Parameter-space analysis

Aluko. M., and H.-C. Chang. Multi-scale analysis of exotic dynamics in surface catalyzed reactions-II. Quantitative parameter space analysis of an extended Langmuir-Hinshelwood reaction scheme, Chem. Eng. ScL, 39, 51-64 (1984). 

What cannot be obtained through local bifurcation analysis however, is that both sides of the one-dimensional unstable manifold of a saddle-type unstable bimodal standing wave connect with the 7C-shift of the standing wave vice versa. This explains the pulsating wave it winds around a homoclinic loop consisting of the bimodal unstable standing waves and their one-dimensional unstable manifolds that connect them with each other. It is remarkable that this connection is a persistent homoclinic loop i.e. it exists for an entire interval in parameter space (131. It is possible to show that such a loop exists, based on the... 

The R-squared value, which indicates how well the three chosen parameters account for the variability in the yield, was 84.2%. The analysis of variance indicates that only temperature and pressure (both P-value = 0.026) have significant impact at 90% confidence level. The P-value of 0.37 for [NaOH] indicates that, within the parameter space examined, the concentration of NaOH does not significantly affect the cyclohexanone yield. Based on the above equation, one can predict the cyclohexanone yield at any given condition within the parameter space chosen. Since [NaOH] does not have a significant effect on the yield, one can fix its value and plot the yield of cyclohexanone as a function of temperature and pressure (Figure 1). 

Equation (118) provides the conceptual basis for all subsequent considerations The nonzero elements of the matrices A and 0% define the new parameter space of the system, that is, the possible dynamic behavior of the system is evaluated in terms of these new parameters. Crucial to the analysis, the elements of both matrices have a well-defined and straightforward interpretation in biochemical terms, making their evaluation possible even in the face of incomplete knowledge about the detailed kinetic parameters of the involved enzymes and membrane transporters. Any further evaluation now rest on a careful interpretation of the two parameters matrices. 

At this point it was clear that the highest potential for increased activity was by substitution in the 2-position of the biphenyl alcohol. We prepared the sequence of compounds shown in Table 1. Substituents were again chosen to maximize the parameter space covered within the relatively stringent synthetic limitations of the biphenyl substitution pattern. The application of regression analysis to the data for these compounds provided no clear relationship between structure and activity when the parameters in our standard data base were used. The best linear fit was found for B4, the STERIMOL maximum radius. However, the correlation coefficient was only 0.625. 

Gray, B. F. and Roberts, M. J. (1988). Analysis of chemical kinetic systems over the entire parameter space. Proc. R. Soc., A416, 391-402. 

The main results of the above PT analysis are shown in the phase diagram in the parameter space of electron density and p (Fig.5). 

It can be seen in the chromatogram of figure 6.11 that four peaks (the three antioxidants plus an unknown impurity) are amply resolved to the baseline. This implies that all values for the peak-valley ratio P are equal to 1 and that the criterion has become very insensitive to (minor) variations in the resolution between the different peak pairs. In the area of the parameter space in which four well-resolved peaks are observed, the only remaining aim of the optimization procedure is to approach the desired analysis time of 4 minutes. The irrelevance of the minimum time tmin is illustrated by the occurrence of the first peak in figure 4.9 well within the value of 1.5 min chosen for this parameter. 

As it has been pointed out in Section 5.2, it is natural to formulate dynamic shape analysis aproaches in terms of the dynamic shape space D described earlier [158]. The reader may recall that the dynamic shape space D is a composition of the nuclear configuration space M, and the space of the parameters involved in the shape representation, for example, the two-dimensional parameter space defined by the possible values of the density threshold a, and the reference curvature parameter b of a given MIDCO surface. 

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