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American Mathematical Monthly -April 2008


April 2008

For subscribers, read The American Mathematical Monthly online.

For subscribers, read The American Mathematical Monthly online.

Prime Number Patterns
By: Andrew Granville
In 2004, Ben Green and Terry Tao announced that they had proved that there are infinitely many arbitrarily long arithmetic progressions of primes. With this mighty tool in hand, we investigate such questions as the number of prime values taken on by higher-degree polynomials at consecutive integers, sets of primes in which all of the averages of subsets of elements are prime, magic squares consisting only of primes, and high-dimensional generalized arithmetic progressions of primes. We illustrate these problems with many examples and try to predict when one should expect to first see examples of different types.


Optimal Congressional Apportionment
By: Robert A. Agnew
After two centuries of experimentation with alternative allocation rules, and considerable political maneuvering, the issue of the "best" method for allocating seats in the U.S. House of Representatives remains unsettled, although the field has narrowed considerably to so-called "divisor" methods. In this article, we introduce two new divisor methods based on the logarithmic and identric means, and we contrast them with existing divisor methods based on the geometric and arithmetic means. In particular, we show that all four of these methods correspond to simple constrained optimization problems where a fixed number of seats is allocated across the states. In this context, we argue that the identric and arithmetic objective functions are the most natural, and that the identric objective function has certain theoretical advantages stemming from information theory and statistical hypothesis testing. Finally, we present comparative decennial House apportionments during the period 1920-1990.


The Carathéodory and Kobayashi Metrics and Applications in Complex Analysis
By: Steven G. Krantz
This article is meant to be a self-contained introduction to the idea of invariant metrics in complex analysis on domains in the plane. Particular attention is paid to the Carathéodory and Kobayashi metrics. Connections with the Schwarz lemma and the uniformization theorem are explored. A number of applications are provided.


An Elementary Proof of Marden's Theorem
By: Dan Kalman
Marden's Theorem establishes a beautiful geometric characterization of the roots of the derivative of a cubic polynomial p, in terms of the roots of p. If the roots of p are distinct noncollinear points in the complex plane, and thus vertices of a triangle, then the roots of the derivative are the foci of an ellipse inscribed in that triangle. This paper presents an elementary proof of Marden's Theorem. Key ingredients are parts of proofs given separately by Bôcher (of Bôcher prize fame) in 1892 and Marden himself in 1945, as well as a property of ellipses dating back to Appolonius. A related paper with dynamic graphics, extended historical remarks, and detailed mathematical background will be available at JOMA.


The Hermite-Hadamard Inequality on Simplices
By: Mihály Bessenyei
The classical Hermite-Hadamard inequality provides a lower and an upper estimation for the integral average of any convex function defined on a closed interval, involving the midpoint and the endpoints of the domain. This inequality plays an important role in research on inequalities and has quite a large technical literature. Among the various generalizations that have been studied is the problem of extending Hadamard's inequality to convex functions of several variables. The most general answer to this problem was given by C. P. Niculescu who noticed that there is a very deep connection between the Hermite-Hadamard inequality and Choquet's Theory: one of its corollaries yields a full extension of Hadamard's inequality to the case when the domain is a metrizable compact convex set.

The aim of the paper is to verify Hadamard's inequality on simplices via an elementary approach, independently of Choquet's Theory. The main result provides a lower and an upper estimation for the integral average of any convex function defined on a simplex involving the centroid and the vertices of the simplex. The key idea of the approach is a volume formula for triangular prisms and its higher-dimensional generalization. Hence, the main result can be considered as a simple consequence of the support properties of convex functions.



A Continued Fractions Approach to a Result of Feit
By: John P. Robertson and Keith R. Matthews,

The Sum of Distances Between Vertices of a Convex Polygon with Unit Perimeter
By: Gerhard Larcher and Friedrich Pillichshammer,

Dynamical Systems and Irrational Angle Construction by Paper-Folding
By: Cayanne McFarlane and Wm. Douglas Withers

Lost (and Found) in Translation: André's Actual Method and its Application to the Generalized
Ballot Problem
By: Marc Renault

On Sums of Positive Integers That Are Not of the Form ax+by
By: Amitabha Tripathi


Applied Linear Algebra
By: Peter J. Olver and Chehrzad Shakiban
Reviewed by: Leslie Hogben and Wolfgang Kliemann,