Adequate second-order model

The mathematical model may not closely fit the data. For example. Figure 1 shows calibration data for the determination of iron in water by atomic absorption spectrometry (AAS). At low concentrations the curve is first- order, at high concentrations it is approximately second- order. Neither model adequately fits the whole range. Figure 2 shows the effects of blindly fitting inappropriate mathematical models to such data. In this case, a manually plotted curve would be better than either a first- or second-order model. 

With nonlinear models, which are aimed at mathematical modeling or adequate description of the optimum region and that as a rule have numerous regression coefficients, rejection of insignificant regression coefficients is not so important as in the phase of linear modeling. For second-order models, an estimate of lack of fit or inadequacy of the model is of particular importance. 

Having obtained an adequate mathematical second-order model of a research subject for k<3, we can obtain a geometric interpretation in a two- or three-dimensional space. To obtain this interpretation, it is necessary to transform the second-order model into a typical-canonical form. Canonic transformation of a regression model is terminated in the form ... 

Validation of our proposed kinetic model is illustrated by the solid and dashed curves shown by the C vs time results shown in Figs. 6-8 and 6-9 for C0 of 50 and 100 mg L-1. Here all model parameters for both the multireaction and second-order models were based on adsorption data only. With the exception of p and 0, initial conditions for this initial-value problem were the only input required. Based on these predictions, we can conclude that both models predicted Cu desorption or release behavior satisfactorily. However, predictions of desorption isotherms were not considered adequate at the initial stages of desorption following adsorption. In addition, the model underpredicted amount sorbed that directly influences subsequent predictions for the desorption isotherms. Discrepancies between experimental and predicted are expected if the amounts of Cu in the various phases (C, Se, Sh and S2) at each desorption step were significantly different. These underpredictions also may be due to the inherent assumptions of the model. Specifically, the models may... 

Finally, based on literature review, most retention experiments were designed for adsorption measurements where desorption data were not always sought. Therefore, kinetic retention models, such as those proposed in this study, which are capable of predicting desorption behavior of heavy in soils based solely on adsorption parameters are of practical importance. Based on our results, the overall goodness of our model predictions are considered adequate and provides added credence to the applicability of our proposed model approaches. Moreover, adsorption as well as desorption results, the second-order model was superior compared to the multireaction model. Furthermore, model formulations with consecutive irreversible retention, for MRM as well as SOTS, provided better Cu description than other model versions. 

There is a slight difference in phase shift between the fifth order model and first-order model, the amplitude ratio is only slightly different for high frequencies. As can be seen, the second-order model approximation is almost perfect. However, depending on the application of the model, a first-order model approximation may be adequate. The step responses of the models are shown in Fig. 26.3. 

Westheimer noted the differences between saccade duration-saccade magnitude and peak velocity-saccade magnitude in the model and the experimental data, and inferred that the saccade system was not linear because the peak velocity-saccade magnitude plot was nonlinear, and the input was not an abrupt step function. Overall, this model provided a satisfactory fit to the eye position data for a saccade of 20°, but not for saccades of other magnitudes. Interestingly, Westheimer s second-order model proves to be an adequate model for saccades of all sizes, if one assumes a different input function as described in the next section. Due to its simplicity, the Westheimer model of the oculomotor plant is still popular today. 

Breakthrough Behavior for Axial Dispersion Breakthrough behavior for adsorption with axial dispersion in a deep bed is not adequately described by the constant pattern profile for this mechanism. Equation (16-128), the partial different equation of the second order Ficldan model, requires two boundaiy conditions for its solution. The constant pattern pertains to a bed of infinite depth—in obtaining the solution we apply the downstream boundaiy condition cf 0 as oo. Breakthrough behavior presumes the existence of... 

A kinetic model which accounts for a multiplicity of active centres on supported catalysts has recently been developed. Computer simulations have been used to mechanistically validate the model and examine the effects on Its parameters by varying the nature of the distrlbultons, the order of deactivation, and the number of site types. The model adequately represents both first and second order deactivating polymerizations. Simulation results have been used to assist the interpretation of experimental results for the MgCl /EB/TlCl /TEA catalyst suggesting that... 

In order to ensure accurate CG potentials, one needs to conduct MD simulations with a reliable atomistic potential model. The most desirable theoretical approach for the atomistic-scale simulations would be to use a level of quantum mechanics (QM) that can treat both intermolecular and intramolecular interactions with acceptable accuracy. Realistically, the minimal QM levels of theory that can adequately treat all different types of chemical forces are second order perturbation theory [32] (MP2)... 

For turbulent flow in single-phase systems, the predicted temperature profile is not changed significantly if the Peclet number is assumed to be infinite. Therefore, in turbulent two-phase systems the second-order terms in Eqs. (9) probably do not have a significant effect on the resulting temperature profiles. In view of the uncertainties in the present state of the art for determining the holdups and the heat-transfer coefficients, the inclusion of these second-order terms is probably not justified, and the resulting first-order equations should adequately model the process. 

Improved mathematical models. First or second order linear equations adequately fit much calibration data. If neither model is appropriate, the following semi-empirical multiple curve procedure may be used. 

In order to get significant results, the initial data must be formed by a set of clearly non-A -representable second-order matrices, which would generate upon contraction a closely ensemble A -representable 1-RDM. It therefore seemed reasonable to choose as initial data the approximate 2-RDMs built by application of the independent pair model within the framework of the spin-adapted reduced Hamiltonian (SRH) theory [37 5]. This choice is adequate because these matrices, which are positive semidefinite, Hermitian, and antisymmetric with respect to the permutation of two row/column indices, are not A -representable, since the 2-HRDMs derived from them are not positive semidefinite. Moreover, the 1-RDMs derived from these 2-RDMs, although positive semidefinite, are neither ensemble A -representable nor 5-representable. That is, the correction of the N- and 5-representability defects of these sets of matrices (approximated 2-RDM, 2-HRDM, and 1-RDM) is a suitable test for the two purification procedures. Attention has been focused only on correcting the N- and 5-representability of the a S-block of these matrices, since the I-MZ purification procedure deals with a different decomposition of this block. 

A second-order regression model is adequate with 95% confidence since Fr=2.35[Pg.385]

Since the obtained regression model adequately describes experimental outcomes, a check was done by a second-order regression model ... 

The analysis of variance (ANOVA) indicated that the second-order polynomial model (above) was statistically significant and adequate to represent the actual relationship between the response (percent weight conversion) and the significant variables, with very small p-value (0.0001) and a satisfactory coefficient of determination (R2 = 0.955). 

It is interesting to note that the work was done in two stages. Initially, experiments were chosen for a first-order mathematical model from the projected second-order design. These were carried out first, to check that there was no problem and that the experimental domain was adequate, before doing the remaining experiments for a predictive model that could be used for optimizing. 

The Landau model for phase transitions is typically applied in a phenomenological manner, with experimental or other data providing a means by which to scale the relative terms in the expansion and fix the parameters a, b, c, etc. The expression given in Equation (9) is usually terminated to the lowest feasible number of terms. Hence both a second-order phase transition and a tricritical transition can be described adequately by a two term expansion, the former as a 2-4 potential and the latter as a 2-6 potential, these figures referring to those exponents in Q present. 

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