We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly useful in generalizing certain areas where the spectral theory of matrices has traditionally played an important role. For illustration, we will discuss a m...

In this paper a new quantity for real tensors, the sign-real spectral radius, is defined and investigated. Various characterizations, bounds and some properties are derived. In certain aspects our quantity shows similar behavior to the spectral radius of a nonnegative tensor. In fact, we generalize the Perron Frobenius theorem for nonnegative tensors to the class of real tensors.

In this paper, we show a new relation between phase transition in Statistical Mechanics and the dimension of space harmonic functions (SHF) for transfer operator. This is accomplished by extending classical Ruelle-Perron-Frobenius theory to realm low regular potentials defined on either finite or infinite (uncountable) alphabets. We also give an example potential having where Perron-Frobenius e...

We consider subshifts of nite type on the symbol space generated by incidence matrices over a countably innnite alphabet. We extend the deenition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of HH older continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay ...

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

Exploring long-term implications of valuation leads us to recover and use a distorted probability measure that reflects the long-term implications for risk pricing. This measure is typically distinct from the physical and the risk neutral measures that are well known in mathematical finance. We apply a generalized version of Perron-Frobenius theory to construct this probability measure and pres...

Our proof is built on Perron-Frobenius theorem, a seminal work in matrix theory (Meyer 2000). By Perron-Frobenius theorem, the power iteration algorithm for predicting top K persuaders converges to a unique C and this convergence is independent of the initialization of C if the persuasion probability matrix P is nonnegative, irreducible, and aperiodic (Heath 2002). We first show that P is nonne...

To study the convergence to equilibrium in random maps, we develop the spectral theory of the corresponding transfer (Perron— Frobenius) operators acting in a certain Banach space of generalized functions (distributions). The random maps under study in a sense fill the gap between expanding and hyperbolic systems, since among their (deterministic) components there are both expanding and contrac...

We give a survey on some recent developments in the spectral theory of transfer operators, also called Ruelle-Perron-Frobenius (RPF) operators, associated to expanding and mixing dynamical systems. Different methods for spectral study are presented. Topics include maximal eigenvalue of RPF operators, smooth invariant measures, ergodic theory for chain of markovian projections, equilibrium state...