In this paper, exact vibration frequencies of circular, annular and sector membranes with a radial power law density are presented for the first time. It is found that in general, the sequence of modes may not correspond to increasing az-imuthal mode number n. The normalized frequency increases with the absolute value of the power index |ν|. For a circular membrane, the fundamental frequency occurs at n = 0 where n is the number of nodal diameters. For an annular membrane, the frequency increases with respect to the inner radius b. When b is close to one, the width 1 – b is the dominant factor and the differences in frequencies are small. For a sector membrane, n – 1 is the number of internal radial nodes and the fundamental frequency occurs at n = 1. Increased opening angle β increases the frequency.

We investigate the growth and the frequency of zeros of the solutions of the differential equation f(n) + Pn–1 (z) f(n–1) + … + P0 (z) f = H (z), where P0 (z), P1(z), …, Pn–1(z) are polynomials with P0 (z) ≢ 0, and H (z) ≢ 0 is an entire function of finite order.

Characterizations of strong ergodicity for Markov chains using mean visit times have been found by several authors (Huang and Isaacson (1977), Isaacson and Arnold (1978)). In this paper a characterization of uniform strong ergodicity for a continuous-time non-homogeneous Markov chain is given. This extends the characterization, using mean visit times, that was given by Isaacson and Arnold.