Nonlinear alternative methods

One shortcoming of Schild analysis is an overemphasized use of the control dose-response curve (i.e., the accuracy of every DR value depends on the accuracy of the control EC o value). An alternative method utilizes nonlinear regression of the Gaddum equation (with visualization of the data with a Clark plot [10], named for A. J. Clark). This method, unlike Schild analysis, does not emphasize control pECS0, thereby giving a more balanced estimate of antagonist affinity. This method, first described by Lew and Angus [11], is robust and theoretically more sound than Schild analysis. On the other hand, it is not as visual. Schild analysis is rapid and intuitive, and can be used to detect nonequilibrium steady states in the system that can corrupt... 

An alternative method of solving the equations is to solve them as simultaneous equations. In that case, one can specify the design variables and the desired specifications and let the computer figure out the process parameters that will achieve those objectives. It is possible to overspecify the system or to give impossible conditions. However, the biggest drawback to this method of simulation is that large sets (tens of thousands) of nonlinear algebraic equations must be solved simultaneously. As computers become faster, this is less of an impediment, provided efficient software is available. 

Alternative methods, such as correcting the nonlinearity though the application of an appropriate physical theory as we described above, may do as well or even better than a Taylor series approximation, but a rigorous theory is not always available. Even in... 

Alternative methods of analysis have been examined and evaluated. Shokoohi and Elrod[533] solved the Navier-Stokes equations numerically in the axisymmetric form. Bogy15271 used the Cosserat theory developed by Green.[534] Ibrahim and Linl535 conducted a weakly nonlinear instability analysis. The method of strained coordinates was also examined. In spite of the mathematical or computational elegance, all of these methods suffer from inherent complexity. Lee15361 developed a 1 -D, nonlinear direct-simulation technique that proved to be a simple and practical method for investigating the nonlinear instability of a liquid j et. Lee s direct-simulation approach formed the... 

More advanced mathematical aspects of aromaticity are given in other references [33, 34]. Some alternative methods beyond the scope of this chapter for the study of aromaticity in deltahedral molecules include tensor surface harmonic theory [35-38] and the related Hirsch 2 N -b 1) electron-counting rule for spherical aromaticity [39]. The topological solitons of nonlinear field theory related to the Skyrmions of nuclear physics have also been used to describe aromatic cluster molecules [40]. 

This alternative is denoted the (55/50) algorithm and it provides the optimum distribution of ratios of Hz (estimatel/f/g such that the number of estimates of Hz that are less than 0.9 Hz are minimal. That is, the (55/50) algorithm best avoids underestimating Hz by more than 10 percent, but at the cost of sometimes allowing larger overestimates of Hz than the (70/30) algorithm. A commercial nonlinear optimization method (Quattro Pro ) was utilized to obtain... 

Curvilinear regression should not be confused with the nonlinear regression methods used to estimate model parameters expressed in a nonlinear form. For example, the model parameters a and b in y = axb cannot be estimated by a linear least-squares algorithm. Information in Chapter 7 describes nonlinear approaches to use in this case. Alternatively, a transformation to a linear model can sometimes be used. Implementing a logarithmic transformation on yt = ax/ produces the model log y = log a + blog x which can now be utilized with a linear least-squares algorithm. The literature [4, 5] should be consulted for additional information on linear transformations. 

The alternative method consists of the proteolysis of the protein-antibody complex. Mass spectrometry allows those sites to be observed that are protected against enzymatic cleavage. Such sites are implied in the protein-antibody interaction. This method allows identification of nonlinear epitopes, whereas the first method is limited to linear epitopes. 

The bond graph method defines the structure and constitutive equations of the system. Standard bond graph elements are used to build a model of the structure of the system. Suitable computer programs are available to generate the governing equations, and alternative methods have also been developed for deriving equivalent block diagrams, which can represent nonlinear systems. 

Conjugated polymers like poly(l,4-phenylene-vinylene), PPV, or more generally Poly(arylene-vinylenes), PAVs have evoked considerable interest as electrically conductive and nonlinear optical materials. More recently, electroluminescence properties of PPV have attracted substantial attention, since it was first reported in 1990. Direct synthesis of PPV has been limited by its insolubility. Hence, the most commonly used routes are based on soluble polymer precursors or soluble conjugated precursors. The latter process is also commonly referred to as the sulfonium-based polyelectrolyte precursor route. PPV thin films from these solution-based routes, however, have problems related to contamination by solvents and oxidative defects in the polymer. C VP is an alternate method for the deposition of high quality thin films of PPV. Reported first by Iwatsuki et al., it was investigated for electroluminescence applications by Staring et al. ... 

Burland et al. refer to the nonlinearity as x -2a< (M,(a) and their introduction of this quantity in the earlier part of their paper would lead to the conclusion that it is the standard susceptibility as defined in eqn (2.4), in which case the that they derive from it would be in the B convention. However, the S5mbol, F, subsequently appears and it is stated that x (—2linear plot of x —2co co,a)) versus w is shown from which the value of (dr z /dw)o in column A has been obtained. An alternative method of analysis of the macroscopic data for F, which does not involve an explicit calculation of the derivative, is discussed below. 

Among the nonlinear methods, there are, besides nonlinear least squares regression, i.e. polynomial regression, the nonlinear PLS method. Alternating Conditional Expectations ACE), SMART, and MARS. Moreover, some Artificial Neural Networks techniques have also to be considered among nonlinear regression methods, such as the back-propagation method. 

The set of differential and algebraic equations given above for modeling multicomponent distillation in a packed column must be integrated numerically in general. The complexity and nonlinearity of the above equations precludes analytical solution in most cases of practical importance. Moreover, because the vapor and liquid streams flow in opposite directions means that, in all but one circumstance—total reflux—several integrations may be required in order to properly solve the equations. An alternative method of solving approximate forms of these equations is discussed in Chapter 14. 

Various extensions of linear state-space approach have been proposed for developing nonlinear models [227, 274]. An extension of linear CVA for finding nonlinear state-space models was proposed by Larimore [160] where use of alternating conditional expectation (ACE) algorithm [24] was suggested as the nonlinear CVA method. Their examples used linear CVA to model a system by augmenting the linear system with pol3momials of past outputs. 

An alternative method of solving nonlinear simultaneous equations within a simulation is based on the properties of equation (2.93). Since the vector function, g, is constant (at zero) throughout all time, it follows that its time differential is also zero at all times ... 

An alternative method of solving the implicit equations associated with complex flow networks is to use the Method of Referred Derivatives discussed in Section 2.11 of Chapter 2. The general set of nonlinear, simultaneous equations... 

Another nonlinear technique that is potentially applicable to thermometric measurements is DPWM [7,9]. Por instance, a Boltzmann plot constructed out of the relative line intensities of a DPWM spectrum can lead to temperature predictions that can be as accurate as CARS in some cases. An alternative method is to fit theoretical simulations to the experimental spectrum. Nonetheless, the versatility of CARS is not equaled by DPWM. In effect, single pulse measurements seems to be limited to some radical species and mode fluctuations of conventional lasers perturb the data severely. To avoid troubles with such laser intensity fluctuations, saturated DPWM is often employed, but the difficulties of spectral S5mthe-sis remain a serious hindrance to a major role of DPWM thermometry. 

The preceding sections outline the three most commonly implemented and currently practical methods for computing nonlinear optical properties. But, as discussed, each of those methods has drawbacks, and a great deal of effort is being expended in the quantum chemistry community to develop alternate methods that are more accurate and more efficient. This section briefly introduces a few of the approaches that currently look most promising. We introduce the reader to these methods but do not provide a detailed discussion. 

While SHAKE/RATTLE is a robust procedure the sequential, iterative character of nonlinear solvers needed for its implementation can lead inefficiencies when it is implemented on a parallel computer. Therefore, a number of the alternative methods avoid iteration. We briefly summarize a few of the popular alternative schemes here. 

With measurements made in a differential reactor, the values of are obtained directly. The important step of a differential data evaluation procedure consists of a comparison between the experimental data and the hypothesis of Equ. 2.53. Usually, this is done graphically using a linearization procedure (see Sect. 2.4), as is shown in Fig. 4.17 for the general case. The function f c) represents the mathematical function that was postulated in constructing the model. With the computer, nonlinear regression methods are also a realistic alternative. 

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