Norm inequalities for functions of matrices

In this paper, we prove several spectral norm and unitarily invariant norm inequalities for matrices in which the special cases of our results present some known inequalities. Also, some of our results give interpolating inequalities which are related to well-known recent inequalities.

Regularity of non-stationary subdivision: a matrix approach

In this paper, we study scalar multivariate non-stationary subdivision schemes with integer dilation matrix M and present a unifying, general approach for checking their convergence and for determining their Hölder regularity (latter in the case [Formula: see text]). The combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non-stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius. As an application, we prove a recent conjecture by Dyn et al. on the Hölder regularity of the generalized Daubechies wavelets. We illustrate our results with several examples.

Norm statement considered harmful: comment on 'evolution of unconditional dispersal in periodic environments'

The mathematical symbol for the norm, which is heavily overloaded with multiple definitions that have both universal and specific properties, lends itself to confusion. This is manifest in the proof of an important theorem for population dynamics by Schreiber and Li on how dispersal increases population growth in a periodic environment. Here the theorem is placed in context, the proof is clarified, and the confusing but inconsequential errors corrected.