Probability Generating Functions pgf

In the ideal case, all reactive groups have the same reactivity irrespective of the shape and size of the hyperbranched molecules and no rings are formed. Then, the distribution of units in different reaction states is expressed by the following probability generating function (pgf), F0n(Z, z) ... 

The distribution of monomer units according to the number of bonds they issue is conveniently expressed through a probability generating function which is a simple tool used in generation of the trees. Thus, the probability generating function (pgf) for the number of bonds issuing from a f-functional monomer in the root Fp(z) is defined as... 

The probability generating function (pgf) for amine and epoxy units f (z) and fg(z), respectively, just describes this distribution... 

A more advanced method is the inverse (discrete) Laplace transformation (or probability generating function [pgf]) method (Asteasuain et al., 2002a,b, 2004). In this method, the CLD is reconstracted from the integrated moment equations, as illustrated in Fig. 10.6. A number and mass probability are first introduced for the living polymer molecules ... 

Principle of Laplace transformations (or probability generating function (pgf) method) to reconstruct the CLD based on the moment equations, k, vector with rate coefficients [A], vector with concentrations of nonmacromolecules At, time step for integration CLD, chain length distribution ... 

F (sz,Sa), F (sz,Sa) probability generating functions (pgf) of the counts of the different kinds of connecting and unreacted functional groups directly linked to, respectively, a monomer unit X , an unreacted group A , or a half-bond Z ... 

Probability generating functions (pgf) are defined as polynomials in the transformation variable z [where the coefficients p represent a probability distribution (that is, a length distribution)) ... 

This type of coupling occurs quite often in probability problems and is called a convolution of two distributions (mostly abbreviated by a star product). Passing to generating functions, the rather complicated sum of products of probabilities is transformed into a simple product of the corresponding pgf-s. 

In Fq(z), Pj is the probability of finding a monomer unit in the root which issues i bonds. This probability is equal to the fraction of units with i reacted functional groups z is an auxiliary (dummy) variable of the generating function through which the operations with the pgf are performed. It is important to note that Pj is just the coefficient at z By operations, the differentiations or rarely integrations are meant. In the derived statistical averages, z is put equal to 1 or 0. Thus... 

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