Let β > 1 be an algebraic number. A general definition of a beta-conjugate of β is proposed with respect to the analytical function fβ(z) = −1 + ∑ i≥1 tiz i associated with the Rényi β-expansion dβ(1) = 0.t1t2 . . . of unity. From Szegö’s Theorem, we study the dichotomy problem for fβ(z), in particular for β a Perron number: whether it is a rational fraction or admits the unit circle as natural...

In this paper, we study the perturbation bound for the Perron vector of an mth-order n-dimensional transition probability tensor P = (pi1,i2,...,im) with pi1,i2,...,im ≥ 0 and ∑ n i1=1 pi1 ,i2,...,im = 1. The Perron vector x associated to the largest Z-eigenvalue 1 of P , satisfies Pxm−1 = x where the entries xi of x are nonnegative and ∑i=1 xi = 1. The main contribution of this paper is to sho...

A matrix A can be tested to determine whether it is eventually positive by ex1 amination of its Perron-Frobenius structure, i.e., by computing its eigenvalues and left and right 2 eigenvectors for the spectral radius ρ(A). No such “if and only if” test using Perron-Frobenius prop3 erties exists for eventually nonnegative matrices. The concept of a strongly eventually nonnegative 4 matrix was wa...

In 1907, Oskar Perron showed that a positive square matrix has a unique largest positive eigenvalue with a positive eigenvector. This result was extended to irreducible nonnegative matrices by Geog Frobenius in 1912, and to irreducible nonnegative tensors and weakly irreducible nonnegative tensors recently. This result is a fundamental result in matrix theory and has found wide applications in ...

We present some advances, both from a theoretical and from a computational point of view, on a quadratic vector equation (QVE) arising in Markovian Binary Trees. Concerning the theoretical advances, some irreducibility assumptions are relaxed, and the minimality of the solution of the QVE is expressed in terms of properties of the Jacobian of a suitable function. From the computational point of...

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.

where {Fn, n ∈ N}, is the natural filtration associated to (Xn). We assume that X0 ∈ R d + and that random vectors ξn are such that for all n, Xn ∈ R d + almost surely. The Perron-Frobenius Theorem [10, pp. 3-4] states that M has a positive Perron root ρ. We call Xn “subcritical” if ρ < 1, “supercritical” if ρ > 1 and “critical” if ρ = 1. In the “subcritical” case, one has P(‖Xn‖ → ∞) = 0 becau...