Noncrossing Partitions in Surprising Locations
by Jon McCammond
Certain mathematical structures make a habit of reoccurring in the most diverse list of settings. Some obvious examples exhibiting this intrusive type of behavior include the Fibonacci numbers, the Catalan numbers, the quaternions, and the modular group. In this article, the focus is on a lesser known example: the noncrossing partition lattice. The focus of the article is a gentle introduction to the lattice itself in three of its many guises: as a way to encode parking functions, as a key part of the foundations of noncommutative probability, and as a building block for a contractible space acted on by a braid group. Since this article is aimed primarily at nonspecialists, each area is briefly introduced along the way.
Linear Independence and Series Expansions in Function Spaces
Ole Christensen and Khadija Laghrida Christensen
We consider complex vector spaces generated by certain special functions and examine whether their linear combinations have unique representations. We consider trigonometric functions, complex exponential functions, and certain more complicated systems of functions (Gabor systems and wavelet systems) that have recently attracted much attention both in pure mathematics and in applied science. We present some open problems related to those systems, problems that are easy to formulate but apparently very difficult to solve. Finally, we introduce frames, which generalize the concept of an orthonormal basis. The motivation for this generalization comes from Gabor analysis, where we show that certain desirable properties are incompatible with the orthonormal basis requirement. We show how the concept of linear dependence for wavelet systems plays a key role in modern constructions of frames having wavelet structure.
Rearranging the Calculus Sequence to Better Serve Its Partner Disciplines
Bernd S. W. Schröder
This paper shows how the calculus-differential equations sequence for disciplines close to mathematics can be rearranged in a multitude of ways to better serve our partner disciplines. One possible rearrangement is described in the context of a college wide effort to reintegrate engineering, mathematics, and the sciences.
A Geometric Proof That e is Irrational and a New Measure of Its Irrationality
by Jonathan Sondow
Pairings and Signed Permutations
by Valerio De Angelis
Iterated Products of Projections in Hilbert Space
by Anupan Netyanun and Donald C. Solmon
Noting the Difference: Musical Scales and Permutations
by Danielle Silverman and Jim Wiseman
On a Congruence modulo a Prime
by Hao Pan
Problems and Solutions
When Computers Were Human
by David Alan Grier
Reviewed by David E. Zitarelli