All Questions

Note: I have edited the post below in order to include sharper (conjectured) inequalities, using $|H_1 \cap H_2|$. Let $[n] = \{1, \dots, n\}$ and let $\sim$ be an equivalence relation on $[n]$. Then $...

Let $H$ denote an $n$ by $n$ hermitian positive semidefinite matrix. Let $G$ and $K$ be two subgroups of the symmetric group $\Sigma_n$. Define $$ f_{G, K}(H) = \sum_{(\sigma, \tau) \in G \times K} \...

A somewhat strange elementary polynomial inequality came up recently in my work, and I wonder if anyone has seen other things that are reminiscent of what follows. Given $n$ non-negative reals $a_1, ...

Let $A$ be a non-negative (all entries $\geq 0$) square matrix. Is it always true that $$ (a_{11}+a_{12}+a_{21}+a_{22})^2\geq 4a_0a_2 $$ where $a_{ij}$ is the permanent of a matrix obtained by ...

Consider the $n$-dimensional Hamming cube, $C = \{-1,1\}^n$. Given an $n \times n$ orthogonal matrix $O$, I'll measure "how close $O$ is to being an isometry of $C$" by the following scoring function:...