Rational Nonlinearity

In a number of models, nonlinearity is introduced as rational terms. Michaelis-Menten reaction kinetics also leads to this type of nonlinearity. Some examples from the literature are given below. 

Abstract models Rossler (those not included in Product Nonlinearity section above.) 

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It is, however, necessary that we try to weave some rational web that allows us to describe a rational nonlinear theory. The general formulation of Green and Rivlin [G16] took the position that the stress tensor is a general hereditary function of the strain history. The stress tensor was expressed as infinite series of integrals ... 

As a summary of nonlinear behavior, it appears possible to eliminate the nonlinear behavior, and at the same time, you typically do not want to operate in that nonlinear behavior regime anyway, so you are both able to, and want to, design out nonlinear behavior. That observation is true generally in aircraft structures, but there are other structures, which are subjected to higher temperatures, for which you simply cannot avoid some of the nonlinear behavior aspects, so you must take them into account in any rational design analysis. 

These structure-function relationships provide extremely useful guidance for the future rational design of molecules and polymers with even higher optical nonlinearities. For non-centrosymmetric molecules such as 95, very high first hyperpolarizabilities /3 that determine the second-order nonfinear optical properties were also measured [140]. 

The above experimental design constitute an excellent set of "preliminary experiments" for nonlinear models with several unknown parameters. Based on the analysis of these experiments we obtain estimates of the unknown parameters that we can use to design subsequent experiments in a rational manner taking advantage of all information gathered up to that point. 

In 1937 Jahn and Teller applied group-theoretical methods to derive a remarkable theorem nonlinear molecules in orbitally degenerate states are intrinsically unstable with respect to distortions that lower the symmetry and remove the orbital degeneracy.37 Although Jahn-Teller theory can predict neither the degree of distortion nor the final symmetry, it is widely applied in transition-metal chemistry to rationalize observed distortions from an expected high-symmetry structure.38 In this section we briefly illustrate the application of Jahn-Teller theory and describe how a localized-bond viewpoint can provide a complementary alternative picture of transition-metal coordination geometries. 

Here C is the concentration vector and D(C) is the diffusivity tensor defined by (3.1.15a). Thus, locally electro-neutral electro-diffusion without electric current is exactly equivalent to nonlinear multicomponent diffusion with a diffusivity tensor s being a rational function of concentrations of the charged species. 

The second important aspect in thermodynamic studies is the determination of the enthalpy. A knowledge of the thermochemistry of epoxy-amine interactions is important also as a prerequisite for rational curing processes as manufacturing methods. The solution of this problem is also important for the application of the calorimetric method to the kinetic investigations. In fact, in the case of reactions with continuously varying concentrations of the donors and acceptors, the observed heat release (Q) may depend nonlinearly on conversion (a) as of the general case... 

The first type, which includes, for example, the problem of strong explosion or propagation of heat in a medium with nonlinear thermal conductivity [3], is characterized by the fact that the exponents are found from physical considerations, from the conservation laws and their dimensionality. In addition, the exponents turn out to be rational numbers. The task of the calculation is to find the dimensionless functions by integration of ordinary differential equations. After this the problem is completely solved, since the numerical constants are determined by normalizing the solution to the conserved quantity (the total energy released in these examples). 

Two of the most important nonlinear optical (NLO) processess, electro-optic switching and second harmonic generation, are second order effects. As such, they occur in materials consisting of noncentrosymmetrically arranged molecular subunits whose polarizability contains a second order dependence on electric fields. Excluding the special cases of noncentrosymmetric but nonpolar crystals, which would be nearly impossible to design from first principles, the rational fabrication of an optimal material would result from the simultaneous maximization of the molecular second order coefficients (first hyperpolarizabilities, p) and the polar order parameters of the assembly of subunits. (1)... 

Pharmacokinetic models. An important advance in risk assessment for hazardous chemicals has been the application of pharmacokinetic models to interpret dose-response data in rodents and humans (EPA, 1996a Leung and Paustenbach, 1995 NAS/NRC, 1989 Ramsey and Andersen, 1984). Pharmacokinetic models can be divided into two categories compartmental or physiological. A compartmental model attempts to fit data on the concentration of a parent chemical or its metabolite in blood over time to a nonlinear exponential model that is a function of the administered dose of the parent. The model can be rationalized to correspond to different compartments within the body (Gibaldi and Perrier, 1982). 

A very rich scenario was, on the other hand, observed with Pt(110) if perturbed under conditions for which regular autonomous oscillations with well-defined frequencies existed (91-93). Typically, these autonomous oscillations were established at fixed external parameters and then one of the partial pressures was periodically modulated by use of a feedback-regulated gas inlet system with frequencies up to 0.5 s l and relative amplitudes around 1% (31, 33). Following the pioneering mathematical treatment of forced nonlinear oscillations by Kai and Tomita (SO), the results can be rationalized in terms of a dynamic phase diagram characterizing the response of the system as a function of the amplitude A and of the period of the pressure modulation Tcx with respect to that of the... 

The development of highly active third-order nonlinear optical materials is important for all-optical signal processing. In contrast to second-order nonlinear optical molecular systems, there are few rational strategies for optimizing the third-order nonlinear optical response of molecular materials. Unlike second-order materials, there exist no molecular symmetry restrictions for the observation of a third-order nonlinear optical response. It is the instantaneous... 

It should be emphasized that for this approach to be successful, several data points per peak and a relatively high SNR are required. Under these conditions, a fit by a rational function provides a robust method for obtaining starting values for the nonlinear least-squares fitting techniques listed in... 

A minor variant of the SI system, called MKS by experimentalists in nonlinear optics, is called SI here and defined below. Alas, there are also other systems "rationalized" or "unrationalized", Heaviside-Lorentz, "atomic" (where me = c = h = 1), and so on. The "rationalized" and "unrationalized" versions differ in how they apportion the pesky factor An (surface area of a sphere with unit radius that is involved in surface integrals) between the various electrical and the magnetic variables. 

A second prominent feature here is the ergodic character (or lack thereof) of the process, depending on the rationality or irrationality of <. This leads inevitably to the fascinating question, Does a real system choose between these values of , and if so, how The boundaries themselves remain neutral with respect to the choice of whenever they are compatible with the flow. Thus, for a slide flow, the walls must be parallel to the slides, whereas for a tube flow, they must be parallel to the tube. In both cases there remains an additional degree of freedom, which is precisely the choice of f. Other examples of indeterminancy arise from the neglect of fluid and particle inertia, as already discussed in Section I (see also the review in Leal, 1980). Whether or not inclusion of nonlinear inertial effects can remove the above indeterminacy, as it often does for the purely hydrodynamic portion of the problem, is a question that lies beyond the scope of the present (linear) Stokesian context. 

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