfu.ca

We are now more than a century past the appearance of Lebesgue's paper announcing his integral. Yet our calculus and introductory real analysis courses continue to present a full treatment of Riemann's integral with no better justification than that the alternative might be too difficult for the students. This is not so. The correct integral on the real line can be introduced at an early stage, requiring no more preparation than a thorough treatment of compactness arguments. This article outlines some of the concerns that a designer of an elementary real analysis course should confront in developing an adequate theory of integration.

**The Probability That a Matrix of Integers is Diagonalizable**

By: Andrew J. Hetzel, Jay S. Liew, and Kent E. Morrison

ahetzel@tntech.edu, liew@jaysern.org, kmorriso@calpoly.edu

In this article, we exhibit the results of an undergraduate research project where we asked the question: *How frequently is an n x n matrix with integer entries diagonalizable over the complex numbers, the real numbers, and the rational numbers, respectively?* Such a frequency is couched in terms of a variant on the number theoretic notion of "natural density." Complete information is given for the frequency of diagonalizability over the complex numbers, and results are provided for the frequency of diagonalizability over the real numbers and the rational numbers if *n* = 2. At the end of the article, we provide three open questions based upon this work that may be suitable for other undergraduate research projects.

**The Nonholonomy of the Rolling Sphere**

By: Brody Dylan Johnson

brody@slu.edu

This article represents a glimpse into the subject of holonomy, which is devoted to understanding the effects of velocity constraints on the motion of mechanical systems. An old problem in this area asks what orientations of a sphere are achievable by rolling the sphere (with no slipping or twisting) along closed paths in the plane. It turns out that any orientation of the sphere is achievable and, for this reason, the system is termed nonholonomic. The main goal of this article is to present an elementary proof of this well-known result using basic dynamics and familiar techniques from ordinary differential equations.

**Hausdorff Dimension, Its Properties, and Its Surprises**

By: Dierk Schleicher

dierk@iu-bremen.de

We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well known. We also give examples where the Hausdorff dimension has some surprising properties: we construct a set *E* in X of positive planar measure and with dimension 2 such that each point in *E* can be joined to ∞ by one or several curves in X such that all curves are disjoint from each other and from *E* and such that their union has Hausdorff dimension 1. We can even arrange things so that each point in ? that is not on one of these curves is in E. These examples have been discovered very recently, where they arise quite naturally in the context of complex dynamics, more precisely in the iteration theory of simple maps such as *z* # π sin(*z*).

**Notes**

By: Ling-Ling Shi and Hao Pan

linglingshi@eyou.com, haopan79@yahoo.com.cn

**A Local Maximal Function Simplifying Measure Differentiation**

By Jürgen Bliedtner and Peter A. Loeb

bliedtne@math.uni-frankfurt.de, loeb@math.uiuc.edu

**A Short Proof of the Transcendence of Thue-Morse Continued Fractions**

By: Boris Adamczewski and Yann Bugeaud

adamczew@math.univ-lyon1.fr, bugeaud@math.u-strasbg.fr

**Extremal Points, Critical Points, and Saddle Points of Analytic Functions**

By Joseph Bak, Pisheng Ding, and Donald Newman

jbak@sci.ccny.cuny.edu, pd260@nyu.edu

**Reviews**

Introduction to Plane Algebraic Curves

By: Ernst Kunz

Reviewed by: Susan J. Colley

sjcolley@math.oberlin.edu

Monthly Homepage**Read the June/July issue of the Monthly online. (This requires MAA membership.) **