MathOverflow

Consider the following simple Markov chain $ X_1\to X_2\to\cdots\to X_n $ where each $X_i$ is $\{-1,1\}$-valued and $X_1\sim\mathrm{Unif}(\{-1,1\})$ (such that the chain is stationary).The flip probability is $\delta$, i.e., $\Pr[X_{i+1} = -x\,|\,X_i...

I believe I've proved that the power semigroup of non-negative integers with addition has a trivial automorphism group. The proof is a bit long, completely elementary and rather unremarkable (as the fact itself probably) so I won't post it here. However...

Can $\mathbb R^2$ be the union of a family of infinitely differentiable, injective, unit-speed images of the real line such that at each point of the plane, precisely two of these curves merge (they coincide from that point on)?

Assign exactly one $0$ and one $1$ to the two ends of each edge of a regular simple undirected bipartite graph of $e$ edges, uniform degree $d$ and number of vertices $2v$. The assignment of the pair $\{0,1\}$ to an edge can be considered as orienting...