Johann Faulhaber and sums of powers

Early 17th-century mathematical publications of Johann Faulhaber contain some remarkable theorems, such as the fact that the $r$-fold summation of $1^m,2^m,...,n^m$ is a polynomial in $n(n+r)$ when $m$ is a positive odd number. The present paper explores a computation-based approach by which Faulhaber may well have discovered such results, and solves a 360-year-old riddle that Faulhaber presented to his readers. It also shows that similar results hold when we express the sums in terms of central factorial powers instead of ordinary powers. Faulhaber's coefficients can moreover be generalized to factorial powers of noninteger exponents, obtaining asymptotic series for $1^{\alpha}+2^{\alpha}+...+n^{\alpha}$ in powers of $n^{-1}(n+1)^{-1}$

The sandwich theorem

This report contains expository notes about a function $\vartheta(G)$ that is popularly known as the Lov\'asz number of a graph~$G$. There are many ways to define $\vartheta(G)$, and the surprising variety of different characterizations indicates in itself that $\vartheta(G)$ should be interesting. But the most interesting property of $\vartheta(G)$ is probably the fact that it can be computed efficiently, although it lies ``sandwiched'' between other classic graph numbers whose computation is NP-hard. I~have tried to make these notes self-contained so that they might serve as an elementary introduction to the growing literature on Lov\'asz's fascinating function

A note on digitized angles

We study the configurations of pixels that occur when two digitized straight lines meet each other