The joint spectral radius of a finite set of real d× d matrices is defined to be the maximum possible exponential rate of growth of products of matrices drawn from that set. In previous work with K. G. Hare and J. Theys we showed that for a certain one-parameter family of pairs of matrices, this maximum possible rate of growth is attained along Sturmian sequences with a certain characteristic r...

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. In [16, 17] 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 th...

We study the smoothness of quasi-uniform bivariate subdivision. A quasi-uniform bivariate scheme consists of different uniform rules on each side of the y-axis, far enough from the axis, some different rules near the y-axis, and is uniform in the y-direction. For schemes that generate polynomials up to degree m, we derive a sufficient condition for C continuity of the limit function, which is s...

Consider a square random matrix with independent and identically distributed entries of mean zero unit variance. We show that as the dimension tends to infinity, spectral radius is equivalent root in probability. This result can also be seen convergence support circular law theorem under optimal moment conditions. In proof we establish reciprocal characteristic polynomial analytic function outs...

‎let $n$ and $k$ be two positive integers‎, ‎$kleq n$ and $a$ be an $n-$square quaternion matrix‎. ‎in this paper‎, ‎some results on the $k-$numerical range of $a$ are investigated‎. ‎moreover‎, ‎the notions of $k$-numerical radius‎, ‎right $k$-spectral radius and $k$-norm of $a$ are introduced‎, ‎and some of their algebraic properties are studied‎.

Overlap-free words are words over the binary alphabet A = {a, b} that do not contain factors of the form xvxvx, where x ∈ A and v ∈ A. We analyze the asymptotic growth of the number un of overlap-freewords of length n as n→∞. We obtain explicit formulas for theminimal andmaximal rates of growth of un in terms of spectral characteristics (the joint spectral subradius and the joint spectral radiu...

X. Zhan has conjectured that the spectral radius of the Hadamard product of two square nonnegative matrices is not greater than the spectral radius of their ordinary product. We prove Zhan’s conjecture, and a related inequality for positive semidefinite matrices, using standard facts about principal submatrices, Kronecker products, and the spectral radius.

We obtain a formula for the essential spectral radius ρess of transfer-type operators associated with families of C1+δ diffeomorphisms of the line and Zygmund, or Hölder, weights acting on Banach spaces of Zygmund (respectively Hölder) functions. In the uniformly contracting case the essential spectral radius is strictly smaller than the spectral radius when the weights are positive.

In this paper, we first give a relation between the adjacency spectral radius and the Q-spectral radius of a graph. Then using this result, we further give some new sharp upper bounds on the adjacency spectral radius of a graph in terms of degrees and the average 2-degrees of vertices. Some known results are also obtained.