Curve-shape fitting

Curve) shape fitting as in electron spin resonance and gas electron diffraction. 

Recorded kinetic curves were fitted to the five-parameter Equation (1). The parameters pj with their errors and the standard deviation of regressions are summarized in Tables 1-6. Comparison of the data confirm the previously reported (refs. 8,12) similarity in the behavior of the two isomers in the presence of strong bases in spite of the different shape of the kinetic curves. The relatively good agreement of exponents p2, P4 computed for the diastereomers at the same temperature and amine concentration demonstrates the validity of the model used. From comparison of Equations (4) and (7) it follows that both reaction must give the same exponent. 

For any method of structure reconstruction by ab-initio methods additional assumptions must be made. The multipole expansion95 method of Harrison, Stuhr-mann, and Svergun ([86], Sect. 5.3 [101], Chap. 6) assumes homogeneous internal density. The shape of the scattering curve is fitted by varying the envelope of the particle. 

Figure 12.1. Hubble diagram of nearby Type la supernovae. The distances are derived from light curve shape corrected luminosities (data from Tonry et al. 2003). The solid line is for an empty universe (Qa = Ojh = 0), the dotted line for an Einstein-de Sitter model (Qa = 0, Hm = 1) and the dashed line for a model with no matter and all cosmological constant (Qa = 1,0 1 = 0). The concordance model (Qa = 0.7, Qaj = 0.3) is shown as the line fitting the data best. The bottom panel shows all distances relative to the empty universe model.

Finally, Table XII showed that for both 3- and 4-substituted TFMS series, poorer correlations were obtained when herbicidal data obtained in the presence and absence of Tween 80 for a given series acting on a particular weed type were pooled and fitted simultaneously. Comparing the appropriate surface plots in Figures 2-3 for Foxtail and Wild Mustard, it is evident that poor correlations of the pooled data would have been obtained. The very different graphical curve shapes obtained for each TFMS series in the presence and absence of Tween 80 illustrate why correlations of the pooled data produced poor structure-activity correlations. 

The dependence of m on (- for a localizing solvent C in mobile phases A/C or A/B/C is given by Eq. (30a). According to our theory of how these solvent-solute-localization effects arise (i.e., competition of localizing solvent and solute molecules for the same site), we expect that/( e) will increase slowly with 6c until the maximum localized-coverage is approached (6c > 0.5). The function/(flc) should then increase sharply and level out for 6c > 0.75. Figure 16 shows actual datafor/(do) in the case of alumina (Fig. 16a) and silica (Fig. 16b). The same function/( c) is shown in each plot, and it is apparent that experimental data fall close to the solid curves (best-fit values of m° for each solvent C). The shape of the solid curve is as predicted, based on the analysis of Fig. 3. 

Fig. 19. The basic shape of the damping curve must fit the intended application. Table II. Mechanical Instrumentation for Measuring the Glass Transition and Damping...

The flat appearance of the E" curve is due to the compressed nature of this particular nomograph scale. Both functions appear to fit equally well and therefore satisfy the criteria of curve shape and shift factor consistency for using the reduced variable time-temperature superposition. Additionally, the criterion of reasonable values for a-j- is satisfied by virtue of using the "universal" WLF equation. 

Fig. 10. A comparison of the experimental and predicted oscillatory CO2 production time series in the CO oxidation reaction over Pt. The theoretical curves are generated by the Sales-Ttirner-Maple model. Note the close fit in both period and curve shape over the 522-550 K temperature range. (From Ref. 272).

Note that the same classification system used before for solubility, partition, and heat of mixing can serve as a rough guide here also. The correlation of curve shape is not completely dependable. For example, though some of Parshad s data (1577) fit Fig. 2-16, he reports curves such as that in (b) for ether-chlorcrfbrm and ether-benzene even though the first would be expected to behave like acetone-chloroform and the second like CCU-ethyl acetate. 

Each of the curves in Figure 2 is similar in shape to a familiar form in process control — the log-log Bode amplitude-ratio curve for a first-order system. The amplitude rate curve is fitted by the equation... 

The mapping of T from the subdomain of temperatures onto the interval [0, 1] is the membership or compatibility function fx T). The form of this function is subject to some relatively nonarbitrary constraints to be consistent with one s primitive notions of hotness, fx T) should increase monotonically and smoothly with T, and it should be roughly sigmoidal in shape. However, since a whole family of curves will fit this description, the choice of numerical values for the parameters of fi(T) will to a considerable extent be an arbitrary one. Therefore, the threshold criteria, which are given by fi(T), are themselves fuzzy. 

Hajek et al. [114] studied the thermal reactions of some ammonium salts of terephthalic and isophthalic acids in an inert-gas fluidized bed between 373 and 473 K. Simultaneous release of both NHj molecules occurred from the diammonium salts, without dehydration or amide formation, during which the reactant crystallites maintained their external shapes and sizes. The nr-time curves were fitted by the contracting volume equation. Rate coefficients for decompositions of diammonium terephthalate, monoammonium terephthalate and diammonium isophthalate at 423 K were in the ratio 7.4 1.0 134 and values of E, (in the same sequence) were 87, 108 and 99 kJ mol. ... 

As with the kinetic analyses of homogeneous rate processes, quantitative comparisons are made between the experimentally measured data for a reaction of interest and the curve shapes of the various rate equations (Table 5.1) to identify the applicable kinetic model. This can be approached in several ways (29,105). One traditional method is to plot graphs of g(a) against time and decide which, from the available expressions (Table 5.1), provides the best (linear) representation, or fit. There is no general agreement on what criteria constitute a best or a satisfactory fit. The... 

There is one distinctive feature of shape fitting—one opportunity to choose for better or worse not shared by intensity or location fitting. This is the choice of how one looks at the curve. In electron spin resonance, for instance, it is commonplace to look at a curve that comes close to representing the first derivative of the absorption. It would be possible to look at a wide variety of other presentations suggestions of sharpening 1 only scratch the surface of the possibilities. All that should be said here is that this is not a simple problem-surely not one for which an a priori calculation can give a satisfactory answer-and that it is hard to see why something close to a first derivative is the best choice. 

A software tool Trend Analysis for analysis the time series was applied. Trend analysis fits a general trend model to time series data and provides forecasts. S-curve is best fitted to our drying case. The S-curve model fits the Pearl-Reed logistic trend model. This accounts for the case where the series follows an S-shaped curve. The model is ... 

Bardsley [28] and Childs and Bardsley [29] have provided a substantial body of mathematical theory to facilitate the categorisation of detailed curve shapes in cases where the data do not fit the linear transformations of Eqn. 4. This approach may be seen as essentially inductive. It is an attempt to set up rigorous procedures for empirical mathematical description of enzymes kinetic behaviour. Such description should in theory define minimum levels of complexity for physical models. Application of this approach will severely test the precision of rate measurements in real cases, and there is a risk that vaUd mechanisms may be ruled out on the basis of apparent subtleties of curve shape that are no more than experimental error. This, however, is certainly no excuse for ignoring genuine non-linearity. 

Figure 6 Sound velocity c, as a function of gas pressure pgas in a two-step aerogel (p=5.3 kg/m ) evacuation of air (open triangles) venting with SFe (filled triangles). The solid curves are fit to the data according to Eq. (5). The shape of the SFe-curve is influenced by the high specific heat and density of this gas.

It is therefore not surprising that equation 31 fits a variety of curve shapes since) with the Langmuir term expsinded to n = 2 (i.e. equation 32), the relation in fact contains the ratio of two quadratics. Nevertheless, and particularly in view of the successful quantitative interpretation of the Langmuir forms of isotherm equations in the instances of the BET model (equations 7, et seq.) and retentions with blended stationary phases in gas chromatography (equations 14, et seq.), the LC relations cannot be dismissed as entirely empirical since, in any event, although such a connection has yet to be established, whatever interpretations are placed on the fitting constants must presumably involve at least the solute activity coefficients in the mobile and stationary phaises (see below) and, most likely, the (finite-concentration) activity coefficients pertinent to the mobile- and stationary-phase components as well. 

Processing any plastics, reinforced or unreinforced, into curved panels is relatively easy and inexpensive. Panels fit the structural theory that curved shaped can be stiffer to bend than flat shapes of the same weight. However, to withstand external pressure, a square section component will usually be heavier than one that is circular and of the same volume. Both single- and double-curvature designs are widely used to ensure a more effective use of RP materials. 

An elementary surface is created as a known shape as a shape generated according to a law using arbitrary curves or fit to existing points or curves, depending on the shape and the available input information. A surface as described by mathematical functions has no inherent boundary curves. It can be restricted by... 

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