In a synchronized network of n nodes, each node will update its parameter based on the system state in a given iteration. It is well-known that the updates can converge to a fixed point if the maximum absolute eigenvalue (spectral radius) of the n × n iterative matrix F is less than one (i.e. ρ(F) < 1). However, if only a subset of the nodes update their parameter in an iteration (due to delays...

A dilation equation is a functional equation of the form f(t) = ∑N k=0 ck f(2t − k), and any nonzero solution of such an equation is called a scaling function. Dilation equations play an important role in several fields, including interpolating subdivision schemes and wavelet theory. This paper obtains sharp bounds for the Hölder exponent of continuity of any continuous, compactly supported sca...

Recently, Uhlig [Numer. Algorithms, 52(3):335-353, 2009] proposed open questions about the ratios between the spectral norm, the numerical radius and the spectral radius of a square matrix. In this note, we provide some observations to answer these questions. Keywords—Spectral norm, Numerical radius, Spectral radius, Ratios