##
December 2000

**Properties of Tournaments Among Well-Matched Players**

by Carolyn Eschenbach, Frank Hall, Rohan Hemasinha, Stephen Kirkland, Zhongshan Li, Bryan Shader, Jeffrey Stuart, and James Weaver

[email protected]

In a round robin tournament, each participant plays one match against each of the other participants. If each match must have a winner, then the outcomes of a tournament can be represented by a 0-1 matrix known as a tournament matrix. Naturally, players can be grouped by their scores the number of matches that they win. In order to more completely rank the players, however, we must determine how players with the same score compare. Surprisingly, a useful ranking scheme can be developed based on certain eigenvector properties of the associated tournament matrix. For such a ranking scheme, how can the overall closeness between well matched players be measured? And for a fixed size, which tournaments have the most closely ranked players? The answers are linked to the open problem of determining which tournament matrices of a fixed size have the maximum spectral radius. We present supporting evidence, including some recently proved special cases, for a famous conjecture by Brualdi and Li. We also present several other open problems that are easily accessible to students who have had an introductory course in matrix theory.

**Is There a Planar Set with Given Width, Diameter, and Inradius?**

by M. A. Hernández Cifre

[email protected]

There is an abundant literature on geometric inequalities for convex figures. These inequalities connect several geometric quantities and in many cases determine the extremal sets that satisfy the equality conditions.

Each new inequality obtained is interesting on its own, but it is also possible to ask if a collection of inequalities involving several geometric magnitudes is large enough to determine the existence of the figure. Such a collection is called a *complete system of inequalities:* a system of inequalities relating all the geometric characteristics such that for any set of numbers satisfying those conditions, a planar figure with these values of the characteristics exists in the given class. Santaló considered the area, the perimeter, the diameter, the minimal width, the circumradius and the inradius of a planar convex K (A, p, d, w, R, and r, respectively). He tried to find a complete system of inequalities concerning either two or three of these measures. He found that for pairs of measures, the known classic inequalities between them are a complete system. However, the situation is different for triples of measures. He gave a list of all the known inequalities involving triples of measures, studying in which cases there were complete systems of inequalities: (A,r, w), (A, r, r), (A, r, R),(A, r, R s), (A, d, w), (r, d, w), (d, r, R). He left the remaining cases as open problems. The cases (d, w, R) and (w, R, r) have been solved recently. In this paper I derive two new inequalities among d, w, and r, and use them to specify a complete system of inequalities for (d, w, r). So, there are still 11 open cases.

**A Unified Interpretation of the Binomial Coefficients, the Stirling Number, and the Gaussian Coefficients**

by John Konvalina

[email protected]

The *binomial coefficients,*Stirling numbers, and the *Gaussian coefficients* are classical combinatorial numbers with diverse interpretations that arise in various areas such as discrete mathematics, combinatorics, algebra, number theory, discrete probability, linear algebra, finite geometry, analysis of algorithms, and numerical analysis. Is there a simple unified combinatorial interpretation of these three classes of numbers? In this expository paper we summarize the various classical interpretations of these numbers followed by a unified combinatorial interpretation. We define *generalized binomial coefficients of the first and second kind* in terms of object (ball) selection from weighted boxes with and without box repetition. By varying the weights of the boxes (constant, linear , and exponential), we obtain as special cases the three classes of combinatorial numbers (binomial coefficients, Stirling numbers, and Gaussian coefficients, respectively).

**Constructible Approximations of Regular Polygons**

by Stefan Treatman and Cameron Wickham

[email protected]

It is no secret that not all regular *n*-gons can be constructed with a straight-edge and compass, but in this paper, we show a simple algorithm for constructing a sequence of polygons that converges to the regular *n* -gon, for any *n* greater than 2. Furthermore, by identifying an *n*-gon with a vector in real *n*-space, we can define this convergence using any vector norm in real *n*-space. Finally, we prove that this convergence is (asymptotically) geometric, independent of norm, with a surprising common ratio.

**Using Integral Transforms to Estimate Higher Order Derivatives**

by David M. Bradley

[email protected]

When doing error analysis for numerical quadrature, achieving good uniform bounds on higher order derivatives of the integrand is paramount. As undergraduates become increasingly adept with programmable calculators, numerical integration schemes such as Simpson’s Rule and the Trapezoidal Rule take on a new relevance. In this article, we offer some non-trivial examples of instances where integral transforms lead to elegant and useful error estimates. Our examples include a selection of some of the classical special functions, as well as Ramanujan’s inverse tangent integral, and an integral arising in the solution of a difference-differential equation relating to sieve theory.

**NOTES**

**A Simple Special Case of Sharkovskii's Theorem**

by Reid Barton and Keith Burns

[email protected]

**The Quadratic Residues -1 and -3**

by William Watkins

[email protected]

**Polynomials Solving Dirichlet Boundary Value Problems**

by Gerd Herzog

[email protected]

**An Extension of Fermat’s Little Theorem, and Congruences for Stirling Numbers**

by Karl Dilcher

[email protected]

**Morley Related Triangles on the Nine-Point Circle**

by Floor van Lamoen

[email protected]

**Archimedean Properties of Parabolas**

by Árpád Bényi, Pawel Szeptycki, and Fred Van Vleck

[email protected]

**PROBLEMS AND SOLUTIONS**

**REVIEWS**

**Women in Mathematics: the Addition of Difference. **

By Claudia Henrion

Reviewed by Marion D. Cohen

[email protected]

**Industrial Mathematics: Modeling in Industry, Science and Government.**

By Charles R. MacCluer

**Applied Mathematical Modeling: A Multidisciplinary Approach.**

By D.R. Shier and K.T. Wallenius

Reviewed by J. David Logan

[email protected]

**EDITOR'S ENDNOTES**