4 problems

Huang–Jebara Bethe permanent kernel conjecture. The function $\kappa_{\operatorname{perm}_{\mathrm{B}}}$ is a positive definite kernel on $\mathcal{X}^n\times\mathcal{X}^n$. The cl…

Let $\mathcal{C}$ be a binary linear code with parity-check matrix ${\bm{H}}\in{\mathbb{F}}_2^{m\times n}$, where $m<n$. Let $\mathcal{I}({\bm{H}})$ be its column-index set, let…

Let $\boldsymbol{\theta}$ be a matrix of indeterminates, let $M\in\mathbb{Z}_{>0}$, and let $\tilde{\Psi}_M$ be the set of degree-$M$ graph-covering permutations. For…

Let $\boldsymbol{\theta}$ be a non-negative $n\times n$ matrix. For $M\in\mathbb{Z}_{>0}$, let $\tilde{\Psi}_M$ be the set of degree-$M$ graph-covering permutations and let…