Parabolic fitness function

Figure 8.6 Left intensities of the CO bands of linear and bridged CO on Pd-Au alloys, as a function of Pd content. The almost parabolic fit to the intensities for CO in twofold positions reflects the probability of finding two adjacent Pd atoms in the Pd-Au surface (data from Kugler and Boudart [20]). Right schematic illustration of how alloying destroys ensembles of Pd where CO adsorbs in the twofold position.

To avoid these problems Brent (ref. 11) suggested a combination of the parabolic fit and the golden section bracketing technique. The main idea is to apply equation (2.20) only if (i) the next estimate falls within the most recent bracketing interval (ii) the movement from the last estimate is less than half the step taken in the iteration before the last. Otherwise a golden section step is taken. The following module based on (ref. 12) tries to avoid function evaluation near a previously evaluated point. 

Figure 4.17. The distribution of full widths at half maximum as a function of Bragg angle obtained using DMSNT (open circles, solid line) and WinCSD (filled triangles, dash-dotted line) algorithms. The lines represent parabolic fit of the two sets of data to illustrate the trend.

The Fukui function defined in Eq. (46) is represented in Fig. 11 for two Na clusters Na4o Naioo- This function was obtained by a finite difference method, that is, using the density of the neutral, positive and negatively charged clusters to perform, at each point f, a parabolic fit of p(f) versus N, from which 0p(r)/0N was then obtained. The densities of Nun, NaJ and Nun come from an extended Thomas-Fermi calculation including the functionals of Eqs. (14), (IS) and (16). 

Fig. 1. Heat of solution of different alkali-halide salts, plotted as a function of the difference between the Pauling radii given in Table 1. A maximimi in the heat of solution is observed for a given cation when the anion radius is larger by about tn = 0.08 nm. Data are from Ref 33, and the lines are parabolic fits.

FIGURE 35.3 Free-energy functions for reactant (AE) and product Ag (AE) of an electron transfer reaction as calculated using umbrella sampling within a simple dipolar diatomic solvent. AG° is the reaction free energy. Solid lines are polynomial fittings to the simulated points. Dashed lines are parabolic extrapolations from the minimum of the curves. (From King and Warshel, 1990, with permission from the American Institute of Physics.)... 

More careful examination of this shape reveals two important facts, (a) Plots of ssq as a function of k at fixed Io are not parabolas, while plots of ssq vs. Io at fixed k are parabolas. This indicates that Io is a linear parameter and k is not. (b) Close to the minimum, the landscape becomes almost parabolic, see Figure 4-6. We will see later in Chapter 4.3, Non-Linear Regression, that the fitting of non-linear parameters involves linearisation. The almost parabolic landscape close to the minimum indicates that the linearisation is a good approximation. 

Fig. 3.38 (a) Neutron reflectivity profile for a PS-PEO diblock (M = 15 kg mol-1,1.5% PEO) end-adsorbed from d-toluene onto quartz (Field et al. 1992a). The symbols indicate measured values, whilst the full line is a fit to a parabolic volume fraction profile, (b) Models for the density profile. The parabolic function was found to give the best fit to the data. 

Fig. 9.11. Focused beam spot size of the 27th harmonic wave as a function of distance from the SiC/Mg multilayer coated off-axis parabolic mirror. The open circles and filled squares correspond to the horizontal and vertical directions respectively. The best-fit curve based on (9.1) is also shown by the solid line...

Fig, 8. A free-energy relationship for mesolytic cleavage of C-C bonds in 7t-radical ions of bicumene derivatives [78, 99, 102]. The solid line represents a hypothetical reaction with no overhead (km = (kbJ /h)exp( — AG J(R T)), i.e. one with only the thermodynamic barrier). The broken lines are the best-fit lines to parabolic (Marcus-type) and hyperbolic (Weller-type) functions. 

A second and more accurate quadrature formula is the one most often used. It is applicable only to an odd number of equally spaced data points and is based on fitting parabolic functions to successive groups of three points. 

Figure 10 Fitted values of and as a function of charge for wollastonite-melt partitioning from Law et al. (2000). Note parabolic dependence of on cation charge and decreasing with increasing charge.

In Eqs. 4.1 to 4.6, A/ typically varies from 6 to 12 when the entire powder diffraction pattern is of concern. In some instances, when profile fitting is applied to short fragments of the powder diffraction pattern, the most suitable background function is that given by Eq. 4.1 with iV = 1 or 2, i.e. a linear or parabolic background. 

Fitting to Scatter Outside the Object In this method, the scatter is estimated by fitting an analytic function (Gaussian or parabolic) to the activity outside the source and interpolating the function to the source. The interpolated scatter contributions are then subtracted from the measured source counts to obtain scatter-corrected data for reconstruction of PET images. This method is based on the assumption that (1) the events outside the source are only scatter events, (2) the scatter distribution is a low-frequency function across the FOV, and (3) they are independent of activity distribution in the source. 

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