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American Mathematical Monthly -March 2010


March 2010

Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2010 to Kenneth A. Ross for Distinguished Service to Mathematics
By: Barbara Faires
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Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries
By: Marvin J. Greenberg
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This survey highlights some foundational history and some interesting recent discoveries in elementary geometry that deserve to be better known, such as the hierarchies of axiom systems, Aristotle’s axiom as a "missing link," Bolyai’s discovery— proved and generalized by William Jagy— of the relationship of "circle-squaring" in a hyperbolic plane to Fermat primes, the undecidability, incompleteness, and consistency of elementary Euclidean geometry, and much more. A main theme is what Hilbert called "the purity of methods of proof," exemplified in his and his early twentieth century successors’ works on foundations of geometry.


Pólya's Theorem on Random Walks via Pólya's Urn
By: David A. Levin and Yuval Peres
[email protected], [email protected]
We give a proof of Pólya's 1921 theorem on the transience of the simple random walk on Z3 using the Pólya urn process (Eggenberger and Pólya, 1923; Pólya, 1931). We give a self-contained exposition of the method of flows, which provides a necessary and sufficient condition for transience of a simple random walk on an infinite graph. The key ingredient to our proof of transience of Z3 is the construction of a flow on Z3 using the Pólya urn process. While the transience result is classical and can be proved in many ways, it is particularly satisfying to derive it from Pólya's urn, a connection which surely was not realized by Pólya himself.


A Congruence Problem for Polyhedra
By: A. Borisov, M. Dickinson, and S. Hastings
[email protected], [email protected], [email protected]
It is well known that to determine a triangle up to congruence requires 3measurements: three sides, two sides and an angle, or one side and two angles. We consider various generalizations of this fact to two and three dimensions. In particular we consider the following question: given a convex polyhedron P, how many measurements are required to determine P up to congruence? We show that, in most cases, the number of measurements required to determine the polyhedron locally is equal to the number of edges of the polyhedron. However, for some polyhedra fewer measurements suffice; in the case of the cube we show that nine carefully chosen measurements are enough. We also prove a number of analogous results for planar polygons. In particular we describe a variety of quadrilaterals, including all rhombi and all rectangles, that can be determined up to congruence with only four measurements, and we prove the existence of n-gons requiring only n measurements. Finally, we show that one cannot do better: for any ordered set of n distinct points in the plane one needs at least n measurements to determine this set up to congruence.


Euclid Meets Bézout: Intersecting Algebraic Plane Curves with the Euclidean Algorithm
By: Jan Hilmar and Chris Smyth
[email protected], [email protected]
Finding the intersection point of two lines in the plane is easy. But doing the same for a pair of algebraic plane curves is more difficult, not least because each point on both curves may be a multiple intersection point. However, we show that, with the help of the Euclidean algorithm for polynomials, this general problem can be reduced to the case of intersecting lines, giving an algorithm for finding these intersection points, with multiplicities. It also yields a simple proof of Bézout's theorem, giving the total number of such points.



A Realization of Measurable Sets as Limit Points
By: Jun Tanaka and Peter F. McLoughlin
[email protected], [email protected]
Starting with a sigma-finite measure on an algebra, we define a pseudometric and show how measurable sets from the Caratheodory Extension Theorem can be thought of as limit points of Cauchy sequences in the algebra.

A Parity Theorem for Drawings of Complete and Complete Bipartite Graphs
By: Dan McQuillan and R. Bruce Richter
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Forty years ago, Kleitman considered the numbers of crossings in good planar drawings of the complete bipartite graph K_{m,n}. Among other things, he proved that, for m and nboth odd, the parities of these numbers of crossings are all the same. His proof was sufficiently controversial that he provided another proof a few years later. In this work, we provide a complete, simple proof based on counting, elementary graph theory, and the Jordan Curve Theorem.

Three Proofs of the Inequality e{n+0.5}
By: Sanjay K. Khattri
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The inequality e>(1+1/n)n is well known. In this work, we give three proofs of the inequality e{n+0.5}. For deriving the inequality, we use the Taylor series expansion and the Hermite-Hadamard inequality. In the third proof, we define a strictly increasing function which is bounded from above by 0.5.


Alfred Tarski: Life and Logic
By: Anita Burdman Feferman and Solomon Feferman
Reviewed by: Anil Nerode
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