American Mathematical Monthly - April 2009

Articles

Transformations Between Self-Referential Sets
By: Michael F. Barnsley
mbarnsley@aol.com
Did you know that there are continuous transformations from a fractal fern onto a filled square? Also, there are functions of a similar wild character that map from a filled triangle onto itself. We prove that these fractal transformations may be homeomorphisms, under simple conditions, and that they may be calculated readily by means of a coupled Chaos Game. We illustrate several examples of these beautiful functions and show how they exemplify basic notions in topology, probability, analysis, and geometry. Thus they are worthy of the attention of the mathematics community, both for aesthetic and pedagogical reasons.

Risk and Return Considerations in The Weakest Link
By: B. Ross Barmish and Nigel Boston
barmish@engr.wisc.edu, boston@engr.wisc.edu
The television game show The Weakest Link involves contestants making a sequence of decisions over time. A number of authors have recognized that this show serves as a laboratory for assessment of human decision-making. To this end, a question arises whether actions taken by the contestants are rational and consistent with the pursuit of optimality. The main objective of this paper is to provide arguments that the models used in the literature to date may result in an erroneous impression of the extent to which contestants' decisions deviate from the optimum. More specifically, we first point out that previous authors, while concentrating on maximization of the expected value of the return, totally neglect the risk component; i.e., the expected return is considered while its variance is not. Subsequently, we expand the analysis of previous authors to include both risk and return and a number of other factors: mixing of strategies and so-called end effects due to fixed round length. It is seen that many strategies, discounted by previous authors as being sub-optimal, may in fact be efficient in the risk-return plane.

Descartes' Rule of Signs, Alternations of Data Sets, and Balanced Differences
By: H. Fejzić, C. Freiling, and D. Rinne
hfejzic@csusb.edu, cfreilin@csusb.edu, drinne@csusb.edu
If two real data sets have different means it is pretty obvious that the data set with the larger mean has an element larger than some element of the other. We show that there is a rather natural, although not so obvious, way to extend this observation to the case of two data sets whose means as well as the first k-1 central moments are equal. Specifically, we find that there must be a decreasing sequence of k+1 elements that alternate between the two data sets with the first element from the data set with the larger kth central moment. This result has an analogue stated in terms of polynomials. Namely, if p(x) and q(x) are polynomials of degree n with positive leading coefficients and all real roots and if p(x)-q(x) is of degree k with positive leading coefficient, then there must be a decreasing sequence of n-k+1 numbers that alternate between the roots of the two polynomials, the largest being a root of q(x). We also use the result to show that a linear functional difference can be decomposed into a sum of balanced differences.

An Optimization Problem with a Surprisingly Simple Solution
By: D. Drinen, K. G. Kennedy, and W. M. Priestley
ddrinen@sewanee.edu, kgracekennedy@alumni.sewanee.edu, wpriestl@sewanee.edu
Suppose you and n of your friends play the following game. A continuous cumulative distribution function arising from a probability density function (pdf) on the real line is given in order to generate a random number, called the target. Each of you will guess what the target number will be and the winner is the one whose guess comes closest. Assume that your friends' guesses and the target are independently generated from the same pdf. Having only this probabilistic knowledge, what guess should you make in order to optimize your chances of winning?

We show, roughly speaking, that if n is very large but "indeterminate," the problem is surprisingly simple. It essentially does not matter what you guess, so long as your guess is reasonable in a sense that we make precise. You will then have virtually the same small chance as each of your friends. Curiously, however, if you know the value of n, you may use properties of the given pdf to arrive at a strategy that gives you roughly a 7% better chance of winning than each of your friends, at least in simple settings such as when the pdf reflects the uniform distribution on [0, 1] or simple transformations thereof. More complicated settings bring new issues and more surprises.

A Simple Proof of a Generalized No Retraction Theorem
By: Ethan D. Bloch
bloch@bard.edu
The world does not need yet another proof of the classical no retraction theoremÂ—many lovely elementary proofs are widely known. What does merit a new proof is a much less well-known generalization of the no retraction theorem to the class of all topological spaces that are homeomorphic to the underlying spaces of finite simplicial complexes. We give a simple proof of the 2-dimensional version of this generalized no retraction theorem; if one were interested only in the classical no retraction theorem, our proof could be simplified even further to give a particularly low-tech proof of that result.

Notes

Central Cross-Sections Make Surfaces of Revolution Quadric
By: Bruce Solomon
solomon@indiana.edu
Intersect a surface of revolution with a plane perpendicular to its axis and you get a circleÂ—of course. Tilt that plane a bit and you still get a closed, convex loop. But if every loop you get that way is also centralÂ—i.e., fixed by reflection through some pointÂ—your surface is quadric: an ellipsoid, hyperboloid, paraboloid, or cylinder. Basic though it is, this result seems new. We give a short, elementary proof.

On Furstenberg's Proof of the Infinitude of Primes
By: Idris D. Mercer
idmercer74@gmail.com
Ever since Euclid first proved that the number of primes is infinite, mathematicians have amused themselves by coming up with alternate proofs. A 1955 proof of Furstenberg used, of all things, topological language. In this note, we provide a new short proof that the number of primes is infinite, using Furstenberg's ideas but rephrased without topological language.

Mean, Meaner and the Meanest Mean Value Theorem
By: J. J. Koliha
j.koliha@ms.unimelb.edu.au
The Mean Value Theorem of the elementary calculus keeps attracting the attention of mathematicians who ponder how to make its proof simple and elegant, how to generalize it, how to use it in proofs of other theorems, and, perversely, how to avoid it. This MONTHLY has carried dozens of articles in which the Mean Value Theorem was the hero or the villain. We present three versions of the theorem in the form on an inequality, and boy, do they deliver a mean kick!

A Curious Proof of Fermat's Little Theorem
By: Giedrius Alkauskas
giedrius.alkauskas@maths.nottingham.ac.uk
We give a proof of Fermat's little theorem that uses neither arithmetic (the Euclidean algorithm) nor algebra (group theory). Rather, it employs the field of formal power series $mathbb{Q}((x))$ and is based on the property that any integer power series with constant term +1 can be uniquely expanded into a certain infinite product. This note is an example of a mathematical joke, though it contains a rigorous proof.