Cubic spline procedure

A method for interpolation of calculated vapor compositions obtained from U-T-x data is described. Barkers method and the Wilson equation, which requires a fit of raw T-x data, are used. This fit is achieved by dividing the T-x data into three groups by means of the miscibility gap. After the mean of the middle group has been determined, the other two groups are subjected to a modified cubic spline procedure. Input is the estimated errors in temperature and a smoothing parameter. The procedure is tested on two ethanol- and five 1-propanol-water systems saturated with salt and found to be satisfactory for six systems. A comparison of the use of raw and smoothed data revealed no significant difference in calculated vapor composition. 

In order to study this question in a more systematic way, we have recently optimized 144 different structures of ALA at the HF/4-21G level, covering the entire 4>/v )-space by a 30° grid (Schafer et al. 1995aG, 1995bG). From the resulting coordinates of ALA analytical functions were derived for the most important main chain structural parameters, such as N-C(a), C(a)-C, and N-C(a)-C, expanding them in terms of natural cubic spline parameters. In fact, Fig. 7.18 is an example of the type of conformational geometry map that can be derived from this procedure. 

In the present study we have extracted the EXAFS from the experimentally recorded X-ray absorption spectra following the method described in detail in Ref. (l , 20). In this procedure, a value for the energy threshold of the absorption edge is chosen to convert the energy scale into k-space. Then a smooth background described by a set of cubic splines is subtracted from the EXAFS in order to separate the non-osciHatory part in ln(l /i) and, finally, the EXAFS is multiplied by a factor k and divided by a function characteristic of the atomic absorption cross section (20). 

The 2D property can be used to increase filtering efficiently [62]. We have filtered FTIR data from the homogeneous catalyzed rhodium hydroformylation of alkenes using a variety of ID and 2D filters. On blocks of 100-1000 spectra, the ID filters i. e. SG, fft, cubic spline, can reduce noise by ca. 10-50%, but the 2D filters, i. e. 2D fft, can reduce the noise level even further, to ca. 85 %+ [63]. The procedure for each block of spectroscopic data can be viewed as Eq. (7)... 

Apb is the scattering length density difference, Q is Porod s invariant, and Y the mean chord length. For the calculation of Yo(r) we approximated I(q) hy a cubic spline. The equations used for the calculation of " pore and " soUd are to be found in [8,30,39-41,47]. Analytical expressions for the descriptors of RES were published in [10,11,13,42,43]. In its most simple variant, the stochastic optimization procedure evolves the two-point probability S2 (r) of a binary representation of the sample towards S2(r) by randomly excWiging binary ceUs of different phases, starting from a random configuration which meets the preset volume fractions. After each exchange the objective function... 

Figure 6-16 Effect of Temperature on SLR of a Cured Gel (20°C, 48 hr) during Heating from 20 to 90°C. Values of AG At values calculated by derivation of a polynomial equation fitted to the data (continuous line), or after cubic spline fit and smoothed using a moving average procedure ( ) ( ) denote G values.

The applicability of the Kalman filter requires an accurate knowledge of the response of each component and an efficient procedure for background removal. Background subtraction has recently been treated with cubic splines polynomials(5,6] as smoothing interpolators between peak valleys and this has proved to be efficient for baseline resolution particularly for very low signal-to-noise ratios [7]. 

A common method of extracting f K) from Eq. 3.82 is to assume a form of the distribution function by differentiation of a smooth fimction describing the data. The function obtained by this method is called the affinity spectrum (AS) and the method, the AS method [71]. The most general approach uses a cubic spline to approximate the data. However, a simpler procedure uses a Langmuir-Freundlich (LF) isotherm model and the AS distribution is derived from the best parameters of a fit of the experimental isotherm data to the LF model [71]. This approach yields a unimodal distribution of binding affinity with a central peak, if the range... 

A commonly used fitting procedure for step (c) is the method oi cubic splines, tor which computer programs exist (see Press et al. Chapter 3 Shoup, Chapter 6). 

To approximate experimental adsorption isotherms and to calculate appropriate derivatives, we apply a procedure of smoothing splines, described by Reinsch [10]. In this approach, the experimental data are approximated by a cubic spline function g(x), which minimizes the following functional... 

The discrepancy between input and output values is, however, somewhat greater than in the previous graphs. For intermediate chain lengths, the differences are smaller than 10 % but increases towards smaller chain lengths, approximately up to 110 to 210 %. It is difficult to point out the exact reason for this increased discrepancy towards smaller chain lengths, but it can most probably be attributed to the use of the cubic spline fit procedure. If the correct fitting function (equation 3.2) would be used to fit the monomer versus time profile over the smallest chain lengths, then the inaccuracy becomes much smaller. Hence, the cause can not be found in inaccurate numbers in the calculated monomer consumption trace. Nevertheless,... 

Calculation was done according to equation 4.25 using a cubic spline fit procedure. 

As an example, the above procedure is applied to Ar and Xe clusters. The DIM models for all clusters of a particular noble gas are fully defined only when the atomic and diatomic fragment matrices are specified. The former were fixed by taking I(Ar) to be 15.76 eV and I(Xe) to be 12.13 eV [14] the latter were defined by taking the points on the curves U,G,U and G for A from the ab initio computations of Bdhmer and Peyerimhoff [15]. The corresponding points for X were taken from Wadt [16]. The Ai2( I ) and Xe2( IJ) interactions were taken from Watts [17]. For each diatomic curve, the points were fitted to a cubic spline function in the asymptotic region, the interaction was represented by A/Rtt, where A and n were chosen to match the spline function at its largest knot. Some relevant input data is collected in Table 1. 

This section is concerned with outlining the procedures used for data analysis, after the collection of intensity data. We start with an intensity function /g p (x) which has been obtained by step scanning at intervals of 0.2 20 (or preferably in equal intervals of x) from say, 3 to 110°. Such a data function has an element of noise arising from the statistical nature of x-ray production, scattering, and recording. This noise may be minimized by the use of an appropriate smoothing function. We have found the procedures using cubic splines described by Dixon et al. particularly suitable for this purpose. The use of cubic splines has the utility that the data function may be easily... 

When y (s) at the grid point is greater than V Sq) +0.2 kcal/mol, the directly computed value is used as input to the spline routine but when V Cs) is less than this the input to the spline routine is calculated from eq. (61). The final vibrational energy level splitting was the same when this procedure was repeated with the cubic term missing in eq. (61) so we assume that the order of the polynomial in eq. (61) is sufficiently high to represent the V (s) curve within 0.2 kcal/mol of its minimum. 

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