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May 2006

**A $1 Problem**

by Michael J. Mossinghoff

mjm@member.ams.org

Suppose you need to design a new $1 coin with a polygonal shape, fixed diameter, and maximal area or maximal perimeter. Are regular polygons optimal? Does the answer depend on the number of sides? We investigate these two isodiametric problems for polygons and describe how to construct polygons that are optimal, or very nearly so, in each case.

**What Can be Approximated by Polynomials with Integer Coefficients**

by Le Baron O. Ferguson

ferguson@math.ucr.edu

A well-known result of Weierstrass states that any continuous function on a closed bounded interval of the real line can be uniformly approximated by polynomials. If we restrict ourselves to polynomials whose coefficients are all integers can anything interesting be said? The answer is yes.

**Periodicity and Predictability in Chaotic Systems**

by Marcelo Sobottka and Luiz P.L. de Oliveira

sobottka@dim.uchile.cl, luna@exatas.unisinos.br

In this paper, we present a simple chaotic system (satisfying Devaney’s definition) that is periodic and computationally predictable under a symbolic representation scheme. The system consists of the restriction of the tent map to the rational numbers of its original domain. The example contradicts the usual belief that chaotic systems are necessarily nonperiodic and nonpredictable. A general discussion on the concept of computational predictability and its relationship with the existence of periodic orbits is included.

**The Simplest Example of a Normal Asymptotic Expansion**

by José Antonio Adell and Alberto Lekuona

adell@unizar.es, lekuona@unizar.es

The central limit theorem has been described as one of the most important results in mathematics, mainly due to its proven application well beyond its own field. The investigation of the rates of convergence in this theorem, taking the form of normal asymptotic expansions, has great interest both from theoretical and practical points of view. However, proofs of these kinds of results are generally intricate, no matter what method is used. Being guided by the principle of "the general is embodied in the concrete," we provide a very simple example of a normal asymptotic expansion in which the technical complexity is reduced to a minimum. This example has the additional advantage of making clear the connections between some familiar notions and tools from probability theory and mathematical analysis. The content is accessible to a nonexpert looking for the "what" and "why" of this amazing research area.

**Notes**

**The Arbitrariness of the Cevian Triangle**

by Mowaffaq Hajja

**Curiosities Concerning Weak Topology in Hilbert Space**

by Gilbert Helmberg

gilbert.helmberg@telering.at

**More Formulas for π**

by Hei-Chi Chan

chan.hei-chi@uis.edu

**On Gauss’s Entry from January 6, 1809**

by Detlef Gröger

groeger.d@t-online.de

**Problems and Solutions**

**Reviews**

Prime Obsession

by John Derbyshire

Reviewed by Jeffrey Nunemacher

jlnunema@cc.owu.edu

**Stalking the Riemann Hypothesis**

by Dan Rockmore

Reviewed by Jeffrey Nunemacher

jlnunema@cc.owu.edu