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American Mathematical Monthly - May 2003

MAY 2003

Enhanced Linking Numbers
by Charles Livingston
We've all seen the magic trick in which a magician takes a pair of linked metal rings and mysteriously separates them. There must be a trick involved; if the rings were really solid they couldn't be pulled apart in this way. Gauss introduced a mathematical formulation of linking called the linking number. In recent years new tools to study links have been developed, and this paper explores one of them, the enhanced linking number. Computations involving the enhanced linking number are surprisingly easy to carry out, yet deep and unexpected results concerning links follow quickly from such computations. After exploring some of these applications of the enhanced linking number, the paper concludes by relating the topic to other recent developments in knot theory.


Admissions and Recruitment
by Mourad Baïou and Michel Balinski,
Are you applying for admissions to a college or university, or seeking an internship at a hospital, or just plain looking for a job? If so, you happen to be an agent in a very special "two-sided market," which often has its rules and regulations. It helps, then, to know how best to play your hand to achieve your aims! Perhaps the rules are "fair" and perhaps not. It may be that one should honestly express one's aims, and yet it may be that (like a potential bride or groom) it is best to hide them. This paper examines the general situation. It describes two typical cases where the regulations are not fair and applicants can enforce better assignments, collusively hiding their true preferences. Yet it establishes that there is exactly one set of rules and regulations that guarantees fairness and excludes the possibility of strategic manipulation: it characterizes the "assignment mechanism" that should be used.


How Many Unit Equilateral Triangles Can Be Generated by N Points in Convex Position?
János Pach and Rom Pinchasi,
Any set of n points in strictly convex position in the plane has at most 2(n-1)/3 triples that induce equilateral triangles of side length one. This bound cannot be improved. The case of general triangles is also discussed.


Random Harmonic Series
by Byron Schmuland
What happens if you multiply each term in the harmonic series by a plus or minus sign, randomly chosen by flipping a fair coin? The resulting random variable, called the "random harmonic series," has some fascinating properties. For instance, although the partial sums are discrete random variables, the random harmonic series has a smooth density g with a peculiar shape. Certain values of the density are also interesting; for example, g(2)=.12499999999999999999999999999999999999999976421683. We explain why g(2) is so close to 1/8, but not quite equal to it.


Four Colors Do Not Suffice
by Hud Hudson
Is there an overlooked counterexample to the celebrated "four-color conjecture"? Well, that depends on exactly how the conjecture is formulated, and unfortunately, formulations that are widely taken to be equivalent may in fact not be equivalent. This paper relates the peculiar history of Zenopia-an island country with six provinces whose boundaries apparently threaten the conjecture, and whose existence teaches us a lesson we may name "the cartographic many-color thesis."

Problems and Solutions


Carleman's Inequality
by John Duncan and Colin M. McGregor,

A Note on the Solvability of a Diophantine Equation Involving Radicals
by M.A. Nyblom

Zeros of the Alternating Zeta Function on the Line Re(s)=1
by Jonathan Sondow


Matrix Groups: An Introduction to Lie Group Theory
by Andrew Baker
Lie Groups: An Introduction Through Linear Groups
by Wulf Rossmann
Reviewed by A.W. Knapp

Mathematical Treks: From Surreal Numbers to Magic Circles
by Ivars Peterson
Mathematical Vistas: From a Room with Many Windows
by Peter Hilton, Derek Holton, and Jean Pedersen
Reviewed by Marion Cohen

Telegraphic Reviews