MathOverflow

Suppose $G$ is a countable group, such as the free group $F_2$. Let $\mu$ be a probability measure in $G$ not supported in any left coset of any proper subgroup of $G$, and let $*^n\mu=\mu*\cdots*\mu$ $n$ times. Finally, let $f:G\to[0,1]$ be a weakly...

Let $p_n$ denote the $n$-th prime. Cipolla's classical asymptotic expansion (see Cipolla 1902; Salvy 1994; Arias de Reyna–Toulisse 2012, [arXiv:1203.5413]) gives, with $L = \log n$ and $M = \log L$,$$\frac{p_n}{n} \;=\; L + M - 1 + \frac{M - 2}{L} -...

For the following planar vector field:$$\begin{cases}\dot{x}=y^2 \\\dot{y}=a_{02}y^2+a_{01}y+a_{10}x\end{cases}$$where $a_{10} \neq 0$, is it possible to prove strictly that the system has analytic first integral iff one of the two following conditions...

A simple manifold is a compact Riemannian manifold with boundary such that all geodesics are minimizing, have no conjugate points, and the boundary is convex (has positive a second fundamental form).
A geodesic net is an embedding of a finite graph in a Riemannian manifold such that each edge is a geodesic and the directions of edges at each vertex do not point in a hemisphere....