Given a positive integer \(m\), the authors exhibit a group with the...

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**Two Dual Assertions: The First on Learning and the Second on Teaching (or Vice Versa)**

by Guershon Harel

harel@math.purdue.edu

Students' ways of thinking influence how they understand mathematical concepts. This assertion is demonstrated by examples of faulty ways of thinking commonly possessed by students and by examples of essential ways of thinking largely absent from students' mathematical behavior. Conversely, it is suggested that students' ways of thinking are influenced by how they come to understand mathematical content.

**Univalent Polynomials and Non-Negative Trigonometric Sums**

by Alan Gluchoff and Frederick Hartmann

gluchoff@ucis.vill.edu

How can you produce interesting examples of polynomials (with complex coefficients) that are one-to-one on the unit disc? How can you find interesting examples of trigonometric series that are non-negative on the real line, or a part of it? How are these questions related? How can computer graphics packages help us visualize the variety of behaviors displayed by univalent polynomials? These are some of the questions examined in the paper. Beginning with the work of Alexander and Fejer in the early part of this century, we examine specific examples of such polynomials and sums and elementary arguments that prove they possess the desired properties. We then show how knowledge about and examples of each of these types of sums contributed to understanding the other. We highlight their interaction as well as more contemporary efforts. Central to the paper is the idea that elementary arguments relating the two topics can be used to produce specific examples of univalent polynomials whose behavior can then be explored with computer graphics packages. This provides more concrete examples mappings to add to the list of the standard conformal maps one encounters in a typical complex variables course. In this way a geometric feel for holomorphic functions, akin to what one experiences in first semester calculus for elementary polynomials and their growth, can be provided as a supplement to the analytic treatment of the topic. And the mappings are fun to explore!!

**Reflections on Reflection in a Spherical Mirror**

by Peter M. Neumann

peter.neumann@queens.ox.ac.uk

How can reflection points in a spherical mirror be found? That is, given a spherical mirror, a point where a light source is placed, and another point where there is an observer, how can a point on the mirror be determined, where a ray of light is reflected from source to observer? Although this question had been treated by Ptolemy in about AD 150, it is widely known as Alhazen's Problem after the arab scholar Ibn al-Haytham, who wrote an extensive account of it almost exactly 1000 years ago.

Many methods are known. Surprisingly, however, it appears that until now no one had thought to ask whether the reflection point might be found by the ruler and compass methods of classical geometry. Once the question is asked the answer is unsurprising: No, not in general. As with the more famous classical problems of squaring circles, trisecting angles, and duplicating cubes, the impossibility can be established by using the beginnings of Galois theory to study the arithmetic nature of the cartesian coordinates of relevant points. This paper is devoted to an exposition of the proof and some discussion of the context and the issues.

**3-Smooth Representations of Integers**

by Richard Blecksmith, Michael McCallum, and J. L. Selfridge

richard@math.niu.edu, selfridge@math.niu.edu

In his famous first letter to G. H. Hardy, Ramanujan gave a remarkably accurate estimate for the number of integers of the form 2^a 3^b up to a given bound x. Eighty years later another notable twentieth century mathematician, Paul Erdos, gave a 3 line proof that every positive integer n can be written as a sum of integers of the form 2^a 3^b , in which no term divides another. This raises many questions, such as: What's special about 2 and 3? How many such representations does a number typically have? (11 has 2 representations 2 + 9 and 3 + 8; 13 has just one representation 4 + 9.) If n is sufficiently large, can you find a representation for n that avoids small terms? This last question was conjectured by Erdos and Lewin, shortly before Paul passed away in the fall of 1996. In this paper, the authors give a method for finding all representations of a given number. Although a number typically has several representations, a thin set of integers have a unique representation, and the authors characterize these. The paper then settles the conjecture of Erdos and Lewin by showing that every large integer has a representation with all terms bigger than any given bound M. The method is constructive --- they manipulate certain matrices to find the desired representation --- and the bound given is best possible. Finally, the introduction of the paper concludes with a connection between representations and the well known 3x+1 problem.

**Aristotle's Fall**

by C. W. Groetsch

groetsch@uc.edu

The historical roots of the law of fall are traced briefly through the works of Aristotle, Galileo, and Newton. The time of fall, as a function of mass, is investigated. For general media it is shown that this function is decreasing, but never in the way specified by Aristotle.

**NOTES**

On Proving the Existence of Complete Ordered Fields

by B. Banaschewski

A Series for ln *k*

by Charles Kicey and Sudhir Goel

ckicey@valdosta.edu, sgoel@valdosta.edu

C^1-Solutions of *x* = *f(x')* Are Convex or Concave

by Gerd Herzog

Gerd.Herzog@math.uni-karlsruhe.de

UNSOLVED PROBLEMS

Parallel Knockouts in the Complete Graph

by Douglas E. Lampert and Peter J. Slater

slater@math.uah.edu

PROBLEMS AND SOLUTIONS

REVIEWS

Visual Complex Analysis. By Tristan Needham.

Complements to the review of *Visual Complex Analysis*.

Reviewed by Frank A. Farris

ffarris@mailer.scu.edu

Linear Algebra Problem Book. By Paul R. Halmos.

Reviewed by Robert Messer

ram@albion.edu

After Math. Puzzles and Brainteasers. By Ed Barbeau.

The Chicken from Minsk. By Yuri B. Chernyak and Robert M. Rose.

New Mathematical Diversions, revised edition. By Martin Gardner.

Reviewed by David Singmaster

zingmast@sbu.ac.uk

TELEGRAPHIC REVIEWS

EDITOR'S ENDNOTES