Tables second derivative approximations

Table A.4 Second derivative approximations [ ( ) fAitmrily spaced n points (n = 3,4) at positions. ., c , each point at an offset h, = P" " ...

Further manipulations can be performed with the Taylor expansions, such as involving more points (e.g., fix + 2Ax), fix 2Ax), etc.). This leads to (1) better approximations for the first and second derivatives and (2) approximations for higher-order derivatives. Table 4.1 shows the first-order correct formulas for the first and second derivatives, while Table 4.2 depicts the second-order correct counterparts. In the tables. Ax = Xj+j - x, = x, - x,, . 

Another method for finding the end point is to plot the first or second derivative of the titration curve. The slope of a titration curve reaches its maximum value at the inflection point. The first derivative of a titration curve, therefore, shows a separate peak for each end point. The first derivative is approximated as ApH/AV, where ApH is the change in pH between successive additions of titrant. For example, the initial point in the first derivative titration curve for the data in Table 9.5 is... 

The second derivative of a titration curve may be more useful than the first derivative, since the end point is indicated by its intersection with the volume axis. The second derivative is approximated as A(ApH/AV)/AV, or A pH/AV. For the titration data in Table 9.5, the initial point in the second derivative titration curve is... 

Numerical analysis now indicates that for the terms which have been evaluated so far P, provides the dominant contribution to a,. This is illustrated in Table IV by the breakdown of alt into contributions from graphs having C = 1-5 (the data were supplied by M. F. Sykes). Hence we are tempted to investigate an approximation in which only the polygon is taken into account, and all other types with C > 1 are ignored. We shall call this the self-avoiding walk approximation to the specific heat Ch (second derivative with respect to W of In Z) and its behavior in the critical region is characterized by the function... 

The Bom exponent n can be evaluated from the results of experimental measurements of the compressibility of the crystal, which der pends on the second derivative, PV/dR2. It is found that for all crystals n lies in the neighborhood of 9. A somewhat better approximation to the experimental values is shown in Table 13-2 for a crystal of mixed-ion type an average of values of this table is to be used (3 for LiF, for example). 

Inverting the matrices and multiplying out the second row with the coefficient vector finally yields the approximation, presented in Table A.2 in Appendix A, together with a few others. It turns out that in the process, the terms in h5 drop out and the final approximation is of 0(/i4), arising from the neglected terms in h6. The formula has been given as early as 1935 by Collatz [169], who also presented some asymmetric forms in his 1960 book [170], and Bickley in 1941 [88]. Noye [423] also provides a number of multipoint second derivatives for use in the solution of pdes. 

Kimble and White were aware that leapfrog methods are unstable and simply remark that this did not seem to apply to their method. Also, they mention the use of 5 points for all approximations but their table of discretisations shows that they used 6 points at the edges for the spatial second derivative. This is no doubt because, as Collatz already mentions in 1960 [170], the asymmetric 5-point second derivative is only third-order, while a 6-point formula is fourth-order, like the symmetrical 5-point ones used in the bulk of the grid. So, for the second spatial derivative at index i = 1, the form 2/2(6) was used, and the reverse, form 2/5 (6) at i = N. 

If this interpretation of the thermodynamics is correct, the solvent activities should be invariant at all concentrations within the critical region. The exact location of this region is given by zero equivalence of the first and second derivatives of the chemical potential (log ai) with respect to concentration. We have chosen to define this region, arbitrarily, as the region where solvent activity exceeds unity since the tedious calculation necessary to establish it exactly seemed unwarranted by the approximate nature of the model. The maximum resin concentration at which the activity initially exceeds unity is defined as the critical concentration. For the three systems under study, these concentrations are listed in Table IV. It follows that y in Equation 1 is given by 1/cnu where is the critical solvent volume fraction. 

In Table3.9, the vibrational contributions for 11, 1, 2 and 3, calculated within the double-harmonic oscillator approximation are presented. The calculations were performed seminumerically, i.e. second derivatives of energy were calculated by differentiation of analytic first derivatives. Thus, the values of harmonic terms presented in Table 3.9 may serve as a reference point for numerical accuracy assessment of NR contributions discussed above. The relative error for [/uq ] ) term for 11 does not exceed 10%. It follows from Table3.9 that diagonal vibrational contributions to a... 

The table is used in much the same manner as are Eqs. 11-19 and 11-20 in the case of capillary rise. As a first approximation, one assumes the simple Eq. II-10 to apply, that is, that X=r, this gives (he first approximation ai to the capillary constant. From this, one obtains r/ai and reads the corresponding value of X/r from Table II-2. From the derivation of X(X = a /h), a second approximation a to the capillary constant is obtained, and so on. Some mote recent calculations have been made by Johnson and Lane [28]. 

United States production of ethylene oxide in 1990 was 2.86 x 10 metric tons. Approximately 16% of the United States ethylene (qv) production is consumed in ethylene oxidation, making ethylene oxide the second largest derivative of ethylene, surpassed only by polyethylene (see Olefin polymers). World ethylene oxide capacity is estimated by country in Table 11. Total world capacity in 1992 was ca 9.6 x 10 metric tons. 

A second successful prediction is that many so-called metastable species (i.e. isomers) are abundant even if they are quite reactive in the laboratory.66 Perhaps the simplest interstellar molecule in this class is HNC, but large numbers of others can be seen in Table 1. It is assumed that most metastable species are formed in dissociative recombination reactions along with their stable counterparts at approximately equal rates, and that both are destroyed by ion-molecule reactions so that the laboratory reactivity, which is normally determined by reactions with neutral species, is irrelevant. Both HCN and HNC, for example, are thought to derive from the dissociative recombination reaction involving a linear precursor ion ... 

As a validation experiment a large number of load speed measurements on Twaron 2200 PpPTA yarn at different temperatures have been carried out. In order to limit the scatter of the data a slight twist was applied to the yarn. Figure 67 shows the fit of the linear relation Eq. 135 with the experimental data. The values for the parameters used in this fit are listed in Table 5. As stated earlier, the linear relationship Eq. 135 was derived for the approximation t<2g. According to Eq. 134, for large values of the load rate the second term should become very small. Indeed, in Fig. 67 the observed data tend to level off for... 

Consider the exact definition of from Eq. (10.32). When atom fe is a sp carbon, we can safely neglect the second- and higher-order terms because the values are small, in favor of the simple approximation, Eq. (10.41). However, we must consider both (T- and rr-electron densities and their variations. The appropriate first derivatives dEf"/dNk)° are indicated in Table 10.3. 

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