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American Mathematical Monthly - April 2002

APRIL 2002

The Manipulability of Voting Systems
by Alan D. Taylor
Consider a voting situation in which there are three or more alternatives, ballots are linear orderings of the alternatives, and each alternative occurs as the unique winner for at least one sequence of ballots. The famous Gibbard-Satterthwaite Theorem asserts that in such a situation—if the voting system always produces a unique winner—then it is non-manipulable if and only if it is a dictatorship. An elegant generalization of this result to the context in which ties for the win are allowed has recently been obtained by Duggan and Schwartz. We give a direct, self-contained, and short proof of the Duggan-Schwartz Theorem, and we present some extensions of it and the original Gibbard-Satterthwaite Theorem.


Beyond the Descartes Circle Theorem
by Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks,,
The Descartes Circle Theorem states that if four circles are mutually tangent in the plane, with disjoint interiors, then their curvatures (or "bends") bi=1/ri satisfy the relation (b1+b2+b3+b4)2 = 2(b12+b22+b32+b42) . We show that similar relations hold involving the centers of the four circles in such a configuration, coordinatized as complex numbers, yielding a Complex Descartes Theorem. These relations have elegant matrix generalizations to the n-dimensional case, in each of Euclidean, spherical, and hyperbolic geometries. These include analogues of the Descartes Circle Theorem for spherical and hyperbolic space.


Two Applications of the Generalized Ptolemy Theorem
by Shay Gueron
Ptolemy’s Theorem, relating between the lengths of the sides and the diagonals of a cyclic quadrilateral, can be generalized to the situation where the vertices of the quadrilateral are replaced by four circles that touch the circumscribing circle. This paper demonstrates how the Generalized Ptolemy Theorem can be used to prove two results in plane geometry. One of these results is a generalization (to an external form) of Thébault’s Theorem, originally proposed as a problem in the MONTHLY, in 1938, and remained unsolved for some 45 years. The second result is a theorem about the common internal and external tangents of two circles that touch a third one.


Inverting the Pascal Matrix Plus One
by Rita Aggarwala and Michael P. Lamoureux,
The Pascal matrix is an old and familiar form that appears in many different areas of mathematics and its applications. Motivated by a problem that arises in order statistics, we demonstrate how to invert any linear combination of this infinite matrix with the identity matrix, and observe the occurrence of the polylogarithm function


Problems and Solutions


A Nowhere Differentiable Continuous Function Constructed By Infinite Products
By Liu Wen

Rino's Flight Around the World in Circles
By Johannes Huisman

A Tile With Surround Number 2
By Casey Mann

On Stirling’s Formula
By Reinhard Michel

An Inductive Proof of a Known Result about Ψ(x)
By Andrea Vietri


The Versatile Soliton
By Alexandre T. Fillippov
Review by J. David Logan