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MONTHLY, February 2005

**Juggling Probabilities**

by Gregory S. Warrington

gwar@alumni.princeton.edu

Consider a person juggling. At any point in time, the juggler is committed to catching the balls currently in the air at various future times. It follows that if the juggler wants to avoid having two balls land at the same time, there are certain heights to which he should not throw. This simple observation was used roughly twenty years ago to develop a mathematical notation for juggling. Building on this theory, we investigate here some random walks that arise when the juggler chooses the throw heights randomly.

**Prehistory of Faá di Bruno’s Formula**

by Alex D. D. Craik

addc@st-and.ac.uk

What is the best way of deriving a power-series expansion of a function of a function, and who originated the theory? Recent papers by H. Flanders (this *Monthly*** 108 **(2001) 559-561), H. W. Gould (*Utilitatis Mathematica* **61 **(2002) 97-106), and W. P. Johnson (this *Monthly*** 109** (2002) 217-234) explore work on this topic by Faá di Bruno (1855) and several of his near-contemporaries. Here, earlier work unnoticed in these papers is described: this anticipates many results wrongly credited to later writers. This earlier work begins with L. F. A. Arbogast’s *Calcul des dérivations *(1800) and is followed by reworkings by T. Knight, J. West, A. De Morgan, and others.

**The Lesser-Known Δ-Function in Number Theory**

by P. Codecà and M. Nair

cod@dns.unife.it, m.nair@maths.gla.ac.uk

The classical Euler function counts the number of positive integers n less than a given integer N and coprime to it. The function Δ(*x,N*) that is considered here is the truncated form of the Euler function for which the range of *n* is shortened to *xN*, with *x* in the unit interval. This function has a rather complicated behavior, as might be expected of a function having a close association with sieve bounds, and indeed, even the Riemann Hypothesis can be expressed in terms of it. We describe a wide variety of existing results on this function and explore some possible avenues for further research in this area. Some seemingly simple conjectures are also stated.

**On the Perimeter and Area of the Unit Disc**

by Juan Carlos Álvarez Paiva and Anthony Thompson jalvarez@duke.poly.edu, tony@mathstat.dal.ca

The much-calculated number π is at once the area and the semiperimeter of the unit disc in the Euclidean plane. What can be said about these numbers when the norm does not come from an inner product and the circle is not round? Is there still a relationship between them? While lengths of curves are prescribed by the norm, the same is not true of area. How is this handled? The authors explore these questions, which all relate to the affine geometry of convex bodies.

**Trigonometric Identities, Linear Algebra and Computer Algebra**

by Ian Doust, Michael D. Hirschhorn, and Jocelyn Ho

i.doust@unsw.edu.au, M.Hirschhorn@unsw.edu.au

While doing some research in Banach space geometry, the authors started doing some of the computations using a computer algebra package. The patterns that emerged from these experiments took a quite unexpected path, leading eventually to some amazing looking identities involving trigonometric functions and Stirling numbers. This article retraces that journey and reflects on how the computer influenced the direction of the authors’ research.

**Notes**

**A Square-Packing Problem of Erdös**

by Connie Campbell and William Staton

campbcm@millsaps.edu, mmstaton@olemiss.edu

**On the Representation of Natural Numbers as Sums of Squares**

by Laurentiu Panaitopol

pan@al.math.unibuc.ro

**The Perron-Frobenius Theorem and Limits in Geometry**

by Jiu Ding and Temple H. Fay

jiu,ding@usm.edu, thfay@hotmail.com

**How to Measure a Volume with a Thread**

by Piotr Hajlasz and Pawel Strzelecki

hajlasz@mimuw.edu.pl, pawelst@mimuw.edu.pl

**Problems and Solutions**

**Reviews**

**An Introduction to Frames and Riesz Bases**

by Ole Christensen

Reviewed by Eric S. Weber

esweber@iastate.edu

**Geometry**

by David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray

Reviewed by Jeffrey L. Nunemacher

jlnunema@cc.owu.edu