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American Mathematical Monthly -December 2007

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December 2007

Read the latest issue of the Monthly online. (This requires MAA membership.)

Practicing Mathematics in the Public Arena: Challenges and Outcomes in Some Prominent Case Studies
By: Charles R. Hadlock
Textbooks and class lectures often touch on the important role of mathematical modeling in dealing with societal issues, but these are generally the "sanitized" versions in which the neat mathematics has been abstracted from the complexities of real world applications. After all, many of us went into mathematics because we liked precision, elegance, and abstraction. This paper presents an insider’s view of the more sordid world of "real" real world applications in the public policy arena, which are often fraught with public and legal controversy, operate in fits and starts, and require communications with the most diverse audiences. This is still a world where mathematics is making important contributions, sometimes by its calculational results but often more generally by the logical framework it imposes on a problem area. Several environmental cases are followed through their complex history, and this may give both students and professionals a glimpse of opportunities they may wish to pursue.

On Direct and Inverse Proportionality
By: Markku Halmetoja, Pentti Haukkanen, Teuvo Laurinolli, Jorma K. Merikoski, Timo Tossavainen, and Ari Virtanen,,,,,
Two quantities are said to be directly proportional if, by whatever positive real number p one of them is multiplied, the other changes by the same factor p. But is it necessary to go through all ps? What more is required than insisting, say, that if one of the quantities doubles, then the second doubles, too? Given a positive real number p different from 1 and a real-valued function f on positive real numbers satisfying f(px)=pf(x) for all x, we study conditions under which f(x)=f(1)x for all x. Similarly, we consider what we need to assume on a function f satisfying f(px)=f(x)/p for all x to ensure that f(x)=f(1)/x for all x.

The Jordan Curve Theorem, Formally and Informally
By: Thomas C. Hales
This article is intended as an elementary introduction to formal methods in mathematics, using the Jordan curve theorem as an extended example. The author has recently completed a formal proof of the Jordan curve theorem, in which every single logical inference has been generated and checked by computer. Contrary to widespread misconceptions about formal proofs, this formal proof is comprehensible and even intuitive, although considerably longer than a conventional proof.

The Poncelet Grid and Billiards in Ellipses
By: Mark Levi and Serge Tabachnikov,
The Poncelet porism is one of the most beautiful theorems of classical projective geometry. Given two nested conics in the plane, suppose that there exists a closed polygonal line which is inscribed in one conic and circumscribed about the other one. The Poncelet porism states that then there exists a one-parameter family of such inscribed-circumscribed Poncelet polygons. Recently Rich Schwartz discovered a refinement of this theorem. Intersecting the extensions of the sides of a Poncelet polygon, one obtains a collection of points, called the Poncelet grid. The Poncelet grid lies in the union of conics and has a rich symmetry group. We survey the theory of mathematical billiards and deduce the Schwartz theorem from properties of the billiard system inside an ellipse.


A Property of Parallelograms Inscribed in Ellipses
By: Alain Connes and Don Zagier,

An Elementary Proof of the Wallis Product Formula for Pi
By: Johan Wästlund

Automorphisms of Finite Abelian Groups
By: Christopher J. Hillar and Darren L. Rhea, drhea@math.berkeley.ed

On the Adjugate of a Matrix
By: António Guedes de Oliveira


Divine Proportions: Rational Trigonometry to Universal Geometry
By: N. J. Wildberger
Reviewed by: Michael Henle

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