MathOverflow

My university has decided that from now on there will be no personal (snail) mailboxes for individual faculty members. Each department as a whole will continue to have its own mailbox.
Of course, each of us is supposed to list his or her professional address on our publications so as to identify the university where we work, for the greater glory of the university. However, mail sent to such an address will not arrive to the individual...

Is there at least one prime $q$ such that $\sigma{(q^{p-1})}$ assumes infinitely many composite values, as $p$ ranges over the primes? If $q=2$, we get the Mersenne numbers, $2^{p}-1$.

Given an irrational rotation $\omega \in \mathbb T^d$, we may define the set$$S_\gamma = \{m \in \mathbb Z^d: \|\langle \omega,m\rangle\|_{\mathbb T} < |m|^{-\gamma}\},$$and the densities$$A_\gamma(N) = \{m \in S_\gamma : |m| \leq N\}.$$I am particularly...

I suspect that I'm missing some category theoretic/universal algebraic terminology which would make this question more efficient. Feel free to edit to include that!
Given a variety (in the sense of universal algebra) $\mathbb{V}$ and a set $X$, let $\mathbb{V}(X)$ be the set of $\mathbb{V}$-algebras with underlying set $X$. Say that a a $(\mathbb{V},X)$-umbrella is a pair $(M,(f_A)_{A\in\mathbb{V}(X)})$ where $M...