7 problems

For a tensor dimension $n_1\times n_2\times\dots\times n_d$, let $\phi(\mathbb{R}^{n_1\times n_2\times\dots\times n_d}_+)$ be the infimum of the spectral-to-Frobenius norm ratio ov…

Let $2\le n=\prod_{k=1}^{s_n}p_k$ be the prime factorization, with $2\le p_1\le p_2\le\dots\le p_{s_n}$. Consider a tensor…

Let $\boldsymbol{\beta}$ denote the lower bound in equation (main) for the ratio of the spectral norm to the Frobenius norm. An unfolded permutation tensor is a tensor obtained by…

Let $G$ be a connected hypergraph, let $G^{m,s}$ denote its generalized power hypergraph with parameters $m$ and $s$, and let $c(G)$ denote the cyclic index of $G$. Spectral index…

Irreducibility characterization conjecture. A nonnegative tensor $\mathbb{A}$ is primitive if and only if $\mathbb{A}$ is irreducible and there exists some $j\in[n]$ such that…

Let $\mathbb{A}$ be a nonnegative primitive tensor of fixed order $m$ and dimension $n$, and let $\gamma(\mathbb{A})$ denote its primitive degree. Shao's conjecture. When $m$ is fi…

Let $\mathbb {A}$ be a nonnegative primitive tensor of fixed order $m$ and dimension $n$, and let $\gamma (\mathbb {A})$ denote its primitive degree. Primitive-degree polynomial bo…