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American Mathematical Monthly -February 2005

MONTHLY, February 2005

Juggling Probabilities
by Gregory S. Warrington
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
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,
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,
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,
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.



A Square-Packing Problem of Erdös
by Connie Campbell and William Staton,

On the Representation of Natural Numbers as Sums of Squares
by Laurentiu Panaitopol

The Perron-Frobenius Theorem and Limits in Geometry
by Jiu Ding and Temple H. Fay

How to Measure a Volume with a Thread
by Piotr Hajlasz and Pawel Strzelecki,

Problems and Solutions


An Introduction to Frames and Riesz Bases
by Ole Christensen
Reviewed by Eric S. Weber

by David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray
Reviewed by Jeffrey L. Nunemacher