For a complex manifold $M$ with differential $d$ and Dolbeault operators $\partial,\bar\partial$, it is clear that $$d\alpha = \partial\alpha + \bar\partial\alpha$$ implies $\partial\alpha=\bar\partial\alpha=0$ for any $\alpha\in A^{p,q}(M)$, since $\partial\alpha\in A^{p+1,q}(M)$ and $\bar\parti...

This question originates from my attempt to study certain closure operators that arise from symmetric, anti-reflexive relations. To be precise, consider a relation $\perp$ on a set $X$ that is symmetric (i.e., $x \perp y$ iff $y \perp x$ for all $x, y \in X$) and anti-reflexive (i.e., $x \not\per...

Prove that if $K$ is a convex cone , then its closure $\overline{K}$ is also a convex cone. Attempt: I start by noting that $K$ being a convex cone means that for any $x, y \in K$ and any non-negative scalars $\alpha, \beta \geq 0$, the combination $\alpha x + \beta y \in K$. The closure $\overli...

I would like to know a reference in the math literature where the Lemma below is proven or at least mentioned, in its full generality and abstract form. I am not asking for a proof (it is easy). Does it perhaps even have a common name? Lemma. Let $f,g \in S_X$ be two permutations such that there...

Suppose that $b\in L^1[0,1]$, and that $I$ is a subinterval of $[0,1]$ centered at $s_0$. we denote by $\widetilde{I}$ the double of $I$, namely, the interval centered at $s_0$ with length $2|I|$. I want to show that there exists universal constant $C>0$, such that \begin{align} |I| \cdot\int_{[0...