The Perron-Frobenius Theorem for irreducible higher order tensors does not guarantee simplicity of the unique positive eigenvalue nor does it guarantee the unique positive eigenvalue is strictly larger than the modulus of any other eigenvalue. Irreducibility of a tensor in relation to a graph is studied. A higher order tensor is essentially positive if it takes the positive cone to its interior...

The celebrated Perron–Frobenius (PF) theorem is stated for irreducible nonnegative square matrices, and provides a simple characterization of their eigenvectors and eigenvalues. The importance of this theorem stems from the fact that eigenvalue problems on such matrices arise in many fields of science and engineering, including dynamical systems theory, economics, statistics and optimization. H...

In this third talk we will discuss properties related to edge expansion. In particular, we will define the Cheeger constant (which measures how easy it is to cut off a large piece of the graph) and state the Cheeger inequalities. We also will define and discuss discrepancy for undirected and directed graphs. We also state the Perron-Frobenius Theorem which is a useful tool in spectral graph the...

We study Ruelle–Perron–Frobenius operators for locally expanding and mixing dynamical systems on general compact metric spaces associated with potentials satisfying the Dini condition. In this paper, we give a proof of the Ruelle Theorem on Gibbs measures. It is the first part of our research on the subject. The rate of convergence of powers of the operator will be presented in a forthcoming pa...

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your perso...

We propose a notion of Frobenius-Perron dimension for certain free Z \mathbb {Z} </mml:semant...

By the use of Perron-Frobenius theory, simple proofs are given of the Fundamental Theorem of Demography and of a theorem of Cushing and Yicang on the net reproductive rate occurring in matrix models of population dynamics. The latter result is further refined with some additional nonnegative matrix theory. When the fertility matrix is scaled by the net reproductive rate, the growth rate of the ...

In this paper we demonstrate the application of the Frobenius-Perron operator to the statistical analysis of chaotic systems. The Frobenius-Perron operator is the main analysis tool whenever statistical properties of the generated and processed signals are of interest. One important eld of application are systems, where information is located in a particular statistic of the chaotic signal. For...