Fourier amplitudes

To assess the extent to which the exponentiated Fourier series has appreciable Fourier amplitudes, and set the sampling grid accordingly, further development of formula (21) is needed. We first rewrite... 

From the computational point of view the Fourier space approach requires less variables to minimize for, but the speed of calculations is significantly decreased by the evaluation of trigonometric function, which is computationally expensive. However, the minimization in the Fourier space does not lead to the structures shown in Figs. 10-12. They have been obtained only in the real-space minimization. Most probably the landscape of the local minima of F as a function of the Fourier amplitudes A,- is completely different from the landscape of F as a function of the field real space. In other words, the basin of attraction of the local minima representing surfaces of complex topology is much larger in the latter case. As far as the minima corresponding to the simple surfaces are concerned (P, D, G etc.), both methods lead to the same results [21-23,119]. 

Figure 6.9 Comparison of the radial distribution functions of the Pt foil (dashed line) and the Pt/Si02 catalyst (solid line) in Ar. Arrows show the positions of the second, third and fourth coordination shells. The Pt foil Fourier amplitude was divided by 2 for scaling purposes. (Reproduced from Reference [30].)...

Equation (2) is, strictly speaking, not suitable for optical fields, which are rapidly varying in time. The damping of the oscillating dipole, and the resultant phase shift, is then conveniently expressed by treating the hyperpolarizabilities as complex, frequency-dependent quantities. For the cubic hyperpolarizability, the relation between the Fourier components of the electric field and the Fourier amplitude of the oscillation of the electric dipole gives... 

The Fourier spectra of concentrations and of the reaction rate are quite similar. The difference is that the strong concentration decay suppresses the Fourier amplitudes at high frequencies. For the concentration motion the change in time of the K (t) acts as an external noise from which the concentration motion selects the main frequencies. 

Representation of Spectral Function in Terms of Fourier Amplitudes... 

Appendix 1. Calculation of Fourier Amplitudes b -i for Librators Appendix 2. Transformation of Integral for Spectral Function of Precessors Appendix 3. Optical Constants of Liquid Water... 

In order to relate these molar quantities to properties of the single molecule we can apply arguments of statistical classical mechanics. At moderate intensity, the electric field gives rise to a dipole density by electronic and atomic translation (or deformation) effects and by rotation (or orientation) effects. We recall that the rotation effects are counteracted by the thermal movement of the molecules and thus they are strongly dependent on the temperature T whereas the translation effects are only slight dependent on T because they are intramolecular phenomena. The general expression to be used to define the Fourier amplitudes (2.165)-(2.167) is ... 

Examples of mathematical methods include nominal range sensitivity analysis (Cullen Frey, 1999) and differential sensitivity analysis (Hwang et al., 1997 Isukapalli et al., 2000). Examples of statistical sensitivity analysis methods include sample (Pearson) and rank (Spearman) correlation analysis (Edwards, 1976), sample and rank regression analysis (Iman Conover, 1979), analysis of variance (Neter et al., 1996), classification and regression tree (Breiman et al., 1984), response surface method (Khuri Cornell, 1987), Fourier amplitude sensitivity test (FAST) (Saltelli et al., 2000), mutual information index (Jelinek, 1970) and Sobol s indices (Sobol, 1993). Examples of graphical sensitivity analysis methods include scatter plots (Kleijnen Helton, 1999) and conditional sensitivity analysis (Frey et al., 2003). Further discussion of these methods is provided in Frey Patil (2002) and Frey et al. (2003, 2004). 

The direct method (DM) for solution of this set of equations was proposed by Atherton et al. [5], and in a somewhat a modified form by Dickinson and Gelinas [4] who solved r sets of equations each of size In consisting of Eq. (1) coupled with a particular j—value of Eq. (2). Shuler and coworkers [5] took an alternative approach in the Fourier Amplitude method in which a characteristic periodic variation is ascribed to each a, and the resulting solution of (1) is Fourier analyzed for the component frequencies. These authors estimate that 1.2r2 5 solutions of Eq. (1) together with the appropriate Fourier analyses are required for the complete determination of the problem. Since even a modest reaction mechanism (e.g. in atmospheric chemistry or hydrocarbon cracking or oxidation) may easily involve 100 reactions with several tens of species, it is seen that a formidable amount of computation can result. 

The first widely used global method was the Fourier Amplitude Sensitivity Test (FAST) (for a review see [83]). In the FAST method, all rate parameters were simultaneously perturbed by sine functions with incommensurate frequencies. Fourier analysis of the solution of the model provided the variance crf(t) of concentration i, and also the variance o- (t) of c, arising from the uncertainty in the /th parameter. Their ratio... 

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