Small nonlinear terms, effect

In both the linear and the nonlinear cases the total variation of the residuals is the sum of the random error, plus the departure from linearity. When the data is linear, the variance due to the departure from nonlinearity is effectively zero. For a nonlinear set of data, since the X-difference between adjacent data points is small, the nonlinearity of the function makes minimal contribution to the total difference between adjacent residuals and most of that difference contributing to the successive differences in the numerator of the DW calculation is due to the random noise of the data. The denominator term, on the other hand, is dependent almost entirely on the systematic variation due to the curvature, and for nonlinear data this is much larger than the random noise contribution. Therefore the denominator variance of the residuals is much larger than the numerator variance when nonlinearity is present, and the Durbin-Watson statistic reflects this by assuming a value less than 2. 

The basic remark is that linearity of the macroscopic law is not at all the same as linearity of the microscopic equations of motion. In most substances Ohm s law is valid up to a fairly strong field but if one visualizes the motion of an individual electron and the effect of an external field E on it, it becomes clear that microscopic linearity is restricted to only extremely small field strengths.23 Macroscopic linearity, therefore, is not due to microscopic linearity, but to a cancellation of nonlinear terms when averaging over all particles. It follows that the nonlinear terms proportional to E2, E3,... in the macroscopic equation do not correspond respectively to the terms proportional to E2, E3,... in the microscopic equations, but rather constitute a net effect after averaging all terms in the microscopic motion. This is exactly what the Master Equation approach purports to do. For this reason, I have more faith in the results obtained by means of the Master Equation than in the paradoxical result of the microscopic approach. 

The terms beyond odE are not linear in E they are referred to as the nonlinear polarization and give rise to nonlinear optical effects. Also note that at small fields the polarization will more nearly approximate a linear response however, with increasing field strength, nonlinear effects become more important. Since a p, y, there were few observations of NLO effects before the invention of the laser with its associated large electric fields. 

Nonlinear optical effects can be introduced into this picture by postulating that the restoring force in equation 1 is no longer linear in the displacement and adding a term, say ar2, to the left hand side of the equation, (3). The differential equation can no longer be solved in a simple way but, if the correction term is assumed to be small relative to the linear term, a straightforward solution follows leading to a modification of equation 3. 

The use of electrochemical data for the actual molecule can accommodate some of the effects of covalency. In general, the observations on LMCT absorptions in the ammine complexes suggest that nonlinear, or cross-term effects make only small contributions to the transition energies. Thus, the absorption maxima of the... 

In isotropic media, only the odd-order susceptibilities are nonzero because of inversion symmetry, hence the lowest order, nonlinear term is the third-order susceptibility. If (as in the gas phase) the number density N is low enough so that local field effects are small, then the macroscopic nonlinear polarization induced by the laser field is simply and this... 

Sen and Davis (20) have performed an interesting analysis of capillary flow in a slot heated from the side (a half zone) by using the matching procedure suggested by Cormack, Leal and Imburger (26) for natural convection in a slot. In this way they take end-effects into account. However, they assume that the nonlinear terms in the momentum equation can be neglected and perform an analysis for small values of d/ , which they use as their perturbation parameter. 

Huid turbulence has had many descriptions. It is random, chaotic, dissipative, and multiple scaled. The turbulent flows describable by the Navier- tokes equations present these properties when the nonlinear terms, which represent the convective effect of fluid motion, become relatively large compared with the other terms, such as the viscous forces. The Re5molds number can be regarded as one such measure for the ratio. At small Reynolds numbers, or when the viscous effects dominate the nonlinearity in the system, the solutions of the Navier-Stokes equations are regular and smooth, a state commonly referred to... 

Another simple approach assumes temperature-dependent AH and AS and a nonlinear dependence of log k on T (123, 124, 130). When this dependence is assumed in a particular form, a linear relation between AH and AS can arise for a given temperature interval. This condition is met, for example, when ACp = aT" (124, 213). Further theoretical derivatives of general validity have also been attempted besides the early work (20, 29-32), particularly the treatment of Riietschi (96) in the framework of statistical mechanics and of Thorn (125) in thermodynamics are to be mentioned. All of the too general derivations in their utmost consequences predict isokinetic behavior for any reaction series, and this prediction is clearly at variance with the facts. Only Riietschi s theory makes allowance for nonisokinetic behavior (96), and Thorn first attempted to define the reaction series in terms of monotonicity of AS and AH (125, 209). It follows further from pure thermodynamics that a qualitative compensation effect (not exactly a linear dependence) is to be expected either for constant volume or for constant pressure parameters in all cases, when the free energy changes only slightly (214). The reaction series would thus be defined by small differences in reactivity. However, any more definite prediction, whether the isokinetic relationship will hold or not, seems not to be feasible at present. 

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