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For any positive integers $r$, $s$, $m$, $n$, an $(r,s)$-order $(n,m)$-dimensional rectangular tensor ${cal A}=(a_{i_1cdots i_r}^{j_1cdots j_s}) in ({mathbb R}^n)^rtimes ({mathbb R}^m)^s$ is called partially symmetric if it is invariant under any permutation on the lower $r$ indexes and any permutation on the upper $s$ indexes. Such partially symmetric rectangular tensor arises naturally in studying directed hypergraphs. Ling and Qi [Front. Math. China, 2013] first studied the $(p,q)$-spectral radius (or singular values) and proved a Perron-Fronbenius theorem for such tensors when both $p,q geq r+s$. We improved their results by extending to all $(p,q)$ satisfying $frac{r}{p} +frac{s}{q}leq 1$. We also proved the Perron-Fronbenius theorem for general nonnegative $(r,s)$-order $(n,m)$-dimensional rectangular tensors when $frac{r}{p}+frac{s}{q}>1$. We essentially showed that this is best possible without additional conditions on $cal A$. Finally, we applied these results to study the $(p,q)$-spectral radius of $(r,s)$-uniform directed hypergraphs.
In this paper, we generalize some conclusions from the nonnegative irreducible tensor to the nonnegative weakly irreducible tensor and give more properties of eigenvalue problems.
The Frobenius-Perron theory of an endofunctor of a $Bbbk$-linear category (recently introduced in cite{CG}) provides new invariants for abelian and triangulated categories. Here we study Frobenius-Perron type invariants for derived categories of comm
A uniform hypergraph $mathcal{H}$ is corresponding to an adjacency tensor $mathcal{A}_mathcal{H}$. We define an Estrada index of $mathcal{H}$ by using all the eigenvalues $lambda_1,dots,lambda_k$ of $mathcal{A}_mathcal{H}$ as $sum_{i=1}^k e^{lambda_i
The spectral properties of the Frobenius-Perron operator of one-dimensional maps are studied when approaching a weakly intermittent situation. Numerical investigation of a particular family of maps shows that the spectrum becomes extremely dense and
In arXiv:1709.07504 Ardila and Aguiar give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on hypergraphs. We g