12 problems

Qi's conjecture. The bound

Conjectured bounds.

Let $f(m,n)$ denote the determinant of the $m$th-order, $n$-dimensional Pascal tensor, for $n\geq 3$. Pascal-tensor determinant conjecture. The quantity $f(m,n)$ is the $m$th power…

Universal state-inversion positivity conjecture. For every $C\in L(\mathscr{H})$ and every $v\in\\{0,1\\}^n$,

Non-negativity and self-adjointness conjecture. The operator $D^{(m)}$ is non-negative and self-adjoint for every $m$. The conjecture concerns the operator arising in the analysis…

Let $f$ be a tangential symmetric tensor field, let $j$ denote contraction with the Kronecker tensor, and let $h$ be the symmetrized second derivative of $f$ as in equation (4.12).…

Small spectral norm conjecture. One may conjecture that $A$ has small spectral norm as an operator on $\ell^{r_1}\times\dotsb\times\ell^{r_k}$ for some choice of $r_i$ satisfying

Let $T$ be a symmetric tensor of arbitrary order, let $T_{i_1,\dots,i_k}$ denote one of its entries, and let $\lambda_{maxabs}$ denote the largest absolute value of an eigenvalue o…

M-PSD to S-PSD conjecture. Every M-PSD tensor can be modified into an S-PSD tensor while preserving the summations of the corresponding entries.

Let $v$ be a compactly supported vector field on $\mathbb{R}^3$, let $I$ denote the relevant line-integral transform, and let $P$ denote the corresponding tensor transform. Euler r…

Let $v$ be a compactly supported vector field on $\mathbb{R}^3$. Write $\operatorname{div}(v\otimes v)$ for the divergence of its quadratic tensor field, $\operatorname{curl}$ for…

Let $T(x)$ be a symmetric tensor of arbitrary rank on an $n$-dimensional real Euclidean space, with vanishing divergence on every index. For a word $w$, let $[w]$ denote its length…