MathOverflow

I understand this is not a research level question but I really want to know, would anyone please help.
This question is related to the signatures that arises in rough path theory.
Is there any reference to understand if the norms of the different levels of signatures are convex function or not? is it possible to derive under what condition the norm ($L^1$ or $L^2$) of the first or second level of signature would be convex? As...

Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic function $\phi: X\to\mathbb{P}^1\mathbb{C} $ ramified at most at $0,1,$ and $\infty$;...

What are the rings $R$ such that the natural map$$\operatorname{Hom}_R(M,N) \to \operatorname{Hom}_{\mathbb{Z}}(M,N)$$is surjective for any $R$-modules $M$ and $N$? In other words, when is the forgetful functor from the category of $R$-modules to the...

I asked this question on mathstackexchange, where more details and attempts are given. Given an $n \times n$ grid of white cells, what is the minimum number of black cells you need to shade such that there exists only one closed, non-intersecting loop...