May 2012 Contents
In the articles in the May edition of The American Mathematical Monthly, we consider the age old problem of finding minimal surfaces of revolution, look at integers whose divisors sum to a perfect power, study functions satisfying the almost-near property, learn about the (un)equal tangents problem, and consider coin denomination arrays which yield unique ways to make change in terms of using the fewest possible number of coins. Our Notes Section considers the Catalan numbers, a characterization of Euclidean space due to Aronszajn, and trigonometric polynomials. James Walsh reviews Randomness and Recurrence in Dynamical Systems by Rodney Nillsen, and as always, our Problems Section will keep you busy.
Lost Mail: The Missing Envelope in the Problem of the Minimal Surface of RevolutionTony Gilbert
This paper exhibits some new features in one of the classical problems of the calculus of variations: finding minimal surfaces of revolution. We give a simple geometric construction to count the number of smooth extremals which connect the two given endpoints. We also give explicit formulae for the envelope of the family of smooth extremals which pass through one endpoint, and for the locus of the second endpoint on which the minimizing smooth extremal gives the same area as the Goldschmidt curve.
Power Values of Divisor SumsFrits Beukers, Florian Luca, and Frans Oort
We consider positive integers whose sum of divisors is a perfect power. This problem had already caught the interest of mathematicians from the 17th century like Fermat, Wallis and Frenicle. In this article we study this problem and some variations. We also give an example of a cube, larger than one, whose sum of divisors is again a cube.
On What Is the Almost-Near PrincipleHassan Boualem and Robert Brouzet
We introduce a class of functions satisfying what we call the Almost-Near property, or briefly the (AN) property. The motivation of this investigation is provided by the phenomenon “almost implies near” which appears in many various situations. We study some elementary properties of these functions and develop several examples (such as polynomial functions, complex exponential and covering maps).
The (Un)equal Tangents Problem
There are two tangent segments to a strictly convex closed plane curve from every point in its exterior. We discuss the following problem: does there exist a curve such that one can walk around it so that, at all moments, the two tangent segments to the curve have unequal lengths?
A Variation on the Money-Changing Problem
Lance Bryant, James Hamblin, and Lenny Jones
The classical money-changing problem is to determine what amounts of money can be made with a given set of denominations. We present a variation on this problem and ask the following question:
For what denominations of money a1, a2, . . . at is there exactly one way, using the fewest number of coins possible, to make change for every amount that can be made?
We provide a solution to this problem when we have at most three denominations.
Flexagons Lead to a Catalan Number Identity
Hexaflexagons were popularized by the late Martin Gardner in his first Scientific American column in 1956. Oakley and Wisner showed that they can be represented abstractly by certain recursively defined permutations called pats, and deduced that they are counted by the Catalan numbers. Counting pats by the number of descents yields the identity
where only the middle third of the summands are nonzero.
Aronszajn’s Criterion for Euclidean Space
R. D. ArthanWe give a simple proof of a characterization of Euclidean space due to Aronszajn and derive a well-known characterization due to Jordan and von Neumann as a corollary.
When Is a Trigonometric Polynomial Not a Trigonometric Polynomial?
Joseph E. Borzellino and Morgan ShermanAs an application of Bézout’s theorem from algebraic geometry, we show that the standard notion of a trigonometric polynomial does not agree with a more naive, but reasonable notion of trigonometric polynomial.
PROBLEMS AND SOLUTIONS
Randomness and Recurrence in Dynamical Systems. By Rodney Nillsen. The Carus Mathematical Monographs Number 31, Mathematical Association of America, Washington, DC, 2010, xviii + 357 pp., ISBN 978-0-88385-043-5, $52.95.
Reviewed by James A. Walsh