arXiv subjects

Galen Novello

Publications and source records attributed to Galen Novello.

3 recordsLinked to original sources

arXiv2025

Weighted Chebyshev Polynomials on Compact Subsets of the Complex Plane

We study weighted Chebyshev polynomials on compact subsets of the complex plane with respect to a bounded weight function. We establish existence and uniqueness of weighted Chebyshev polynomials and derive weighted analogs of Kolmogorov's criterion, the alternation theorem, and a characterization due to Rivlin and Shapiro. We derive invariance of the Widom factors of weighted Chebyshev polynomials under polynomial pre-images and a comparison result for the norms of Chebyshev polynomials corresponding to different weights. Finally, we obtain a lower bound for the Widom factors in terms of the Szegő integral of the weight function and discuss its sharpness.

math.CV↗

arXiv2022

Reply to comment on "Failure of the simultaneous block diagonalization technique applied to complete and cluster synchronization of random networks"

We respond briefly to a comment [1, arXiv:2110.15493] recently posted online on our paper [2, arXiv:2108.07893]. Complete and cluster synchronization of random networks is undoubtedly a topic of interest in the Physics, Engineering, and Nonlinear Dynamics literature. In [3] we study both complete and cluster synchronization of networks and introduce indices that measure success (or failure) of application of the SBD technique in decoupling the stability problem into problems of lower dimensionality. Our usage of the word `failure' indicates that the technique does not produce a decomposition which results in a system which is easier to analyze, not that the technique fails in correctly decoupling these problems.

eess.SY↗

arXiv2022

Failure of the simultaneous block diagonalization technique applied to complete and cluster synchronization of random networks

We discuss here the application of the simultaneous block diagonalization (SBD) of matrices to the study of the stability of both complete and cluster synchronization in random (generic) networks. For both problems, we define indices that measure success (or failure) of application of the SBD technique in decoupling the stability problem into problems of lower dimensionality. We then see that in the case of random networks the extent of the dimensionality reduction achievable is the same as that produced by application of a trivial transformation.

cond-mat.dis-nn↗