Correlation partial

The multidensity Ornstein-Zernike equation (70) and the self-consistency relation (71) actually describe a nonuniform system. To solve these equations numerically for inhomogeneous fluids one needs only an appropriate generalization of the Lowett-Mou-Buff-Wertheim equation (14). Such a generalization, employing the concept of the partial correlation function has been considered in Refs. 34,35. 

Eq. (101) is the multidensity Ornstein-Zernike equation for the bulk, one-component dimerizing fluid. Eqs. (102) and (103) are the associative analog of the singlet equation (31). The last equation of the set, Eq. (104), describes the correlations between two giant particles and may be important for theories of colloid dispersions. The partial correlation functions yield three... 

We apply the singlet theory for the density profile by using Eqs. (101) and (103) to describe the behavior of associating fluids close to a crystalline surface [120-122], First, we solve the multidensity OZ equation with the Percus-Yevick closure for the bulk partial correlation functions, and next calculate the total correlation function via Eq. (68) and the direct correlation function from Eq. (69). The bulk total direct correlation function is used next as an input to the singlet Percus-Yevick or singlet hypernetted chain equation, (6) or (7), to obtain the density profiles. The same approach can be used to study adsorption on crystalline surfaces as well as in pores with walls of crystalline symmetry. 

Figure 3-17. A partial correlation diagram for d showing weak- (left) and strong-field (right) limits for spin-triplet terms.

Figure 4-6. Partial correlation diagram for the spin-quartets of d ions in octahedral symmetry.

In the first example, the predicted oral absorption for a series of ACE inhibitors has been compared with published values of human bioavailability. For the generation of calculated absorption, a sigmoidal curve between observed human absorption and PSA for a series of reference compounds was used [25], The predicted oral absorption for ACE inhibitors is plotted against the calculated PSA values is shown in Fig. 19.6 however, as expected, only a partial correlation existed between predicted absorption and observed in vivo bioavailability. 

Qualitatively, all proposals indicate a linear dependence on ml (linewidth over a hyperfine pattern increases from low to high field or vice versa cf. Figure 9.4) plus a quadratic dependence on m, (outermost lines more broadened than inner lines). Multiple potential complications are associated with the lump parameters A, B, C, notably, their frequency dependence (Froncisz and Hyde 1980), partial correlation with g-strain (Hagen 1981), and low-symmetry effects (Hagen 1982a). The bottom line quantitative description of these types of spectra has been for quite some time, and still is, awaiting maturation. 

The correlation coefficients between a 10 year monthly mean time series of volatilisation rates and SST, 1 Om wind speed and pollutant concentration are used to elucidate which of the parameters drives the volatilisation rate changes and causes the deviations from the long term mean. All of the parameters do not vary independently. Since both SST and wind speed influence the volatilisation rate in a nonlinear manner, it is not intuitive whether an increase in wind speed leads to an increase in volatilisation rate. A raise in wind speed that coincides with a decrease of the sea surface temperature can lead to a negative linear correlation coefficient between volatilisation rate and wind speed. For that reason the partial correlation coefficient is calculated in addition to the simple linear correlation coefficients. It explains the relation between a dependent and one or more independent parameters with reduced danger of spurious correlations due to the elimination of the influence of a third or fourth parameter, by holding it fixed. One important feature of the partial correlation coefficient is, that it is equal to the linear correlation coefficient if both variables... 

Partial correlation coefficients are denoted by R12.34, linear correlations by R12 substituting 1,2,3,4 with sea surface temperature (t), 10m wind speed (u), volatilisation rate (v) and pollutant concentration (c). Indices after the dot refer to variables held fixed in the calculation. 

In the northern hemisphere the coefficient of determination of the partial correlation between the pollutant concentration in the dissolved phase and the volatilisation rate excluding wind speed and SST, R2V ut, is very low in comparison to both coefficients that omit the pollutant concentration (Figure 2.27). Hence the apparently high correlation between pollutant concentration and volatilisation rate, shown as yellow to green colour in Figure 2.26 in some areas in the Pacific Ocean is not caused by a causal relation between them, but spurious. Both partial correlation coefficients are much lower there. The differences between values of R, uc andR2v ut (Figure 2.27c)... 

Relations between linear and partial correlation coefficients are described in Appendix A2... 

The partial correlation coefficient between two variables 1,2 with a third variable 3, whose influence on 1 and 2 shall be excluded from the correlation is derived from... 

Fig. B.l Relations between the linear and partial correlation coefficients for two variables and a controlling variable. Lines denote causal relations between the variables.

Fig. 2.27 Absolute differences of the coefficients of determination of partial correlations a)...

Positive slopes in partial correlations of small series for which the Taft equation is suitable. For a resonance parameter r = 0.8. 

The cause of the cell cycle specificity of the bisindole alkaloids may be associated with the ability of these compounds to interact with the protein tubulin and thereby inhibit the polymerization (and depolymerization) of microtubules (16,17). In this respect the cellular pharmacology of vinca alkaloids is similar to that of other cytotoxic natural products such as colchicine or podophyllotoxin. On closer inspection, however, Wilson determined that the specific binding site on tubulin occupied by vinblastine or vincristine is chemically distinct from the site occupied by the other natural products (18). Subsequent experiments have determined that the maytansinoids, a class of ansa-macrocycles structurally distinct from the bisindoles, may bind to tubulin at an adjacent (or overlapping) site (19). A partial correlation of the antimitotic activity of these compounds with their tubulin binding properties has been made, but discrepancies in cellular uptake probably preclude any quantitative relationship of these effects (20). 

The superscript indicates the CL if it is <99 9 o 0. Partial correlation coefficients. The matrix below gives the parameters corresponding to the subscripts. 

Table 6, for column headings, ce indicates that the partial correlation coefficient is that for ctc... 

Ca represents the contribution of the polarizability parameter a to the regression equation r is the partial correlation coefficient of cri with a. 

In summary, the calculation of pressure drops by the Lockhart-Marti-nelli method appears to be reasonably useful only for the turbulent-turbulent regions. Although it can be applied to all flow patterns, accuracy of prediction will be poor for other cases. Perhaps it is best considered as a partial correlation which requires modification in individual cases to achieve good accuracy. Certainly there seems to be no clear reason why there should be a simple general relationship between the two-phase frictional pressure-drop and fictitious single-phase drops. As already pointed out, at the same value of X in the same system, it is possible to have two different flow patterns with two-phase pressure-drops which differ by over 100%. The Loekhart-Martinelli correlation is a rather gross smoothing of the actual relationships. 

Fig. 14. Partial correlation of absorption results for the liquid phase controlling, slug, and annular flow, for CO2 in water.

The data used are presented in Table 2, and the results of the correlations in Table 3. The steric term was included to account for the observation that conformation in molecules of the type XCH2CH2Z where Z is constant is a function of both ct,x and ux. Unfortunately as is shown by the partial correlation coefficients, ux is significantly linear in ctx. The results nevertheless support the general validity of equation 4. Further support is provided by the correlation of aliphatic nx values with equation 5. Again, the data used and the results of the correlations are presented in Tables 2 and 3 respectively. 

Most of the information on this subject refers to solvent effects on the substitution of tetraalkyltins by mercuric chloride (see Chapter 6, Section 1.5, p. 70) and to the iododemetallation of tetraalkyltins and tetraalkylleads (see Chapter 8, Section 6, p. 173). Data on solvent effects for a number of reactions are summarised in Table 12. There is only a partial correlation of the rate coefficients with any of the usual parameters51 of solvent polarity in particular, the solvents acetonitrile and acetone increase the value of the various rate coefficients to a much greater extent than would be predicted on the basis of solvent dielectric constant (e), or Z values, or 2sT values, etc.51. This may be seen especially for the iododemetallation of tetramethyllead, reaction (6) (R = Me)... 

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