All Questions

Assume that we have a bi-Lipschitz mapping $f:\bar{\mathbb{B}}^n(0,1)\to\mathbb{R}^n$. The mapping need not be smooth anywhere and it may happen that it cannot be extended to a homeomorphism of a ...

As undergraduates, we learn that conditional convergence of infinite series is highly sensitive to the order structure on $\mathbb N$: if $\sum_{n=0}^\infty x_n$ conditionally converges and $x \in \...

Let $\Omega$ be an open subset of $S^2$ with $\overline{\Omega} \neq S^2$. Suppose a continuous tangent vector field $G$ is defined on $\partial \Omega$ such that $|G(y)| = 1$ for all $y \in \partial \...

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 $...

$\DeclareMathOperator\Diff{Diff}$Setting: Let $(M,g)$ be a compact and connected $C^{\infty}$-Riemannian manifold. Let $d_g$ denote the induced shorted path metric and equip $C^{\infty}(M)$ with the ...