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Register a new userFrobenius-Perron theory for projective schemes
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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 commutative and noncommutative projective schemes. In particular, we calculate the Frobenius-Perron dimension for domestic and tubular weighted projective lines, define Frobenius-Perron generalizations of Calabi-Yau and Kodaira dimensions, and provide examples. We apply this theory to the derived categories associated to certain Artin-Schelter regular and finite-dimensional algebras.
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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 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 the eigenfunctions become concentrated in the vicinity of the intermittent fixed point. Analytical considerations generalize the results to a broader class of maps near and at weak intermittency and show that one branch of the map is dominant in determination of the spectrum. Explicit approximate expressions are derived for both the eigenvalues and the eigenfunctions and are compared with the numerical results.
In this paper we provide nearly linear time algorithms for several problems closely associated with the classic Perron-Frobenius theorem, including computing Perron vectors, i.e. entrywise non-negative eigenvectors of non-negative matrices, and solving linear systems in asymmetric M-matrices, a generalization of Laplacian systems. The running times of our algorithms depend nearly linearly on the input size and polylogarithmically on the desired accuracy and problem condition number. Leveraging these results we also provide improved running times for a broader range of problems including computing random walk-based graph kernels, computing Katz centrality, and more. The running times of our algorithms improve upon previously known results which either depended polynomially on the condition number of the problem, required quadratic time, or only applied to special cases. We obtain these results by providing new iterative methods for reducing these problems to solving linear systems in Row-Column Diagonally Dominant (RCDD) matrices. Our methods are related to the classic shift-and-invert preconditioning technique for eigenvector computation and constitute the first alternative to the result in Cohen et al. (2016) for reducing stationary distribution computation and solving directed Laplacian systems to solving RCDD systems.
We prove that a quasi-bialgebra admits a preantipode if and only if the associated free quasi-Hopf bimodule functor is Frobenius, if and only if the relative (opmonoidal) monad is a Hopf monad. The same results hold in particular for a bialgebra, tightening the connection between Hopf and Frobenius properties.
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