Evolution error analysis

As we checked in practical applications, the evolution of the collective variables can be modelled with a Langevin equation also for the Lagrangian metadynamics introduced in Sect. 2.1. Therefore, the error analysis performed in this section can be applied also for Lagrangian metadynamics. 

The slow inihation results typically in a positive curvature (acceleration) of the kinetic plot, but in a negative curvature for the molar mass evolution plot (see e.g. the slow initiation case in the LA/Al(0 Pr)3 tetramer system [125a]). Beste and Hall [125b], and later also Pepper [125c], each described methods of trial and error analysis which allowed the determination k and kp values on the basis of experimental [M] versus time data (an example is provided in Ref. [125d]). More recently, however, less-cumbersome computational methods starting from kinetic Equations 1.27a and b have more often been employed (see e.g. Ref [125a]). 

The second considered example is described by the monostable potential of the fourth order (x) = ax4/4. In this nonlinear case the applicability of exponential approximation significantly depends on the location of initial distribution and the noise intensity. Nevertheless, the exponential approximation of time evolution of the mean gives qualitatively correct results and may be used as first estimation in wide range of noise intensity (see Fig. 14, a = 1). Moreover, if we will increase noise intensity further, we will see that the error of our approximation decreases and for kT = 50 we obtain that the exponential approximation and the results of computer simulation coincide (see Fig. 15, plotted in the logarithmic scale, a = 1, xo = 3). From this plot we can conclude that the nonlinear system is linearized by a strong noise, an effect which is qualitatively obvious but which should be investigated further by the analysis of variance and higher cumulants. 

Thus, from a parabolic fit to the REDOR evolution data, the second moment can be evaluated. As mentioned in Section 1, this analysis has to be restricted to the initial part of the evolution curves AS/Sq <0.3, as exemplified in Figure 2. However, the first order approximation entails a systematic imderestimation of M2, as shovm by Bertmer and Eckert. Numerous variations of the original REDOR pulse sequence have been established to adapt the technique to specific needs. To accoimt for pulse imperfections and other experimental errors, Chan and Eckert introduced compensated REDOR. In this approach, an /-channel 7r-pulse in the centre of the pulse sequence cancels the reintroduction of the 7-S dipolar couplings hence the echo amplitudes are solely attenuated by the... 

At this point, we have to verify the eorreetness of the selection of the unification relations. When S sSint we can conclude that our selection for the unification relations is good in this case, we can also note that the calculations have been made without errors. Otherwise, if computation errors have not been detected, we have to observe that the selected interactions for the unification of blocks are strong and then they carmot be used as unification interactions. In this case, we have to carry out a new experimental research with a new plan. However, part of the experiments realized in the previous plan can be recuperated. Table 5.68 contains the synthesis of the analysis of the variances for the current example of an esterification reaction. We observe that, for the evolution of the factors, the molar ratio of reactants (B) prevails, whereas all other interactions, except interaction AC (temperature-reaction time), do not have an important influence on the process response (on the reaction conversion). This statement is sustained by all zero hypotheses accepted and reported in Table 5.68. It should be mentioned that the alcohol quality does not have a systematic influence on the esterification reaction efficiency. Indeed, the reaction can be carried out with the cheapest alcohol. As a conclusion, the analysis of the variances has shown that conversion enhancement can be obtained by increasing the temperature, reaction time and, catalyst concentration, independently or simultaneously. 

To be of any practical use, the artificial boundary conditions have to be stable in particular, they should not depend on rounding errors. The stability analysis is made on the linearized problem by computing the time evolution of some solution norms. 

The evolution of the fluorescence intensity of CD-St (at 357 nm) as a function of oxazine concentration contains information on the stability constant of the complex and on the energy transfer efficiency. In fact, the latter is directly related to the asymptotic value of the fluorescence intensity (corresponding to full complexation). After correction for the inner filter effect, data analysis provides the value of (2.4 + 0.1) X 10 for the association constant / = [(CD-St)2 oxazine]/[2CD-St] [oxazine]. Moreover, the asymptotic value of the fluorescence intensity is very close to zero within experimental error which means that the energy transfer efficiency is close to 1. However, a full characterization of these photoinduced processes requires time-resolved techniques (see below). 

Time evolution of the concentration of 3S and products formed following reduction for ca. 6 s, at this time ( = 0 s) the potentiostat was switched to open circuit. The concentration change is obtained by multicomponent analysis of the dilferential absorption IR spectra. The values of the concentration change for 3S are olfset by 2.1 mM. The error bars are drawn at the 3 e.s.d. level. The mass balance corresponds to the sum of the concentration changes. The inset shows the change in concentration of 3S on a root timescale. Modified from ref. 19. 

Numerical errors do, however, have some influence in the case of neutral experiments, especially for very large input perturbations. If the input flux contains a very large fraction of marked (labeled) compound, the dynamics of the marker is close to saturation and in this area small variations of the input flux lead fo relatively large variations in the output, which produces numerical errors in the deconvolution of eq. (12.105). This effect explains the spike in fig. 12.3 for large perturbations. It follows that, even for response experiments of type (b), it is not recommended to use very large input perturbations. Nevertheless, it seems that the admissible input perturbations that produce reasonable results are much larger than the admissible perturbations for experiments of type (a) for which the analysis is based on linearized evolution equations. 

In conclusion, the linearization of the evolution equations for the analysis of chemical and biochemical networks is unpredictably limited. For the linearization to be valid it is necessary to use small perturbations, for which the experimental errors are very large. In the papers that have appeared on this subject, insufficient (or no) attention has been given to error accumulation and propagation [25]. The response approaches developed here avoid linearization and hence are to be preferred. 

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