Generating functionals

During Stages II and III the average concentration of radicals within the particle determines the rate of polymerization. To solve for n, the fate of a given radical was balanced across the possible adsorption, desorption, and termination events. Initially a solution was provided for three physically limiting cases. Subsequentiy, n was solved for expHcitiy without limitation using a generating function to solve the Smith-Ewart recursion formula (29). This analysis for the case of very slow rates of radical desorption was improved on (30), and later radical readsorption was accounted for and the Smith-Ewart recursion formula solved via the method of continuous fractions (31). 

Table 2.5-2 Mean Variance and Moment-Generating Functions for Several Distributions ...

Table 2.5-2 provides a convenient summary of distributions, means and variances used in reliability analysis. This table also introduces a new property called the generating function (M,0). 

Generating functions are used in calculating moments of distributions for power series expansions. In general, the nth moment of a distribution,/fxj is E x ") = lx" f x) dx, where the integration is over the domain of x. (If the distribution is discrete, integration is replaced by summation.)... 

A generating function is defined by equation 2.5-47. To illustrate it use. Table 2.> 2 gives the generating function for an exponential distribution as -A/(0-X). Each moment i.s obtained by successive differentiations. Equation 2.5-48 shows how to obtain the first moment. By taking the limit of higher derivatives higher moments are found. 

In restructured electric markets, the vertical electric monopoly vill no longer be the sole provider of electricity. The generation, transmission, distribution, and customer service functions will be separated. The upstream generation function will be competitive, allowing new, any power producer to produce and sell electricityin any service territoi"y. The transmission and distribution functions will continue to be regulated, but will be required to allow access to power suppliers and marketers. This separation or unbundling of the industi"y is necessai"y to provide nondiscrimina-tory access for all suppliers of electricity. Customers will have their choice of electric suppliers. 

In our algebraic formalism, the time evolution of this system is represented by multiplying the generating function by the dipolynomial... 

But since x — 1) contains a constant term, if A x 1) is a conventional polynomial, then A x-, )/ x — 1) must also be a conventional polynomial. Thus, all reachable configurations represented by the generating function 1) have the form... 

Let t x) be the generating function of the topologically different planted trees. 

The definitions of and are, from a purely geometric-combinatorial point of view, somewhat artificial. However, p is related to R like to T. p will be derived from R, and R is the coefficient of x in the power expansion of the generating function... 

The series (2) of Sec. 3, too, is a generating function the collection of figures comprises the planted trees which are topologically different. The nodes of the rooted trees play the role of the balls in the figure there is only one category of balls, and thus the series depends only on one variable. Figure 1 indicates how the figures (planted trees) of the same content (number of nodes) are combined in the coefficients. 

That is, F(x,y,z) is the generating function of the number of nonequivalent configurations. The solution of our problem consists in expressing the generating function F(x,y,z) in terms of the generating function /(x,y,z) of the collection of figures and the cycle index of the permutation group H. 

It is easy to see that a combination with no repetitions gives rise to exactly two transitivity systems with respect to Ag. Summarizing the results, we have the rule the number of different transitivity systems of configurations with respect to Ag is the sum of the respective numbers of combinations with and without repetitions. Therefore, the generating function of the permutations which are nonequivalent with respect to Ag is... 

We note that (1.12) relates to the cycle index (1.16) like (1.15) (taking (1.12) and (1.14) into account) to (1.17). The following definitions allow us to state the rules on the construction of the generating functions in a unified way. To introduce the functions /(x), f(x,y) into the cycle index means putting... 

Theorem. The generating function for the configurations [] which are nonequivalent with respect to H is obtained by substituting the generating function of [4>] in the cycle index of U. 

Let jp(5 ) be the number of those configurations with content k,tl,m) which remain invariant under the permutation S in (1.6). Thus, the generating function of interest is... 

We apply the assumption to the special collection of figures whose generating function is... 

According to the nature of the congruence, topological, spatial, or planar, the generating functions of the planted C-H trees are given by... 

Making use of the relationship discussed above, "the number of noncongruent planted trees equals the number of nonequivalent configurations of three planted trees", of the generating function and the main theorem of Chapter 1 (Sec. 16) and taking the special case n 0 into account, we establish for each of the three situations an equation ... 

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