The Commandant's Corollary
by C. W. Groetsch
groetsch@email.uc.edu
Galilieo's parabolic model of the trajectory of a point particle in a resistance-less medium is considered by many to be the genesis of the science of dynamics. While discussions of the Galiean model are widespread in the undergraduate curriculum, elementary results on resisted trajectories seldom find their way into calculus courses. One such result is a barely known, and quite pretty, theorem of a fin de siècle French artillery commander that relates the general resisted trajectory to two enveloping Galilean trajectories. This note is an exposition of that theorem based on results from first year calculus.
Using Statistics to Determine Causal Relationships
by Jerome Reiter
jerome.p.reiter@williams.edu
Increasingly, causal questions are being answered with statistics. For both scientists and consumers, it has become important to understand how valid causal studies can be designed and how suspicious studies can be identified. This paper aims to further these understandings by explaining the statistical principles and techniques that underlie valid studies of causal relationships. It explains how randomized experiments yield reliable estimates of causal effects and describes techniques used to estimate causal effects in non-randomized studies.
Paradoxes Involving Conflicts of Interest
by James Case
jcase66777@aol.com
We argue that the boundary separating genuine paradox from mere subtlety advanced significantly in 1944, with the publication of The Theory of Games and Economic Behavior, by J. von Neumann and O. Morgenstern. Much of what had previously seemed paradoxical suddenly seemed less so. The thesis is developed with the aid of specific examples, including the original "Holmes-Moriarty paradox", the so-called "backward induction paradox", and the "surprise examination" or "hanging paradox".
Start Where They Are: Geometry as an Introduction to Proof
by J. E. McClure
mcclure@math.purdue.edu
All of us who teach mathematics majors would like our students to be able to do proofs, but it is a common experience that students have great difficulty making the transition from their early mathematical training to an environment in which proof is emphasized. In response to this difficulty, many mathematics departments have created "bridge" courses, usually focusing on a set theory or the elements of analysis. Another common approach is to use linear algebra as the students' first encounter with formal proof. Both of these methods can be used with reasonable success to train students to write elementary proofs that obey the rules, but I argue that neither of them is the best foundation for further progress because neither really addresses the root causes of our students' difficulties. Students who are learning about axiomatic formal proof for the first time have underlying, often unarticulated, confusions that are not easily perceived by professional mathematicians to whom formal proof has become second nature, and my contention is that the only truly satisfactory way to meet and answer these confusions is to begin with Euclidean geometry. Since there has been a trend for many years to de-emphasize or even eliminate formal proof at the high school level, I suggest that for most students the natural role of a bridge course is to provide a thorough study of the use of proofs in Euclidean geometry. I describe one effective way to organize such a course, and I also argue that Euclid's Elements should be one of the tools we use to teach axiomatic methods.
NOTES
A Simple Construction for Tournaments With Every k Players Beaten by a Single Player
by Jurek Tyszkiewicz
jty@mimuw.edu.pl, jty@cse.unsw.edu.au
Alternative Formulations of the Twin Prime Problem
by Maria Suzuki
marias@caltech.edu
An Easy Reduction of an Isoperimetric Inequality on the Sphere to Extremal Set Theory
by Alexander Russell
acr@cs.berkeley.edu
Rings Without Maximal Ideals
by Peter Malcomson and Frank Okoh
petem@math.wayne.edu, okoh@math.wayne.edu
A Proof of a Conjecture on Egyptian Fractions
by Thomas R. Hagedorn
hagedorn@math.tcnj.edu
THE EVOLUTION OF...
Two Letters by N. N. Luzin to M. Ya. Vygodskii
by S. S. Demidov
PROBLEMS AND SOLUTIONS
REVIEWS
Mapping Time: The Calendar and Its History.
By E. G. Richards
Reviewed by John P. Pratt
john@johnpratt.com
TELEGRAPHIC REVIEWS