## November 2005

Function Series, Catalan Numbers and Random Walks on Trees
by Ibtesam Bajunaid, Joel M. Cohen, Flavia Colonna, and David Singman
bajunaid@hotmail.com, jcohen@umd.edu, fcolonna@gmu.edu, dsingman@gmu.edu
The delight of finding unexpected connections is one of the rewards of studying mathematics. In this paper, we link several superficially unrelated entities. We use the generalized Catalan numbers of parameter k as the coefficients of a power series in a certain polynomial of degree k. These series converge and yield continuous–but not differentiable–functions on intervals. These functions turn out to determine the transience of various random walks on trees. They also satisfy functional equations that, by means of the Lagrange-Bürmann inversion formula, lead back to the Catalan numbers.

Virtual Empirical Investigation: Concept Formation and Theory Justification
by Dan Kalman
kalman@american.edu
This article presents the what and why of interactive computer activities for mathematics instruction. It discusses the pedagogical framework for such activities, describes several samples, and explains how these activities can be easily created using mathwright software. Mathwright was created by James White, and featured in MAA projects WELCOME and IMTP. Readers can access samples at www.dankalman.net/mathwright, http://www.dankalman.net/mwweb, and at the mathwright library

A Differential Forms Perspective on the Lax Proof of the Change of Variables Formula
by Nikolai V. Ivanov
ivanov@math.msu.edu
In two recent articles in the MONTHLY Peter Lax suggested a new beautiful approach to the change of variables formula and applied it to give a very simple proof of the Brouwer fixed-point theorem. In this article we recast the Lax proof from the differential forms perspective. We use only the most elementary part of the theory of differential forms; in particular, the integration of differential forms is not a prerequisite for our approach. Our approach transforms some mysterious parts of the Lax proof into completely straightforward and natural calculations with differential forms. We present a fairly detailed comparison of our proof with Lax’s. Such a comparison is very instructive, for it sheds light on both the efficiency of the differential form theory and the brilliance with which Lax uses classical analysis.

The Modular Tree of Pythagoras
by Roger C. Alperin
alperin@math.sjsu.edu
The Pythagorean triples have an amazing tree structure rooted at the familiar 3-4-5 triangle. Starting at the root, all the triples are obtained by simple recursive rules based on a free subgroup of the modular group.

Notes

The Behavior of Solutions near a Stable or Semistable Stationary Point
by Ray Redheffer
reeeniii@aol.com

More Nested Square Roots of 2
by M. A. Nyblom
michael.nyblom@rmit.edu.au

A Simple Proof of Cohen’s Theorem
by A. R. Naghipour
arnaghip@ipm.ir

A Short Proof for the Krull Dimension of a Polynomial Ring
by Thierry Coquand and Henri Lombardi
coquand@cs.chalmers.se, Henri.Lombardi@univ-fcomte.fr

An Elementary Proof of Joris’s Theorem
by Robert Myers
robmyers@Math.Berkeley.edu

Evolution of…
Some Remarks on “Mathematics at the Turn of the Millennium”

by Michael Monastyrsky

Problems and Solutions

Reviews

Mathematical Models in Biology
by Elizabeth S. Allman and John A Rhodes
Reviewed by David J. Logan
dlogan@math.unl.edu

Count Down: Six Kids Vie for Glory at the World’s Toughest Math Competition
by Steve Olson
Reviewed by Virginia M. Warfield
warfield@math.washington.edu