Semigroup inequalities, stochastic domination, Hardy's inequality, and strong ergodicity
For the classical L p -spaces of signed measures on N, we devise a framework in which bounds for a sub-Markovian semigroup of interest can be obtained, up to a constant factor, from bounds for another tractable semigroup that dominates stochastically the first one. The main tools are the Hardy inequality, the definition of related auxiliary L p spaces suited to take advantage of the domination, and the proof that the norms are equivalent to the classical ones if the reference measure is quasi-geometrically decreasing. We illustrate the results using birth-death and single-birth processes.
The classification of birth and death processes
In the applications one is given the matrix A and it is required to construct P(t) and to study the properties of the corresponding stochastic process. The existence, uniqueness, and the analytic properties of P(t) have been discussed in detail in [I]. The objective of this paper is to use the results of [1] to establish equivalences between properties of the stochastic process and properties of the sequences { Xn }, { n,u } and to evaluate, in terms of these sequences, some of the interesting probabilistic quantities associated with the process.
Synthesis of quasi-stationary inhomogeneous impedance body of arbitrary shape
Formulas were obtained for the reflection factors of a plane wave from the inhomogeneous impedance plane, which reflects the incident wave in the given direction. The problem of synthesis of impedance inhomogeneous rotation body was solved for an arbitrary shaped generatrix for the given scattering diagram for co- and cross polarizations. Impedance distributions were obtained in explicit form. Restrictions on the class of implementable scat- terplots were formulated.