In how many ways can a pet photographer pose \(C\) cats, \(D\) dogs,...

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**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 continuousÂ–but not differentiableÂ–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 ofÂ…
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