MathOverflow

Suppose $X$ is a DM stack equipped with a perfect obstruction theory, denote $\mathcal{O}^{vir}_{X}$ be the virtual structure sheaf defined by the given perfect obstruction theory. I'm wondering when $\mathcal{O}^{vir}_{X}$ will be a perfect complex...

The lemniscates are a continuum.
$$\varpi_s = \frac{2}{s}\cdot\frac{\Gamma(1/s),\sqrt{\pi}}{\Gamma(1/s + 1/2)}$$
The even cases: The Lemniscate Constants$$\pi (\varpi_2), \varpi (\varpi_4), \varpi_6, \varpi_8, …$$

Let $A,B$ be Hopf *-algebras. A dual pairing of $A$ and $B$ is a bilinear form $\langle\cdot,\cdot\rangle$ on $A\times B$, such that (among other things)$$\langle a^*,b \rangle = \overline{\langle a,S(b)^* \rangle},$$where $S$ is the antipode of $B$...

I am looking for a reference for the following separation principle.
Let $(C_n)_{n \in \mathbb{N}}$ and $(D_n)_{n \in \mathbb{N}}$ besequences of closed and non-empty sets in the unit interval satisfying $C_n \cap D_n=\emptyset$ for all $n\in \mathbb{N}$. Then there is $g\in \mathbb{N}^\mathbb{N}$ with $d(C_n, D_n)&#...