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December 2007

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**Practicing Mathematics in the Public Arena: Challenges and Outcomes in Some Prominent Case Studies**

By: Charles R. Hadlock

chadlock@bentley.edu

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

markku.halmetoja@mantta.fi, pentti.haukkanen@uta.fi, teuvo@lyseo.edu.ouka.fi, jorma.merikoski@uta.fi, timo.tossavainen@joensuu.fi, ari.virtanen@uta.fi

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 *p*s? 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

tchales@gmail.com

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

levi@math.psu.edu, tabachni@math.psu.edu

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.

**Notes**

**A Property of Parallelograms Inscribed in Ellipses**

By: Alain Connes and Don Zagier

alain.connes@college-de-france.fr, zagier@mpim-bonn.mpg.de

**An Elementary Proof of the Wallis Product Formula for Pi**

By: Johan Wästlund

jowas@mai.liu.se

**Automorphisms of Finite Abelian Groups**

By: Christopher J. Hillar and Darren L. Rhea

chillar@math.tamu.edu, drhea@math.berkeley.ed

**On the Adjugate of a Matrix**

By: António Guedes de Oliveira

agoliv@fc.up.pt

**Reviews**

**Divine Proportions: Rational Trigonometry to Universal Geometry**

By: N. J. Wildberger

Reviewed by: Michael Henle michael.henle@oberlin.edu

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