Convergence of dynamics and the Perron–Frobenius operator

We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron–Frobenius operator. Our main result states that strong convergence of the powers of the Perron–Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by uniform mixing-like properties of the system.     

Stochastic nonlinear Perron–Frobenius theorem

We establish a stochastic nonlinear analogue of the Perron–Frobenius theorem on eigenvalues and eigenvectors of positive matrices. The result is formulated in terms of an automorphism T of a probability space and a random transformation D of the non-negative cone of an n-dimensional Euclidean space. Under assumptions of monotonicity and homogeneity of D, we prove the existence of scalar and vector measurable functions α > 0 and x > 0 satisfying the equation αTx = D(x) almost surely. We apply the result obtained to the analysis of a class of random dynamical systems arising in mathematical economics and finance (von Neumann–Gale dynamical systems).     

Extensions of Perron–Frobenius theory

The classical Perron–Frobenius theory asserts that, for two matrices $A$ and $B$ , if $0\le B \le A$ and $r(A)=r(B)$ with $A$ being irreducible, then $A=B$ . It has been extended to infinite-dimensional Banach lattices under certain additional conditions, including that $r(A)$ is a pole of the resolvent of $A$ . In this paper, we prove that the same result holds if $B$ is irreducible and $r(B)$ is a pole of the resolvent for $B$ . We also prove some other interesting extensions of the theorem for infinite-dimensional Banach lattices.     

A Perron–Frobenius type result for integer maps and applications

Positivity - It is shown that for certain maps, including concave maps, on the d-dimensional lattice of positive integer points, ‘approximate’ eigenvectors can be found. Applications in...     

Vecteurs de Perron–Frobenius et produits Gamma

This Note presents formulas to express the coordinates of Perron–Frobenius vectors of Cartan matrices (finite or affine) as products of Gamma values, for each finite irreducible root system of rank r.     

Vertex maps on graphs – Perron–Frobenius theory

The goal of this paper is to describe the connections between Perron–Frobenius theory and vertex maps on graphs. In particular, it is shown how Perron–Frobenius theory gives results about the sets of integers that can arise as periods of periodic orbits, about the concepts of transitivity and topological mixing and about horseshoes and topological entropy.     

On Multivalent Functions with Negative and Missing Coefficients

Let P_k(p, A, B) be the class of functions f(z) = z~p-sum from n=k to ∞(|α_(n+′p|Z~((n+)~-)p) k≥2 analytic in the unit disc E={z:|z|<1} and satisfying the condition |(zf′(z)/f(z)-p)/(Ap-Bzf (z)/f(z))|<1. for z∈E and -1≤B相似文献   

The Perron–Frobenius theorem – a proof with the use of Markov chains

The Perron–Frobenius theorem for an irreducible nonnegative matrix is proved using the matrix graph and the ergodic theorem of the theory of Markov chains. Bibliography: 7 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 5–16.     

Perron–Frobenius theorem for hypermatrices in the max algebra

Recently, there have been many intriguing new developments in the study of hypermatrices and their associated eigenvalue problems. In particular, results coming from the matrix setting when studying the max algebra have shown especially attractive combinatorial features. We now extend this max algebra setting into the realm of hypermatrices. Considering that the max algebra has shown particular significance in optimization problems for the matrix setting, we look to examine and extend these results in the higher order conditions. Furthermore, we establish some algebraic properties for hypermatrices and then proceed to extend the Perron–Frobenius Theorem for this setting and prove the existence of a unique eigenvalue. We continue by stating a result from Nussbaum, that the Min–Max theorem holds, and provide a proof for completeness. For strongly increasing hypermatrices, an iterative algorithm which converges to our unique eigenvalue is given. Finally, we conclude with an analysis of our results in the hypergraph setting.     

Non-existence of Some Nearly Perfect Sequences,Near Butson–Hadamard Matrices,and Near Conference Matrices

In this paper we study the non-existence problem of (nearly) perfect (almost) m-ary sequences via their connection to (near) Butson–Hadamard (BH) matrices and (near) conference matrices. We refine the idea of Brock on the unsolvability of certain equations in the case of cyclotomic number fields whose ring of integers is not a principal ideal domain and get many new non-existence results for near BH matrices and near conference matrices. We also apply previous results on vanishing sums of roots of unity and self conjugacy condition to derive non-existence results for near BH matrices and near conference matrices.     

On Graphs with Zero Determinant of Adjacency Matrices

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On the Matrices of Clebsch–Gordan Coefficients

We consider the matrices of Clebsch–Gordan coefficients. It turns out that these matrices are convenient in order to state, prove, and use many facts of the theory of representations of the groups SO(3) and SU(2).     

On Some Properties of L Type Intuitionistic Fuzzy Family

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Some Properties of Gromov–Hausdorff Distances

The Gromov?CHausdorff distance between metric spaces appears to be a useful tool for modeling some object matching procedures. Since its conception it has been mainly used by pure mathematicians who are interested in the topology generated by this distance, and quantitative consequences of the definition are not very common. As a result, only few lower bounds for the distance are known, and the stability of many metric invariants is not understood. This paper aims at clarifying some of these points by proving several results dealing with explicit lower bounds for the Gromov?CHausdorff distance which involve different standard metric invariants. We also study a modified version of the Gromov?CHausdorff distance which is motivated by practical applications and both prove a structural theorem for it and study its topological equivalence to the usual notion. This structural theorem provides a decomposition of the modified Gromov?CHausdorff distance as the supremum over a family of pseudo-metrics, each of which involves the comparison of certain discrete analogues of curvature. This modified version relates the standard Gromov?CHausdorff distance to the work of Boutin and Kemper, and Olver.