All Questions

Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map whose rank at any point of $\R^m$ is at most $d$. Here and below, smooth means infinitely differentiable. Can we ...

It has been known that given two probability distributions $\mu_1$ and $\mu_2$ (let us say, they are smooth for simplicity), there is a hyperplane that divides the domain into two regions (denoted as $...

I could not answer or find references of this question, even for the following special case: On $S^2$ (the two-sphere equiped with the standard Riemannian metric), is every positive smooth function ...