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.
Rotational Analysis of the CrO Bands
The (0, 1), (0, 0) and (1, 0) bands of the CrO, which appear in the red and yellow region, are analysed rotationally, and we have been able to show that these bands owe to the transition 5 Π- 5 Π. We have shown the theoretical formulae for the quintet state, which can be applied in the case of nearly (a), and obtained the molecular constants as follows,