Crofton's Differential Equation
by Bennett Eisenberg and Rosemary Sullivan
Consider the situation where n points are chosen at random in a domain of area v and the problem is to find the expected value of some function of the set of points. M. W. Crofton, a nineteenth century mathematician, introduced a technique for simplifying the solution of such problems. It involves solving a differential equation that relates the given expectation to a similar expectation, where one of the points is selected on the boundary of the region and the others are selected inside. By modern standards the statement and proof of the result are very imprecise.
In this paper we give a rigorous derivation of a version of Crofton's differential equation. This version is somewhat easier to apply than the original. More importantly, the form of solution of the equation shows an intimate connection between Crofton's technique and a standard technique for computing expectations in probability. Furthermore, a dimensional analysis shows that in many situations there is a simple algebraic relation between the two expectations in the differential equation.
The paper also describes how Crofton's differential equation can be viewed as a generalization of the fundamental theorem of calculus and is connected to some sophisticated results in modern differential geometry.
What is the Correct Way to Seed a Knockout Tournament?
by Allen J. Schwenk
Many sports competitions use seeding to schedule an elimination (or knockout) tournament. We identify three axioms that ought to be satisfied by any seeding method we might use, namely delayed confrontation, sincerity rewarded, and favoritism minimized. We proceed to show that there is a unique method called cohort randomized seeding that satisfies these axioms. Unfortunately, this is not the method in common use.
As a Continuous Function of x and
by Ali Enayat
We address the following question dealing with the ubiquitous epsilon-delta definition of continuity: can continuity be verified in a continuous manner? In other words, suppose f is a continuous function from some metric space into another. Is there always a continuous function delta of two variables x and epsilon that demonstrates the continuity of the function f?
A Commutative Multiplication of Number Triplets
by Frank R. Pfaff
In October 1843 Hamilton redoubled his efforts to define a multiplication of triplets having all of the properties of complex numbers. On his way he dropped the commutative law and later was forced to replace triplets with quadruplets; this latter revision acting as the algebraic analogue of what he considered as the "paradoxical" idea of 4-dimensional space. Prior to his discovery of quaternions he did define a commutative multiplication of triplets for the case that the corresponding vectors lay in a plane containing the real axis. However, he never pursued this further.
Using this result, which the author rediscovered, we show that if a line in 3-space is selected and a plane containing this line is complexified with the chosen line as the real axis, then each plane containing this line inherits a complex structure in a natural way. The 3-space of triplets can then be considered as a collection of complex planes with a common real axis, much like a rolodex file. A commutative multiplication of two triplets whose plane does not contain the line chosen is defined, and this is used to define a commutative multiplication of arbitrary triplets. But, alas, the structure obtained by using the usual arithmetic together with this product is rather bizarre and, although it is consistent it very likely would have had little or no appeal to Hamilton.
The Rendezvous Number of a Symmetric Matrix and a Compact Connected Metric Space
by Carsten Thomassen
On the Equation a (a + d)(a + 2d)( a + 3d) = x2
by Tamas Erdélyi
On Nonlinear Summability
by Gerd Herzog
The Anti-Social Fermat Number
by Florian Luca
A Fractal Example in Ordinary Differential Equations
by Anatole Beck
Problems and Solutions
Cake-Cutting Algorithms: Be Fair If You Can.
By Jack Robertson and William Webb
Reviewed by Francis Edward Su