On the adjoint representation of sln\mathfrak{sl}_n and the Fibonacci numbers

We decompose the adjoint representation of $\mathfrak{sl}_{r+1}=\mathfrak {sl}_{r+1}(\mathbb C)$ by a purely combinatorial approach based on the introduction of a certain subset of the Weyl group called the \emph{Weyl alternation set} associated to a pair of dominant integral weights. The cardinality of the Weyl alternation set associated to the highest root and zero weight of $\mathfrak {sl}_{r+1}$ is given by the $r^{th}$ Fibonacci number. We then obtain the exponents of $\mathfrak {sl}_{r+1}$ from this point of view.Comment: 9 page

Strengthening Communication Skills through Cadet Created Homework Videos

The wide availability of online tutorial videos has revolutionized the way students learn mathematics. In this project we take the approach of having students create their own homework videos by using the screen-capture and video recording capabilities of an iPad. The aims of these videos were twofold. First, through the creation of homework videos cadets conveyed technical information; this helped strengthen their communication skills. A second objective of this project was the development of an online library of videos that cadets could study from in preparation for homework and exams. In this paper we examine data gathered during this project to determine the benefits of replacing traditional handwritten homework assignments with videos. In addition, we provide some observations as to the usefulness of the videos in strengthening communication skills as well as what effects an online library of videos has on the study habits of students