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Mathematics Magazine - February 2003

ARTICLES

Dr. David Harold Blackwell, African American Pioneer
Nkechi Agwu, Luella Smith, and Aissatou Barry
This article is a biography of Dr. David Harold Blackwell, an African American educational pioneer and eminent scholar in the fields of mathematics and statistics. The article traces the educational and professional journey of Dr. Blackwell from early childhood to retirement, highlighting the contributions that earned him world recognition, as well as racial incidents or controversies that were part of this journey. It emphasizes the connections of mentoring, professional development, and professional service in Dr. Blackwell's rise to fame and to his work as a caring educator of students and young professionals. Lastly, it reflects on distinguishing character traits that fostered Dr. Blackwell's success - leadership, humility, courage and passion for one's field, and the social and psychological consequences of any form of discrimination on society.

A Tale of Three Circles
Charles Delman and Gregory Galperin
Any three pairwise-intersecting circles in the plane are geodesics in a conformal model of some classical geometry (spherical, Euclidean, or hyperbolic), and there is a surprisingly simple criterion for determining which geometry applies. We accordingly classify the angle sum of a curvilinear triangle whose sides are arcs of circles. In the natural course of discussing this topic, we give a self-contained and concrete description of the three classical geometries and many useful conformal transformations.

NOTES

Power Distribution in Four Player Weighted Voting Systems
John Tolle
With the aid of two hypothetical examples, we show that it is not always possible to apportion votes among four parties in such a way that a prescribed hierarchy of power results. Power is determined by computing the Banzhaf power index for each player, resulting in a power distribution for the weighted voting system. We show that in the absence of a condition called veto power, only five power distributions are possible for four player systems.

Self-Similar Structure in Hilbert's Space-Filling Curve
Mark McClure
Hilbert's space-filling curve is a continuous function that maps the unit interval onto the unit square. In this note, we discuss how modern notions of self-similarity illuminate the structure of this curve. In particular, we show that Hilbert's curve has a basic self-similar structure and may be generated using an iterated function system or IFS. Furthermore, its coordinate functions display a generalized type of self-similarity called digraph self-affinity and may be described using an appropriately generalized iterated function system.

A Dynamical Systems proof of Fermat's Little Theorem Kevin M. Iga
Fermat's Little Theorem is an important result in number theory, and has many algebraic proofs. We offer here a visual proof of Fermat's Little Theorem using an easy observation about a particular dynamical system on the unit interval [0, 1].

Using Tangent Lines to Define Means
Brian C. Dietel and Russell A. Gordon
Suppose that f is concave up on (0,∞). If a and b are distinct positive numbers, then the tangent lines to f at a and b meet at a point c between a and b. If f(x)=x2, then c is the arithmetic mean of a and b; if f(x)=1/x, then c is the harmonic mean of a and b. We will show that any function whose concavity has constant sign on (0, ∞) generates a mean in this way and that the means exhibit a surprising degree of simplicity and elegance.

Characterization of Polynomials Using Divided Differences
Elias Y. Deeba and Plamen Simeonov
This note is intended to give an alternative proof for characterizing polynomials via divided differences. The result is a generalization of the following geometric property: if for A function f the secant line through any pair of pints (a,f(a)) and (b,f(b)) is parallel to the tangent line through the point ((a+b)/2,f((a+b)/2)), then f is a quadratic function.

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