The Berger-Wang formula for the Markovian joint spectral radius

The Berger-Wang formula establishes equality between the joint and generalized spectral radii of a set of matrices. For matrix products whose multipliers are applied not arbitrarily but in accordance with some Markovian law, there are also known analogs of the joint and generalized spectral radii. However, the known proofs of the Berger-Wang formula hardly can be directly applied in the case of Markovian products of matrices since they essentially rely on the arbitrariness of appearance of different matrices in the related matrix products. Nevertheless, as has been shown by X. Dai the Berger-Wang formula is valid for the case of Markovian analogs of the joint and the generalized spectral radii too, although the proof in this case heavily exploits the more involved techniques of multiplicative ergodic theory. In the paper we propose a matrix theory construction allowing to deduce the Markovian analog of the Berger-Wang formula from the classical Berger-Wang formula.Comment: 13 pages, 29 bibliography references; minor corrections; accepted for publication in Linear Algebra and its Application

An explicit Lipschitz constant for the joint spectral radius

In 2002 F. Wirth has proved that the joint spectral radius of irreducible compact sets of matrices is locally Lipschitz continuous as a function of the matrix set. In the paper, an explicit formula for the related Lipschitz constant is obtained.Comment: 9 pages, corrected typos and reference

On accuracy of approximation of the spectral radius by the Gelfand formula

The famous Gelfand formula $\rho(A)= \limsup_{n\to\infty}\|A^{n}\|^{1/n}$ for the spectral radius of a matrix is of great importance in various mathematical constructions. Unfortunately, the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities $\|A^{n}\|^{1/n}$ to $\rho(A)$. In the paper this deficiency is made up to some extent. By using the Bochi inequalities we establish explicit computable estimates for the rate of convergence of the quantities $\|A^{n}\|^{1/n}$ to $\rho(A)$. The obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets.Comment: Corrected typos, improved estimate for the kay constant in the paper, added reference

Polynomial reformulation of the Kuo criteria for v-sufficiency of map-germs

In the paper a set of necessary and sufficient conditions for \textit{v-}sufficiency (equiv. \textit{sv-}sufficiency) of jets of map-germs $f:(\mathbb{R}^{n},0)\to (\mathbb{R}^{m},0)$ is proved which generalize both the Kuiper-Kuo and the Thom conditions in the function case ($m=1$) so as the Kuo conditions in the general map case ($m>1$). Contrary to the Kuo conditions the conditions proved in the paper do not require to verify any inequalities in a so-called horn-neighborhood of the (a'priori unknown) set $f^{-1}(0)$. Instead, the proposed conditions reduce the problem on \textit{v-}sufficiency of jets to evaluating the local {\L}ojasiewicz exponents for some constructively built polynomial functions.Comment: 16 pages, 25 bibliography references, corrected misprints Accepted for publication in DCDS-B, added acknowledgments to the supporting gran

Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets

The problem of construction of Barabanov norms for analysis of properties of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. The method of Barabanov norms was the key instrument in disproving the Lagarias-Wang Finiteness Conjecture. The related constructions were essentially based on the study of the geometrical properties of the unit balls of some specific Barabanov norms. In this context the situation when one fails to find among current publications any detailed analysis of the geometrical properties of the unit balls of Barabanov norms looks a bit paradoxical. Partially this is explained by the fact that Barabanov norms are defined nonconstructively, by an implicit procedure. So, even in simplest cases it is very difficult to visualize the shape of their unit balls. The present work may be treated as the first step to make up this deficiency. In the paper two iteration procedure are considered that allow to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.Comment: 17 pages, 36 bibliography references, 3 figures; shortened version, new LaTeX style, fixed typos, accepted in DCDS-