Curve-fitting procedure

Write the determinant for a sixth-degree curve fitting procedure. 

The equation of Knox et al. was not derived theoretically from a basic dispersion model, but the constants of the equation were determined by a curve fitting procedure... 

The results obtained were probably as accurate and precise as any available and, consequently, were unique at the time of publication and probably unique even today. Data were reported for different columns, different mobile phases, packings of different particle size and for different solutes. Consequently, such data can be used in many ways to evaluate existing equations and also any developed in the future. For this reason, the full data are reproduced in Tables 1 and 2 in Appendix 1. It should be noted that in the curve fitting procedure, the true linear velocity calculated using the retention time of the totally excluded solute was employed. An example of an HETP curve obtained for benzyl acetate using 4.86%v/v ethyl acetate in hexane as the mobile phase and fitted to the Van Deemter equation is shown in Figure 1. 

Table 2. Experimental Values for the Dispersion Equation Coefficients Obtained by a Curve Fitting Procedure...

It is now necessary to identify the correct functions of the capacity factor (k") to be used in equation (12) to evaluate the diffusivity. Knox [7] suggested the following approach which required a polynomial curve fitting procedure to identify the necessary constants. Rearranging equation (12),... 

Other strength criteria are described by Sendeckyj [2-28]. Tennyson, MacDonald, and Nanyaro addressed the next logical step in a curve-fitting procedure, namely a third-order polynomial fit to failure data [2-29], However, the added complexity of their criterion has limited its use even though they identified some loading conditions under which their criterion is necessary to properly describe the actual failure behavior. 

Ri seems to be related to solute size, but it is obtained by an iterative curve-fitting procedure. The parameters M and D are characteristic of the solvent. For water Af = 0. A solvophobicity parameter Sp is then defined for the solvent having value M... 

FIGURE 11.12 Overextrapolation of data, (a) Nonlinear curve fitting techniques estimate an ordinate maximal asymptote that is nearly 100% beyond the last available data point, (b) The curve fitting procedure estimates a maximal asymptote much closer to the highest available data point. A useful rule is to reject fits that cause an estimated maximal asymptote that is >25% the value of the highest available data point. 

The two-layered laser flash method has been applied to some molten systems above 1000 K. In Fig. 30 an example of curve fitting is shown for molten calcium aluminosilicate at 1723 K. An analysis in which the radiative component is taken into account gives a good fit. The thermal conductivity and the radiative component parameter can be determined simultaneously by a curve-fitting procedure. 

Spectra of s.o. samples differed markedly from those of a.p. samples and were unaffected by a subsequent evacuation up to 673 K (Fig. 4, a). Spectra consisted of a composite envelope of heavily overlapping bands at 980-1070 cm-, with two weak bands at 874 and 894 cm-. Irrespective of the preparation method, the integrated area (cm- ) of the composite band at 980-1070 cm- was proportional to the V-content up to 3 atoms nm-2. An analysis of spectra by the curve-fitting procedure showed the presence of several V=0 modes. The relative intensity of the various peaks contributing to the composite band depended only on the V-content and did not depend on the method used for preparing the catalysts. Samples with V > 3 atoms nm-2 R-spectra features similar to those of pure V2O5 (spectrum 8 in Fig. 4, a). 

For a given value of a the probability is maximum when the sum in the exponent is minimum. Thus, the minimization of xf becomes the criterion for the most probable value obtainable from n equally reliable measurements. This result is the basis for the various curve-fitting procedures that are commonly used in the analysis of experimental data. 

Armentrout et al.69 have developed a method with which the kinetic shift can be taken into account in the curve fitting procedure. The fitting procedure, which corrects for the kinetic shift, includes only the fraction of the precursor ions which... 

A nonlinear curve fitting procedure of the experimental (Eq. 4.28) to the theoretical (Eq. 4.27) 2D autocovariance function can serve to perform some fundamental characterization of the 2D separation. The total volume (Vy) and the peak height dispersion (/a() can be readily measured in the chromatogram, thus the number of components (m) and the peak widths (a, and ay) can be estimated (Marchetti et al., 2004). 

Swain (7) has discussed the general problem of determining rate constants from experimental data of this type and some of the limitations of numerical curve-fitting procedures. He suggests that a reaction progress variable for two consecutive reactions like 5.3.2 be defined as... 

Currie, J. N., 11 8 Curry powder, 23 166-167 Curtain coating, 7 20-21 method summarized, 7 5t shear rates, 7 32t Curtain spin process, 17 473-474 Curtius degradation, 2 573 Curved surgical needles, 24 206-207 Curve-fitting procedures, 14 237 Cusping, in fine art X-radiography,... 

We will not concern ourselves here with problems associated with line broadening, overlapping peaks, and background subtraction. There are, however, examples discussed later where both deconvolution and curve fitting procedures are shown to be essential in unraveling the contributions of differently bonded species of the same molecule to the total photoelectron yield. Carley and Joyner (14) have discussed recently deconvolution procedures for photoelectron spectra. 

For pure WO3 the fine structure of the W(4f) peak shows considerable changes after reduction of the crystal. With the nonreduced stoichiometric crystal only two levels 4f7yj and 4f5/2 are observed, but after reduction the superposition of two doublets induces broadened features. Shifts due to the transformation of W to are revealed by a curve fitting procedure however, more drastic reduction can only be effected by hydrogen and argon ion-bombardment when W " " and W° states appear (Fig. 19). 

SEC calibration methods which employ a series of narrow MWD standards are based upon a peak position method and traditionally have been the most widely practiced calibration procedures. The peak position method simply correlates the peak elution volume of each standard to its nominal molecular weight or size value. A curve fitting procedure (usually a least squares regression) is used to obtain a working calibration curve. The serious limitation of polymer chemical types for which a series of narrow MWD standards covering a wide molecular weight range can be obtained led to the development of experimental approaches which could be applied to polymer chemical types other than that of the narrow MWD standards employed in calibration. 

In the absence of sufficient data to develop a robust, biologically based model for quantitative risk assessment, a single curve-fitting model for each type of data set is preferred. It is noted that many different curve-fitting models have been developed, and those that fit the observed data reasonably well may lead to several-fold differences in estimated risk at the lower end of the observed range. Therefore, the US-EPA uses a standard curve-fitting procedure for tumor incidence data. 

Curve fitting is an important tool for obtaining band shape parameters and integrated areas. Spectroscopic bands are typically modeled as Lorenzian distributions in one extreme and Gaussian distributions in the other extreme [69]. Since many observable spectroscopic features lie in between, often due to instrument induced signal convolution, distributions such as the Voight and Pearson VII have been developed [70]. Many reviews of curve fitting procedures can be found in the literature [71]. 

A rate equation characterizes the rate of reaction, and its form may either be suggested by theoretical considerations or simply be the result of an empirical curve-fitting procedure. In any case, the value of the constants of the equation can only be found by experiment predictive methods are inadequate at present. 

The upper curve, which is the result of a curve fitting procedure to the points shown, is the HETP curve. The column was 25 cm long, 9 mm in diameter and packed with 8.5 micron (nominal 10 micron) Partisil silica gel. The mobile phase was a solution of 4.8 Sw/v ethyl acetate in n-decane. The minimum of the curve is clearly indicated and it is seen that the fit of the points to the curve is fairly good. As a result of the curve fitting procedure the values of the Van Deemter constants could be determined and the separate contributions to the curve from the multipath dispersion, longitudinal dispersion and the resistance to mass transfer calculated. 

It should be noted that the constants of the equation were arrived at by a curve fitting procedure and not derived theoretically from a basic dispersion model as a consequence the Knox equation has limited use in column design. It Is, however, extremely valuable in accessing the quality of the packing. This can be seen from the diagram shown in figure 2. 

A) term on Dm, by definition, and as a result of the curve fitting procedure ... 

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