Formula, recursion

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). 

This is a recursion formula for the exact case. We would like to be able to apply this to any number n of CSTRs in series and find an analytical and then quantitative result for comparison to the exact PFR result. To do this weneedrecursive programming. There are threeprogrammingstylesin Mathematica Rule-Based,Functional,and Procedural.Wewill attackthisprobleminrecursionwith Rule-Based,Functional,and Procedural programming. WecanbeginbylookingattherM/e-tosed recursioncodesforCaandCbinanynCSTRs. 

The method to derive T was given by Cayley, who established the functional equation in the form of (1) for the function t(x). Cayley s computations of R are more laborious. Henze and Blair have derived the recursion formula (2.56) by direct combinatorial considerations, without knowledge of the functional equation. Here, (2.56) is a consequence of the functional equation (4). 

The functional equations (1 ), (4), (7), (8), (2.22), which have been established earlier and proved in the present paper, not only summarize the recursion formulas for the numbers R, S, Q, R but allow also general inferences (e.g.. Sec. 60), in particular on the asymptotic behavior (in Chapter 4). 

Polya 3, 4, 5. The last paper contains a direct proof for the equivalence of the recursion formulas based on combinatorial considerations and on the functional equation (4). 

Rather similar was the paper [PolG36a] which also derives asymptotic formulae for the number of several kinds of chemical compounds, for example the alcohols and benzene and naphthalene derivatives. Unlike the paper previously mentioned, this one gives proofs of the recursion formulae from which the asymptotic results are derived. A third paper on this topic [PolG36] covers the same sort of ground but ranges more broadly over the chemical compounds. Derivatives of anthracene, pyrene, phenanthrene, and thiophene are considered as well as primary, secondary, and tertiary alcohols, esters, and ketones. In this paper Polya addresses the question of enumerating stereoisomers -- a topic to which we shall return later. 

A table of binomial coefficients is given below (table IV). The numbers resulting from the successive partial differentiations, are given in table V, which can easily be extended with the use of the recursion formula... 

The ratio of consecutive terms in either series solution mi or U2 is given by the recursion formula with 5 = 0 as... 

A power series solution of equation (G.27) yields a recursion formula relating i+4, Ui+2, and ak, which is too complicated to be practical. Accordingly, we make the further definition... 

The recursion formula is obtained by setting the coefficient of each power of (jl equal to zero... 

Since the coefficient of each power of p must vanish, we have for the recursion formula... 

While the Hermite polynomials can be developed with the use of the recursion formula [Eq. (90)], it is more convenient to employ one of their fundamental... 

This equation is of the form of Eq. (15) and hence can be solved by the power-series expansion p) = J2k akPi- The resulting recursion formula is... 

Unlike the previous two examples, this is a one-term recursion formula. Hence, the series that is constructed from the value of no is a particular solution of Eq. (135). Once again, however, because of the problem of convergence, the series must be terminated after a finite number of terms. The condition for it to break off after the term in pv is given by... 

This result is the recursion formula which allows the coefficient an+2 to he calculated from the coefficient a . Starting with either ao or a an infinite series can be constructed which is even or odd respectively. These two coefficients are of course the two arbitrary constants in the general solution of a second-order differential equation. If one of them, say ao is set equal to zero, the remaining series will contain the constant at and be composed only of odd powers of On the other hand if a 0, the even series will result. It can be shown, however that neither of these infinite series can be accepted as... 

As this relation is correct for all values of s, the coefficients in brackets must vanish The result yields the important recursion formula for the Hermite polynomials,... 

Here again the indices n are independent in each summation, so that n can be replaced by n + 2 in the first term. Then, by posing the coefficient of z" equal to zero, the recursion formula becomes... 

The function y(x) can now be developed in a power series following the method presented in Section 5.2.1. The recursion formula for the coefficients is then of the form... 

One may proceed stepwise in this fashion to develop a general recursion formula for the concentration leaving reactor j in an n reactor cascade. 

The last term on the right in Equation 1.22 represents a doubly reduced matrix element, which can be calculated by recursive formula in terms of the coefficients of fractional parentage [4, 14], tabulated in the work of Nielson and Koster [27]. Finally, Equation 1.18 is rewritten as... 

In this way, all the coefficients Ar of the Frobenius series can be determined step by step. The recursion formula generates two independent series for odd and even values of r. For... 

Bessel functions have many interesting properties that will be presented here without proof, e.g. the recursion formula... 

The integral in equation 19.4-35 can be evaluated analytically using a recursion formula... 

The value of cAo depends upon the input function (whether step or pulse), and the initial condition (cAj(0)) for each reactor must be specified. For a pulse input or step increase from zero concentration, cA,-(0) is zero for each reactor. For a washout study, Ai(O) is nonzero (Figure 19.5b), and cAo must equal zero. For integer values of N, a general recursion formula may be used to develop an analytical expression which describes the concentration transient following a step change. The following expressions are developed based upon a step increase from a zero inlet concentration, but the resulting equations are applicable to all types of step inputs. 

For a second-order reaction fy cannot be expressed in explicit form as in 20.1-7, but a recursion formula can be developed that makes the calculation straightforward. 

For a more complex kinetics scheme, a combination of the explicit and recursion-formula approaches may be required. 

SMILES (Simplified Molecular Input Line Systems), 6 3-6 Smith, Adam, 24 364 Smith—Ewart kinetics, 14 715 Smith-Ewart recursion formula, 14 715 Smith-Ewart theory, 25 571, 572 VDC polymerization and, 25 697... 

The recursive formulas that have been presented exhibit some advantages over classical batch processing. First, they avoid the inversion of the normal coefficient matrix, since we would usually process a few equations at a time. Obviously, when only one equation is involved each time, the inversion degenerates into computing the reciprocal of a scalar. Furthermore, these sequential relationships can also be used to isolate systematic errors that may be present in the data set, as will be shown in the next chapter. 

To solve the least squares problem for the estimate of the measurement errors we need to invert the covariance matrix <. It is possible to relate to through a simple recursive formula. Let us recall the following matrix inversion lemma (Noble, 1969) ... 

When measurements are added or deleted, this recursive formula exhibits the advantage of avoiding the inversion of large matrices. Particularly, when only one measurement is added or deleted at a time, the inversion degenerates into computing the reciprocal of a scalar. 

The application of the matrix inversion lemma leads to recursive formulas similar to those previously obtained in the conventional approach set out in Section 6.3. 

In order to estimate the vector i in the presence of gross errors, we need to invert the covariance matrix, < , as Eq. (7.22) indicates. It is possible, though, to relate to balance residuals in the absence of gross errors) through the simple recursive formula (6.32), which was presented in the previous chapter. In this case we obtain the following relation ... 

Therefore, in order to compute the value of the least squares objective resulting from a given set of suspect measurements, we can use the recursive formula of Eq. (7.29) for exploiting the information already available from previous calculations, as is codified by the value... 

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