Superposition plot

We have accordingly examined the j8-lactam class of antibiotics [67]. Geometrical details from a total of 114 jS-lactams containing the fragment 56 were retrieved from the CSD (Version 5.05) and categorised into biologically active (80) and inactive (30) compounds. The superposition plot of 12 randomly selec-... 

Fig. 5a,b. Superposition plot of 12 randomly chosen jS-lactams containing fragment 56 from a the active diagonal b the inactive diagonal of the Woodward (h) - Cohen (c) scatterplot (Taken from Nangia et aL [67])... 

As an alternative to t-T superposition, plot of the elastic stress tensor component as a function of the viscous one has been used, e.g., (cth — CT22) versus G12 or G versus G". For systems in which the t-T is obeyed, such plots provide a temperature-independent master curve, without the need for data shifting and calculating the three shift factors. Indeed, from Doi and Edwards tube model, the following relation was derived ... 

In this work, we advance from reducing variables to numerical curve fitting. Numerical fitting methods afford strong advantages over reducing variables and superposition plots. Numerical fits reveal weak dependences not readily apparent to the naked eye. Furthermore, reducing variables can only lead to superposition... 

Ueda and Kataoka measured shear thinning and other viscoelastic properties of polyisobutylene polybutene(44), as seen in Figure 13.31. At small shear rates, stretched-exponential forms are found S tends to 0.5 at larger c, while a increases with increasing c. The power-law exponent x increases with increasing c no limiting behavior to x is apparent. Correspondingly, one cannot form a completely accurate superposition plot for jy(/c). 

Fifth, as noted in the introduction, classical reduction approaches were found by Ferry(l) and by Pearson(2) to be insufficient to encompass the concentration dependences of the solution viscoelastic functions, because the functions change shape as well as scale. The regular trends uncovered in this chapter, while material-dependent, significantly lift the phenomenological obscurity described by Ferry and Pearson. The trends may not lead to superposition plots, but they do offer a systematization scheme suitable for interpolation and extrapolation. 

If the set of experimental points plotted in Fig. 70 is compared with that of Fig. 51, it will be seen that the situation is very similar. Nor is this close resemblance accidental. It will be recalled that in equation (161) the term Be represents the type of term that would be characteristic of a solute of any kind, and that the situation in Fig. 51 arises from the superposition, on this term, of another term arising from the ionic character of the solute. When the interionic effects had been eliminated, ourin terest turned, in Chapter 9, to a discussion of the B-coefficients... 

Fig. 31.2. Geometrical example of the duality of data space and the concept of a common factor space, (a) Representation of n rows (circles) of a data table X in a space Sf spanned by p columns. The pattern P" is shown in the form of an equiprobabi lity ellipse. The latent vectors V define the orientations of the principal axes of inertia of the row-pattern, (b) Representation of p columns (squares) of a data table X in a space y spanned by n rows. The pattern / is shown in the form of an equiprobability ellipse. The latent vectors U define the orientations of the principal axes of inertia of the column-pattern, (c) Result of rotation of the original column-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the score matrix S and the geometric representation is called a score plot, (d) Result of rotation of the original row-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the loading table L and the geometric representation is referred to as a loading plot, (e) Superposition of the score and loading plot into a biplot.

Fig. 32.7. CFA biplot resulting from the superposition of the score and loading plots of Figs. 32.6a and b. The coordinates of the products and the disorders are contained in Table 32.9.

In addition to the programs to select the optimum discussed previously, graphic approaches are also available and graphic output is provided by a plotter from computer tapes. The output includes plots of a given response as a function of a single variable (Fig. 11) or as a function of all five variables (Fig. 12). The abscissa for both types is produced in experimental units, rather than physical units, so that it extends from—1.547 to + 1.547 (see Table 5). Use of the experimental units allows the superpositioning of the single plots (see Fig. 11) to obtain the composite plots (see Fig. 12). 

Figure 9.24 shows temperature-reduced plots of 2d, Nc and 5 versus Tf + ATg. All the data conform well to a reduced curve like Figure 9.23. Interestingly, when the authors used both Tg and the melting temperature (Tm) [61] depressions to conduct superposition, they recognized that the reduced curve is nicely constructed but there is no significant difference compare with the case of Tf+ ATg. This indicates that Tg depression is important in optimizing foam processing conditions but Tm depression is not a significant factor for processing because the Tf range is still below Tm after C02 saturation.

Figure 15 gives the superposition of RR (full line) and RY (dotted plot) spectral densities at 300 K. For the RR spectral density, the anharmonic coupling parameter and the direct damping parameter were taken as unity (a0 = 1, y0 = ffioo), in order to get a broadened lineshape involving reasonable half-width (a = 1 was used systematically, for instance, in Ref. 72). For the RY spectral density, the corresponding parameters were chosen aD = 1.29, y00 = 0.85angular frequency shift (the RY model fails to obtain the low-frequency shift predicted by the RR model) and a suitable adjustment in the intensities that are irrelevant in the RR and RY models. 

As indicated in Figure 10.2, there is a distinct change in the slope of the line at carbon numbers 8 to 12, and this has also been observed by other researchers.2-3 This change in the slope cannot be explained by the ASF model, which is based on the premise that the chain growth probability factor (a) is independent from the carbon number. Some further developments of the ASF model by Wojciechowski et al.3 made use of a number of abstract kinetic parameters for the calculation of a product spectrum. Although it still predicts a straight line for the a plot, they suggested that the break in the line is due to different mechanisms of chain termination and could be explained by the superposition of two ideal distributions. This bimodal distribution explained by two different mechanisms... 

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