MathOverflow

I'm interested in solutions of the Liouville equation $$-\Delta u = e^u$$ in $\mathbb{R}^N$.
In this article, Farina proves that for $2\leq N \leq 9$ there is no stable solution. He also remarks that this result is sharp since for $N\geq 10$ there are radial stable solutions. Furthermore, he also shows that for $N=2$ there exist stable solutions...

,If in a expression, by changing the sign +,- in front of zero, we get a different result. For example
$$\cot^{-1}\left(\frac{1}{{-0}}\right)={\pi}$$ and$$\cot^{-1}\left(\frac{1}{{0}}\right)={0}$$if you know an example similar to mine, show it
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A filter $F$ on a collection $P$ of subsets of $X$ is fine when for all $x \in X$, $\{ z \in P : x \in z\}$ is in $F$.
Suppose $U$ is a fine ultrafilter on the finite subsets of natural numbers. Does there exist $A \in U$ that is linearly ordered by subset?...

Let $V$ be a complete (integral) variety over a field $k$. I am happy to assume that $k$ is algebraically closed.
Is $V$ always embeddable into some toric variety $T$ (edit: as a closed subscheme)? If not, what are sufficient/necessary conditions for this?...