Spline fit

Figure C2.17.13. A model calculation of the optical absorjDtion of gold nanocrystals. The fonnalism outlined in the text is used to calculate the absorjDtion cross section of bulk gold (solid curve) and of gold nanoparticles of 3 mn (long dashes), 2 mn (short dashes) and 1 mn (dots) radius. The bulk dielectric properties are obtained from a cubic spline fit to the data of [237]. The small blue shift and substantial broadening which result from the mean free path limitation are...

FIG. 16 Negative normal stress T.. as a funetion of substrate separation from grand eanonial ensemble Monte Carlo simulations at T = 1.00, ii = —11.0, and = 0.0 ( )(Pbuik = 0.486) the solid line represents a eubie spline fit to the dis-erete data to guide the eye. Also shown are three isobars —T% = 0.000, 0.598, and 1.196, indieated by horizontal lines. Interseetions between the isobars and the eurve Tzzi z) correspond to stable and metastable phases of the eonfined fluid (see text) (from Ref. 66). 

Figure 2 (a) The pair interaction V as a function of distance PdsoRhso alloy, (b) Spinodal curve for Pdj.Rhi j alloy system. The points indicate calculated points while the solid line is the cubic spline fit through the points. 

It was concluded that the filler partition and the contribution of the interphase thickness in mbber blends can be quantitatively estimated by dynamic mechanical analysis and good fitting results can be obtained by using modified spline fit functions. The volume fraction and thickness of the interphase decrease in accordance with the intensity of intermolecular interaction. 

There are also forms of nonlinear PCR and PLS where the linear PCR or PLS factors are subjected to a nonlinear transformation during singular value decomposition the nonlinear transformation function can be varied with the nonlinearity expected within the data. These forms of PCR/PLS utilize a polynomial inner relation as spline fit functions or neural networks. References for these methods are found in [7], A mathematical description of the nonlinear decomposition steps in PLS is found in [8],... 

Figure 9 and Fig. 10 show calculated channeling dips for the (110) axis and 111 plane, respectively, for the (3He, ap) nuclear reaction and several different possible deuterium sites. These calculations are for the conditions of the experiment but in each case assume a unique site. The curves are formed by drawing a spline fit through calculated points 0.1° apart. 

The first figure compares the plot of this equation with a cubic spline fit of the original data. 

Fig. 5 Noisy infrared data (2048 sample points) with a spline fit to the curved base line.

The original data are shown in Fig. 5. A spline fit (De Boor, 1978) has been made to the curved base line. These data are adjusted to a zero base line in accordance with the transmittance relation Tm = U(x)/U0(x), where Tm is the measured transmittance, U0(x) the unabsorbed flux, and U(x) the observed flux after absorption by the gas in the sample. The relation between... 

The simplest case, that of two large infrared lines, is shown in Fig. 31(a). A smooth curve was fitted to the base line as shown. A spline-fitting computer program developed by De Boor (1978) was used to obtain this fit very conveniently. After the fit was obtained, the data were adjusted to a flat base line, as shown in Fig. 31(b), and the data field was extended by padding with zeros to yield an overall data field of 28 = 256 points. Taking the Fourier transform, we obtained the interferogram function shown in Fig. 32. (Even though it was not obtained directly from the interferometer as recorded... 

A more complicated set of spectral lines is shown in Fig. 34(a). A spline fit is made to the base line as shown, which allows adjustment to a flat base line, shown in Fig. 34(b). Taking the Fourier transform of this yields the interferogram function shown in Fig. 35. Truncating this function after the sharp dropoff that signals the end of the interferogram, we take the Fourier... 

Figure 1 graphically depicts the numerical data relevant to our application listed by Lundberg et al. Different sets of curves off vs. X are provided for individual values of d /d0. Discrete data were provided in the numerical tables of the original work to produce the continuous traces in Figure 1, a cubic spline fitting was used. 

Quasiclassical trajectory Calculations on a spline-fitted ab initio surface 459... 

Plots of all available data points in relative units are shown in Figs. 9-11. Average reference profiles of the different species were obtained by means of a spline fit procedure applied to the respective data points. Table 1 contains pertinent information for every individual halocarbon on tropospheric abundances. With the help of these, absolute profiles can be calculated for any of the substances shown. 

Figure 9. Average vertical profiles of fully halogenated methanes plotted in raltive units with respect to tropospheric abundances measured between ground surface and 8 km altitude. All available data points are shown along with "average curves" spline fitted to these individual points. All data were measured at 44°N.

If our database obtained so far is considered representative for middle, high and low northern latitudes, latitude-height cross-sections of halocarbons can be derived based on the assumption of smooth meridional mixing on sloped surfaces and zonal symmetry. By applying a spline fit to the data obtained at the 3 discrete latitudes, isolines of 90, 80, 70%, - with respect to tropospheric abundance for CFC-12, CFC-11 and CH3CCI3 are shown in figs. 14-16. 

Data array calculations with cubic spline fitting and numerical integration appears to yield data of much greater accuracy. 

Spline Fitting in One Variable. By definition, a cubic spline function in one variable consists of a set of polynomial arcs of degree three or less joined smoothly end to end. The smoothness consists of continuity in the function itself and in its first and second derivatives. 

Spline Fitting in Two Variables. The methods described in the previous section may be extended to functions of two variables (9). The problem now is to find a surface S(x,y) which interpolates data values in a rectangular region a xsb, c ysd. By analogy with the previous section, we specify h "interior" knots in the x direction and l in the y direction. These knots, k, k2 ", kh and nl n2 ni divi< e... 

Figure 9.8. Atmospheric CO2 increase in the past 200 years as indicated by measurements on air trapped in old ice from Siple Station, Antarctica, and by the annual mean values from Mauna Loa Observatory. The line is a spline fit through all data. (After Siegenthaler and Oeschger, 1987.)...

In spite of the flexibility introduced into equation (62) by using 2D splines to describe DM(a, v), a comparison of classical trajectories run on a model potential with those run on a rotated Morse-cubic spline fitted to that model does not show good point-by-point agreement (see ref. 54 and references therein). 

By solving these equations using the method proposed by Mauri[10], a and (3 can be easily calculated, then the discrete components compositions and continuous fraction distribution in the outlet can be obtained. In the calculation, the distribution function was calculated by a cubic spline fit method by Ying [9],... 

Ying, Xugeng Ye Ruqiang Hu ying Phase Equilibria for Complex Mixtures. Continuous-thermodynamics Method Based on Spline Fit. Fluid Phase Equilibria, 53, 407-414(1989). 

Fig. 17 Energy as a function of a T-T distance and b T-T-T angle used in the simulation procedure (calculated as smoothing spline fits to Boltzmann equilibrium interpretations of the histogrammed data taken from 32 representative zeolite crystal structures). Only the central portions are shown, c The contribution to the energy sum for the merging of two symmetry-related atoms merging is only permitted when the two atoms are at less than a defined minimum distance [84], Reproduced with the kind permission of the Nature Publishing Group (http //www.nature.com/)...

Suppose now, that the potential V is spline fitted We consider the general polydimensional case and that the number of mesh points in all directions are equivalent to infinity (N -> < ). This situation is always attainable from a polydimensional grid of computed values in a sufficient quantity by subdivision of the step in each direction and preliminary interpolation of purely numerical supplementary values to be used in the spline expression, which therefore is ... 

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