General Recursion

Even for non-regular polymer graphs and especially catacondensed species there are elegant results, e,g., described in chap. 6 of ref. [82]. This scheme of Gordon Davison [133] has a neat pictorial presentation which might be illustrated for a catacondensed polyhex chain  

---

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 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. 

Treacy, M.M.J., Newsam, J.M., and Deem, M.W. (1991) A general recursion method for calculating diffracted intensities from crystals containing planar faults. Proc. R. Soc. Lond., A443, 499-520. 

From the particular expressions (287)-(289), we can immediately write down the general recursion for the solution xT of the system (284). With this aim, let us introduce the two auxiliary vectors V and W as follows ... 

The estimation results are shown in Table 1. In this table, r denotes the correlation coefficient between measurements and estimates, and RMSE is the root mean square error. The results show that neither recursive PLS nor JIT modeling functions well. In general, recursive PLS is suitable only for slow changes in process characteristics. On the other hand, the reason for the poor performance of JIT modeling seems that JIT modeling does not take account of correlation among variables when a local model is built. 

The first of these, 17-22a, is the indicial equation. From it we see that s is equal to - -m or —to, inasmuch as a0 is not equal to zero. In order to obtain a solution of the form of Equation 17-20 which is finite at the origin, we must have s positive, so that we choose s = + to. This value of s inserted in Equation 17-226 leads to the conclusion that ai must be zero. Since the general recursion relation 17-22c connects coefficients whose subscripts differ by two, and since ai is zero, all odd coefficients are zero. The even coefficients may be obtained in terms of Oo by the use of 17-22c. 

Equation 4.77 for a nonisothermal CSTR establishes the temperature at which a stirred reactor operates for a given set of parameter values. This is also true for adiabatic operation. The only difference is that the heat exchange term UA T - T ) vanishes. In either case, the equation is transcendental and not amenable to extension to a CSTR sequence as a single generalized equation for N reactors. On the other hand, for a first-order reaction, a general recursion formula can be written for N reactors in series. This requires that the temperature of each stage is known to enable calculation of the rate constant. 

Overall the general recursion of eqn. (2) is applicable beyond the case of Kekule structures here elaborated for illustration. The related so-called conju-gated-circuit method turns out to have quite neat (related) linear re-cursions. Generally many sub-graph enumeration problems turn out to be of a linear recursive nature. 

FIGURE 4.1 Heilbronner s general recursion expression for the characteristic polynomial of a tree illustrated on a graph of 3-methylhexane. The recursion applies also to the bridge bond of polycyclic graphs. 

We recognize that the more general recursion defined in eq. (2.85), facilitates the development of hybrid approaches. 

---