Derivative Approach

The time-derivatives can be estimated analytically from the smoothed data 

The best and easiest way to smooth the data and avoid misuse of the polynomial curve fitting is by employing smooth cubic splines. IMSL provides two routines for this purpose CSSCV and CSSMH. The latter is more versatile as it gives the option to the user to apply different levels of smoothing by controlling a single parameter. Furthermore, IMSL routines CSVAL and CSDER can be used once the coefficients of the cubic spines have been computed by CSSMH to calculate the smoothed values of the state variables and their derivatives respectively. 

Having the smoothed values of the state variables at each sampling point and the derivatives, q we have essentially transformed the problem to a usual linear regression problem. The parameter vector is obtained by minimizing the following LS objective function 

However, an important question that needs to be answered is what constitutes a satisfactory polynomial fit An answer can come from the following simple reasoning. The purpose of the polynomial fit is to smooth the data, namely, to remove only the measurement error (noise) from the data. If the mathematical (ODE) model under consideration is indeed the true model (or simply an adequate one) then the calculated values of the output vector based on the ODE model should correspond to the error-free measurements. Obviously, these model-calculated values should ideally be the same as the smoothed data assuming that the correct amount of data-filtering has taken place. 

Finally, the user should always be aware of the danger in getting numerical estimates of the derivatives from the data. Different smoothing cubic splines or polynomials can result in similar values for the state variables and at the same time have widely different estimates of the derivatives. This problem can be controlled 

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A comparison of the theory for EOM-CC properties, which empahsize eigenstates and generalized expectation values, and the derivative approach of CCLR has been presented. The usual form of perturbation theory for properties, employ only lower-order wavefunctions in their determination. CCLR involves consideration of wavefunctions of the same order as the energy of interest, but this ensures extensivity of computed properties. 

The derivative approach described previously can be readily extended to ODE models where the unknown parameters enter in a nonlinear fashion. The exact same procedure to obtain good estimates of the time derivatives of the state variables at each sampling point, t can be followed. Thus the governing ODE... 

In this section we shall only present the derivative approach for the solution of the pyrolytic dehydrogenation of benzene to diphenyl and triphenyl regression problem. This problem, which was already presented in Chapter 6, is also used here to illustrate the use of shortcut methods. As discussed earlier, both state variables are measured and the two unknown parameters appear linearly in the governing ODEs which are also given below for ease of the reader. 

The tartrate (or TADDOL) derived approach to catalyst design has also been applied to the enantioselective a-hydroxylation of p-ketoesters. In this case, an enantiospecihc titanium(IV) complex combines with a sulfonyloxaziridine as the... 

Robust peptide-derived approaches aim to identify a small drug-like molecule to mimic the peptide interactions. The primary peptide molecule is considered in these approaches as a tool compound to demonstrate that small molecules can compete with a given interaction. A variety of chemical, 3D structural and molecular modeling approaches are used to validate the essential 3D pharmacophore model which in turn is the basis for the design of the mimics. The chemical approaches include in addition to N- and C-terminal truncations a variety of positional scanning methods. Using alanine scans one can identify the key pharmacophore points D-amino-acid or proline scans allow stabilization of (i-turn structures cyclic scans bias the peptide or portions of the peptide in a particular conformation (a-helix, (i-turn and so on) other scans, like N-methyl-amino-acid scans and amide-bond-replacement (depsi-peptides) scans aim to improve the ADME properties." ... 

The basic idea in the HEN derivation approach of Floudas et al. (1986) consists of the following steps ... 

The above-related situation can also be improved by using a different algorithm for training. One of the most efficient minimization algorithms is the Levenberg-Marquardt (LM) [56,59]. It is between 10 and 100 times faster than gradient-descent, given it employs a second-derivative approach, while GDM employs only first-derivative terms. As the calculation of the Hessian matrix (matrix of the second derivatives of the error in... 

It is of some note that many of the models may be (and often were) obtained by-passing the derivational approach here. Basically each model may be viewed as represented by the first terms in a graph-theoretic cluster expansion [80]. Once the space on which the model to be represented is specified, the interactions in the orthogonal-basis cases are just the simplest additive few-site operators possible. For the nonorthogonal bases the overlaps are just the simplest multiplicative operators possible, while the associated Hamiltonian operators are the simplest associated derivative operators. These ideas lead [80] to proper size-consistency and size extensivity. Similar sorts of ideas apply in developing wavefunction Ansatze or ground-state energy expansions for the various models. 

Potentiometric titration curves normally are represented by a plot of the indicator-electrode potential as a function of volume of titrant, as indicated in Fig. 4.2. However, there are some advantages if the data are plotted as the first derivative of the indicator potential with respect to volume of titrant (or even as the second derivative). Such titration curves also are indicated in Figure 4.2, and illustrate that a more definite endpoint indication is provided by both differential curves than by the integrated form of the titration curve. Furthermore, titration by repetitive constant-volume increments allows the endpoint to be determined without a plot of the titration curve the endpoint coincides with the condition when the differential potentiometric response per volume increment is a maximum. Likewise, the endpoint can be determined by using the second derivative the latter has distinct advantages in that there is some indication of the approach of the endpoint as the second derivative approaches a positive maximum just prior to the equivalence point before passing through zero. Such a second-derivative response is particularly attractive for automated titration systems that stop at the equivalence point. 

As there is an empirical relationship between the values of approach and range, once the optimum approach (A0) is determined, the corresponding optimum range (R0) can be obtained from the empirically derived approach versus the range curve (on the right of Figure 2.16), which is used as the optimum set point for TDIC-1 in Figure 2.16. 

Using a direct partial derivative approach for the objective function, instead of the Lagrangian multiplier as was used in Eqs. (92) to (95), determine the optimum values of x and y involved in Eqs. (92) to (95). 

The polymer-derived approach is a viable method for the preparation of ceramic fibers for reinforcement applications. During the past decade, significant advances have been made in materials, processes, and characterization capabilities. This approach holds great promise for future advances however, focus must be placed on addressing the key issues that have been identified. 

The fractional derivative technique is used for the description of diverse physical phenomena (e.g., Refs. 208-215). Apparently, Blumen et al. [189] were the first to use fractal concepts in the analysis of anomalous relaxation. The same problem was treated in Refs. 190,194,200-203, again using the fractional derivative approach. An excellent review of the use of fractional derivative operators for the analysis of various physical phenomena can be found in Ref. 208. Yet, however, there seems to be little understanding of the relationship between the fractional derivative operator and/or differential equations derived therefrom (which are used for the description of various transport phenomena, such as transport of a quantum particle through a potential barrier in fractal structures, or transmission of electromagnetic waves through a medium with a fractal-like profile of dielectric permittivity, etc.), and the fractal dimension of a medium. 

The discussion of the pros and cons of the finite-field approach made it clear that we need an analytical formulation of the derivatives of the molecular energy with respect to the external fields in order to maintain computational efficiency and numerical stability. The phrase analytical derivative approaches is often used to denote methods where closed-form expressions have been derived for the part of tire molecular properties that regards the motions of the electrons, i.e. the part related to the first term in Eq. (99). 

The main advantages of the energy derivative approach compared to the finite-... 

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