On the block counting process and the fixation line of exchangeable coalescents
We study the block counting process and the fixation line of exchangeable coalescents. Formulas for the infinitesimal rates of both processes are provided. It is shown that the block counting process is Siegmund dual to the fixation line. For exchangeable coalescents restricted to a sample of size n and with dust we provide a convergence result for the block counting process as n tends to infinity. The associated limiting process is related to the frequencies of singletons of the coalescent. Via duality we obtain an analog convergence result for the fixation line of exchangeable coalescents with dust. The Dirichlet coalescent and the Poisson‐Dirichlet coalescent are studied in detail.
On the basis of bias\|flow analysis and its countermeasures, a control scheme is given for the three fuzzy controller to be built by three pipes absolute temperature\|difference as the input, the openness the valve as the controlled object. After realizing this control scheme, the problem of bias\|flow has be resolved perfectly. Now coked pipes are burned once a year, but no coking occurs. We use this control scheme from beginning to now for seven years. There is no need to change the heating\|pipes at all. A new design method of discrete fuzzy controller presented in this paper can resolve the bias\|flow problem in other fluid distribution control system.