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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
A link L is Brunnian if L is a link of n (≥3) components such that L is not the unlink of n components but such that every proper sublink of L is an unlink. The most famous Brunnian link is called the Borromean rings. In this article we shed new light on the Freedman and Skora result that shows that no Brunnian link can be constructed of round components. We then extend it to two different traditional generalizations of Brunnian links.

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
The polar decomposition of a Möbius transformation of the complex open unit disc represents it as a rotation of the disc preceded, or followed, by a hyperbolic translation. The latter, in turn, gives rise to the Möbius addition in the disc, just as Euclidean translations of a Euclidean space give rise to the common vector addition. Unlike vector addition, however, Möbius addition is neither commutative nor associative. A simultaneous measure of the extent to which Möbius addition deviates from both commutativity and associativity is provided by the so called gyrations. The gyrations are special rotations of the disc that "repair" the breakdown of commutativity and associativity in Möbius addition, giving rise to its gyrocommutative and gyroassociative laws. The disc and Möbius addition in the disc thus give rise to a group-like structure, naturally called a gyrogroup. A gyrogroup is, accordingly, a group in which the associative law has been replaced by the two gyroassociative laws (left and right), and a gyrocommutative gyrogroup is a commutative group in which, in addition, the commutative law has been replaced by the gyrocommutative law. It turns out that finite and infinite gyrogroups, both gyrocommutative and non-gyrocommutative, abound in group theory. The importance in physics of the gyrocommutative gyrogroup structure that Möbius addition encodes rests on the result that Einstein's addition of relativistically admissible velocities, hitherto considered structureless, turns out to be a gyrocommutative gyrogroup operation, just as the addition of classical velocities is a commutative group operation.

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
sennajawa@yahoo.com, jmyciel@euclid.colorado.edu, greg.ob@bigpond.net.au, halinstan@yahoo.com

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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