For x in R, f(x) = 1 is such a function : f(f(x)) = 1. But I need to find another, different function, such that g(g(x)) = 1 for all x in R. Any idea?

The vertices are published authors -- published in print or arXive, or authors of mathematical books. The ordered edges are triples $$ (r\,\,p\,\,a) $$ where $\,p\,$ is a paper or book authored or co-authored by $\,a\,$, and $\,r\,$ is an author referred to in paper/book $\,p\,$ and such that $...

Consider the moduli space $M_g$ of compact Riemann surfaces (i.e., smooth complete algebraic curves over $\mathbb{C}$) of genus $g$ for some $g>1$. I'm interested in knowing how Riemann proved that $M_g$ has dimension $3g-3$. A modern proof involves deformation theory and Riemann-Roch theorem. ...

A book embedding of a graph $G$ consists of placing the vertices of $G$ on a spine and assigning edges of the graph to pages so that edges in the same page do not cross each other. The book thickness $bt(G)$ is the minimum number of pages in which the graph $G$ can be embedded. I wondered whethe...