# American Mathematical Monthly -February 2008

## February 2008

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A Peculiar Connection Between the Axiom of Choice and Predicting the Future
By: Christopher Hardin and Alan D. Taylor
chardin@smith.edu, taylora@union.edu
We consider the problem of how one might try to guess values of a function based only on knowledge of the function on a subset of the domain, without any assumptions about the function being analytic or even continuous. At the level of a single point, this is a hopeless problem. When one considers a collection of many points, however, it is often possible, using the Axiom of Choice, to guess in such a way that most of the guesses are correct.

In particular, there exists a strategy which, for an arbitrary function f from the reals into some set, will for all but countably many t correctly guess the value of f on an interval [t, t + d), d > 0, given only the values of f(x) for x t. If one interprets this function as the evolution of a system over time, this means that, in principle, one can almost always predict an interval of the system's future based only on its past, without any assumption of continuity.

Fair Majority Voting (or How to Eliminate Gerrymandering)
By: Michel Balinski
michel.balinski@shs.polytechnique.fr
The lack of competitiveness in the elections to the U.S. House of Representatives has reached alarming proportions. The cause is political gerrymandering: "the practice of dividing a geographical area into electoral districts, often of highly irregular shape, to give one political party an unfair advantage by diluting the opposition's voting strength" (Black's Law Dictionary). New computing technology is directly implicated. The problem can be overcome. A new electoral system is proposed that maintains single-member districts yet eliminates the possibility of gerrymandering. A characterization justifies the idea.

Brunnian Spheres
By: Hugh Nelson Howards
howards@wfu.edu

Hilbert's Inequality and Witten's Zeta-Function
By: Jonathan M. Borwein
jborwein@cs.dal.ca
We explore a variety of pleasing connections between analysis, number theory, and operator theory, while revisiting a number of beautiful inequalities originating with Hilbert, Hardy and others. We first establish the Hilbert inequality and then apply it to various multiple zeta values. In consequence we obtain the norm of the classical Hilbert matrix, in the process illustrating the interplay of numerical and symbolic computation with classical mathematics.

From Möbius to Gyrogroups
By: Abraham A. Ungar
abraham.ungar@ndsu.edu

Notes

Bounds on the Zeros of the Derivative of a Polynomial with All Real Zeros
By: Aaron Melman
amelman@scu.edu

Qualitative Analysis of a Differential Equation
By: Uri Elias
elias@tx.technion.ac.il

Closed Curves on Spheres
By: W. Holsztynski, J. Mycielski, G. C. O'Brien, and S. Swierczkowski

Sequences Generated by Polynomials
By: E. F. Cornelius Jr. and Phill Schultz
efcornelius@comcast.net, schultz@maths.uwa.edu.au

Evolution ofÂ…
On Ordering the Natural Numbers or The Sharkovski Theorem

By: Krzysztof Ciesielski and Zdzislaw Pogoda
krzysztof.ciesielski@im.uj.edu.pl, zdzislaw.pogoda@im.uj.edu.pl

Reviews

Letters to a Young Mathematician
By: Ian Stewart
Reviewed by: Marion Cohen
mathwoman199436@aol.com

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