Level nonlinear response, form

Distillation columns that produce products with parts-per-million (ppm) levels of impurities offer some challenging control problems. These columns exhibit very nonlinear responses to changes in manipulated variables and disturbances. The nonlinear effects take two forms nonlinear steady-state gains and nonlinear dynamics. 

The second step of the dose-response assessment is an extrapolation to lower dose levels, i.e., below the observable range. The purpose of low-dose extrapolation is to provide as much information as possible about risk in the range of doses below the observed data. The most versatile forms of low-dose extrapolation are dose-response models that characterize risk as a probability over a range of environmental exposure levels. Otherwise, default approaches for extrapolation below the observed data range should take into account considerations about the agent s mode of action at each tumor site. Mode-of-action information can suggest the likely shape of the dose-response curve at these lower doses. Both linear and nonlinear approaches are available. 

Pharmacokinetic models involving nonlinear kinetics of the Michaelis-Menten form have the important extrapolation characteristic of being linear at low dose levels. This low dose linearity contrasts with the low dose nonlinearity of the multihit and Weibull models. Each model, pharmacokinetic, multihit, and Ifeibull, has the desirable ability to describe either convex (upward curvature) or concave (downward curvature) dose-response relationships. Other models, stich as the log normal or multistage, are not consistent with concave relationships. However, the pharmacokinetic model differs from the multihit and Heibull in that it does not assume the nonlinear behavior observed at high dose levels will necessarily correspond to the sane nonlinear behavior at low dose levels. 

Bioimmittance is frequency dependent. In dielectric or electrolytic models there is a choice between a step (relaxational) and sinusoidal (single-frequency) waveform excitation. As long as the step response waveform is exponential and linear conditions prevail, the information gathered is the same. At high voltage and current levels, the system is nonlinear, and models and parameters must be chosen with care. Results obtained with one variable cannot necessarily be recalculated to other forms. In some cases, one single pulse may be the best waveform because it limits heat and sample destruction. 

Cascading. In most cases, the distinction between second- and third-order nonlinearities is evident from the different phenomena each produce. That distinction blurs, however, when one considers the cascading of second-order effects to produce third-order nonlinear phenomena (51). In a cascaded process, the nonlinear optical field generated as a second-order response at one place combines anew with the incident field in a subsequent second-order process. Figure 2 shows a schematic of this effect at the molecular level where second-order effects in noncentrosymmetric molecules combine to yield a third-order response that may be difficult to separate from a pure third-order process. This form of cascading is complicated by the near-field relationships that appear in the interaction between molecules, but analysis of cascaded phenomena is of interest, because it provides a way to explore local fields and the correlations between orientations of dipoles in a centros5nnmetric material (52). 

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