We examine the history of a group of important results in general topology.
Grade School Triangles
By: Jack S. Calcut
In grade school, students learn a standard set of Euclidean triangles. Among this set, the usual 45–45–90 and 30–60–90 triangles are the only right triangles with rational angles and side lengths each containing at most one square root. Are there any other such right triangles? We answer this question completely and present an elegant complement called Ailles’ rectangle that deserves to be in every geometry teacher’s toolkit.
A Better Way to Deal the Cards
By: Mark A. Conger and Jason Howald
Most of the work on card shuffling assumes that all the cards in a deck are distinct, and that in a well-shuffled deck all orderings need to be equally likely. We consider the case of decks with repeated cards and decks which are dealt into hands, as in Bridge and Poker. We derive asymptotic formulas for the randomness of the resulting games. Results include the influence of where a poker deck is cut, and the fact that switching from cyclic dealing to back-and-forth dealing will improve the randomness of a bridge deck by a factor of 13.
Tanvolutes: Generalized Involutes
By: Tom M. Apostol and Mamikon A. Mnatsakanian
The classical involute of a plane base curve intersects every tangent line at a right angle. This paper introduces a tanvolute, which intersects every tangent line at any given fixed angle. This minor change in the definition of a classical concept leads to a wealth of new examples and phenomena that go far beyond the original situation. Our treatment is based on two differential equations relating arclength functions for the base curve and its tanvolute, the tangent-length function from the base to the tanvolute, and the fixed angle. The parameters in the differential equations contribute many essential features to the solution curves. Even when the base curve is relatively simple, for example a circle, the variety in the shapes of the tanvolutes is remarkably rich. To illustrate, as a circle shrinks to a single point, its tanvolute becomes a logarithmic spiral! An application is given to a generalized pursuit problem in which a missile is fired at constant speed in an unknown tangent direction from an unknown point on a base curve. Surprisingly, it can always be intercepted by a faster constant-speed missile that follows a specific tanvolute of the base curve.
The Seventieth William Lowell Putnam Mathematical Competition
By: Leonard F. Klosinski, Gerald L. Alexanderson, and Loren C. Larson
Wronskians and Linear Independence
By: Alin Bostan and Philippe Dumas
We give a new and simple proof of the fact that a finite family of analytic functions has a zero Wronskian only if it is linearly dependent.
A Simple Combinatorial Algorithm for de Bruijn Sequences
By: Abbas M. Alhakim
This paper presents a combinatorial method to construct a de Bruijn sequence for any order n. The method is similar to that of the well-known prefer-one algorithm. The resulting sequences are compared and both methods are shown to be special cases of a unifying graph construction that is capable of generating all de Bruijn sequences.
Central as the gamma function is, it is surprising that its only known simple values at rational points are and for integral m. There are, however, numerous formulas that relate specific values of to other functions, such as elliptic or hypergeometric functions—or to other specific values of itself. Among the latter there are products that have very simple values. This paper shows that a large class of -products at rational points have values that are just products of powers of 2 and . The number of factors in these products is surprisingly small.
Dimension, Linear functionals, and Norms in a Vector Space
By: Miyeon Kwon
Using the axiom of choice, we prove a generalized converse of the well-known fact that if X is a finite-dimensional vector space, then any linear functional on X is continuous with respect to all norms defined on X. We also show that an infinite-dimensional real or complex vector space X has exactly inequivalent norms.
Graph Theory: A Problem Oriented Approach.
By: Daniel A. Marcus
Reviewed by: : Marion Deutsche Cohen