We give a new proof of the well known Wedderburn's little theorem (1905) that a finite‎ ‎division ring is commutative‎. ‎We apply the concept of Frobenius kernel in Frobenius representation theorem in finite group‎ ‎theory to build a proof‎.

We discuss the characterization of chaotic behaviours in random maps both in terms of the Lyapunov exponent and of the spectral properties of the Perron-Frobenius operator. In particular, we study a logistic map where the control parameter is extracted at random at each time step by considering nite dimensional approximation of the Perron-Frobenius operator.

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, which is closely related to the Stein-Rosenberg theorem in numerical linear algebra, is further refined with some additional nonnegative matrix theory. Whe...

Frobenius published two proofs of a theorem which characterizes irreducible and fully indecomposable matrices in an algebraic manner. It is shown that the second proof, which depends on the Frobenius-Konig theorem, yields a stronger form of the result than the first. Some curious features in Frobenius’s last paper are examined; these include his criticisms of a result due to D. K&rig and the la...

Markov chain Monte Carlo, or MCMC, is a way to sample probability distributions that cannot be sampled practically using direct samplers. This includes a majority of probability distributions of practical interest. MCMC runs a Markov chain X1, X2, . . ., where Xk+1 is computed from Xk and some other i.i.d. random input. From a coding point of view, a direct solver is X = fSamp();, while the MCM...

Recently, Storm [10] defined the Ihara-Selberg zeta function of a hypergraph, and gave two determinant expressions of it by the Perron-Frobenius operator of a digraph and a deformation of the usual Laplacian of a graph. We present a new determinant expression for the Ihara-Selberg zeta function of a hypergraph, and give a linear algebraic proof of Storm’s Theorem. Furthermore, we generalize the...

We compute some dependence coefficients for the stationary Markov chain whose transition kernel is the Perron-Frobenius operator of an expanding map T of [0, 1] with a neutral fixed point. We use these coefficients to prove a central limit theorem for the partial sums of f ◦ T , when f belongs to a large class of unbounded functions from [0, 1] to R. We also prove other limit theorems and momen...

We start by stating a version of the Perron–Frobenius theorem. Let A be a d × d stochastic matrix, where here we use this to mean that the entries of A are non-negative, and every column sums to 1: Aij ∈ [0, 1] for all i, j, and ∑d i=1Aij = 1 for all j. Thus the columns of A are probability vectors. Such a matrix A describes a weighted random walk on d sites: if the walker is presently at site ...

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...

We extend the notions of irreducibility and periodicity of a stochastic matrix to a unital positive linear map © on a finite-dimensional C∗algebra A and discuss the non-commutative version of the Perron-Frobenius theorem. For a completely positive linear map © with ©(a) = ∑ l Ll ∗aLl, we give conditions on the Ll’s equivalent to irreducibility or periodicity of ©. As an example, positive linear...