MathOverflow

Let $(\Omega, \mathcal{A}, P)$ be a probability space and let $Y:(\Omega, \mathcal{A})\rightarrow (\mathcal{Y}, \mathcal{G})$ and $X:(\Omega, \mathcal{A})\rightarrow (\mathcal{X}, \mathcal{F})$ be random variables.
Let $\mu$ and $\nu$ be $\sigma$-finite measures on $\mathcal{X}$ and $\mathcal{Y}$, respectivley with $P_{X, Y}\ll\mu\otimes\nu$. Denote by $f_{X, Y}$ the Radon-Niodym derivative. Assume that $P_{Y|X=x}$ and $P_{X|Y=y}$ exist. Call $f_{Y|X}$ the Radon...

In https://arxiv.org/pdf/1009.1965 they prove the following bound for the heat kernel (Theorem 1.2): for positive constants $C_1, C_2$
\begin{equation*} | \nabla_{x} p_{t}(x, y) | \leq C_1 \frac{1}{t^{\frac{d+1}{2}}} {\mathrm e}^{-|x-y|^{2}/(C_2 t)},\quad x, y \in \Omega, \quad t > 0; \end{equation*}...

Let us assume that there is a tree with root 0, and an infinite countable number of nodes at the first level (Say the set of all odd numbers: 1, 3, 5...)Each of these nodes spawns a single path of natural numbers starting from the node at the first level...

Let $P$ be a polytope and $E$ be the set of its extremal points.
Question. Is it true that we always can find a vertex $v\in E$ and a facet $F$ of the polytope $\operatorname{conv}(E\setminus\{v\})$ such that for every vertex $u\in E\setminus\{v\}$ the segment $[u,v]$ meets the facet $F$?...