Nonlinear dependence

Figure Bl.5.2 Nonlinear dependence of tire polarization P on the electric field E. (a) For small sinusoidal input fields, P depends linearly on hence its hannonic content is mainly tiiat of E. (b) For a stronger driving electric field E, the polarization wavefomi becomes distorted, giving rise to new hannonic components. The second-hamionic and DC components are shown.

Surface tensions for aqueous solutions are more difficult to predict than those for nonaqueous mixtures because of the nonlinear dependence on mole fraction. Small concentrations of the organic material may significantly affect the mixture surface tension value. For many binary organic-water mixtures, the method of Tamura, Kurata, and Odanfi maybe used ... 

From the experimental results and theoretical approaches we learn that even the simplest interface investigated in electrochemistry is still a very complicated system. To describe the structure of this interface we have to tackle several difficulties. It is a many-component system. Between the components there are different kinds of interactions. Some of them have a long range while others are short ranged but very strong. In addition, if the solution side can be treated by using classical statistical mechanics the description of the metal side requires the use of quantum methods. The main feature of the experimental quantities, e.g., differential capacitance, is their nonlinear dependence on the polarization of the electrode. There are such sophisticated phenomena as ionic solvation and electrostriction invoked in the attempts of interpretation of this nonlinear behavior [2]. 

Nonlinear dependence between ee values of chiral auxiliary compound or ligand and product of reaction with participation or formation of heterocycle 98AG(E)2922. 

While Eq. (36) is valid for 9 = 1, a qualitatively similar equation is obtained at any value of 9. Since the condition 9 - 1 is difficult to reach experimentally, the value of AEaB0 (adsorption potential shift) is often estimated by means of extrapolation to 9 = 1. This procedure is very delicate and the result is often misleading. The variation of EOm0 with 9 may be linear or nonlinear, depending on lateral interactions between... 

Fig. 1.10. The nonlinear dependence of inverse correlation time (1/t,) on gas density n (k = zjz = nyzc) in comparison with impact relaxation rate 1/t linear in n.

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. 

A case in point is a nonlinear dependence of the coefficient of heat conductivity upon the temperature. From the formula for D it is easily seen that we formally have D = oo for the linear heat conductivity when (7 = 0 meaning that the velocity of heat conducting turns out to be infinite. 

GENERALIZATION TO ODE MODELS WITH NONLINEAR DEPENDENCE ON THE PARAMETERS... 

Fig. 10 Urinary excretion of riboflavin (A, B) and ascorbic acid (C, D) in humans as a function of oral dose. Graphs A and C illustrate the nonlinear dependence of absorption on dose, which is suggestive of a saturable specialized absorption process. Graphs B and D represent an alternative graph of the same data and illustrate the reduced absorption efficiency as the dose increases. (Graphs A and C based on data in Ref. 39 and graphs B and D based on data in Ref. 40.)...

Kubinyi, H., Quantitative structure-activity relationships. IV. Nonlinear dependence of biological activity on hydrophobic character a new model, Arzneim. Forsch. (Drug Res.) 26, 1991-1997 (1976). 

The nature of the optimization problem can mm out to be linear or nonlinear depending on the mass transfer model chosen14. If a model based on a fixed outlet concentration is chosen, the model turns out to be a linear model (assuming linear cost models are adopted). If the outlet concentration is allowed to vary, as in Figure 26.35a and Figure 26.35b, then the optimization turns out to be a nonlinear optimization with all the problems of local optima associated with such problems. The optimization is in fact not so difficult in practice as regards the nonlinearity, because it is possible to provide a good initialization to the nonlinear model. If the outlet concentrations from each operation are initially assumed to go to their maximum outlet concentrations, then this can then be solved by a linear optimization. This usually... 

Even nowadays, a DNS of the turbulent flow in, e.g., a lab-scale stirred vessel at a low Reynolds number (Re = 8,000) still takes approximately 3 months on 8 processors and more than 17 GB of memory (Sommerfeld and Decker, 2004). Hence, the turbulent flows in such applications are usually simulated with the help of the Reynolds Averaged Navier- Stokes (RANS) equations (see, e.g., Tennekes and Lumley, 1972) which deliver an averaged representation of the flow only. This may lead, however, to poor results as to small-scale phenomena, since many of the latter are nonlinearly dependent on the flow field (Rielly and Marquis, 2001). 

The nonlinear dependence of the reaction rate on the partial pressure of CO suggests that there are stimulated either a single reaction step with the established nonlinearity or at least two steps, one of which has a linear and the other one a nonlinear (nperiodic operation of the reactor at the 180°C level and at a middle oxidation state (pretreatment at p /p = 5,0).has been accomplished (see Figure 9) with a H fl/N testing mixture followed by a C0/N recuperation mixture. After an intermediate activity of the catalyst has been attained by this periodic operation, CO has been added in the testing mixture. 

Finally, Fig. 5.37 displays the bond-order-bond-length relationship for O- H and H H bonds over the entire range of the proton-transfer reaction. Both curves display nonlinear dependences tending to Rm oo as ab- 0, and to the standard single-bond distance as The points for the H-bonded complexes (those... 

The separation of synthetic red pigments has been optimized for HPTLC separation. The structures of the pigments are listed in Table 3.1. Separations were carried out on silica HPTLC plates in presaturated chambers. Three initial mobile-phase systems were applied for the optimization A = n-butanol-formic acid (100+1) B = ethyl acetate C = THF-water (9+1). The optimal ratios of mobile phases were 5.0 A, 5.0 B and 9.0 for the prisma model and 5.0 A, 7.2 B and 10.3 C for the simplex model. The parameters of equations describing the linear and nonlinear dependence of the retention on the composition of the mobile phase are compiled in Table 3.2. It was concluded from the results that both the prisma model and the simplex method are suitable for the optimization of the separation of these red pigments. Multivariate regression analysis indicated that the components of the mobile phase interact with each other [79],... 

If there is a nonlinear dependence of kapp on [Lq] then the reaction cannot be a simple bimolecular process. However, the extended equation described inO Section 2.6.1 will give rise to a hyperbolic curve as shown in Figure 10-5b, where Ki is given by k i/ki, and kapp is described by ... 

Figure 18.2. Nonlinear dependence of volume on molality for dilute solutions of sulfuric acid (H2SO4) in water (H2O). Based on data of I. M. Klotz and C. F. Eckert, J. Am. Chem. Soc. 64, 1878 (1942).

Thermodynamic state functions change with temperature this will be true if values of the heat capacity of any component is nonzero (which is almost always true). Whenever the heat capacity is not a constant, the various thermodynamic state functions will show nonlinear dependencies on temperature. 

Depressed catalytic activity occurring when an inhibitor binds more than once to a single enzyme form (or forms). While standard double-reciprocal plots are usually linear, secondary replots of the data (i.e., plots of slopes and/or intercepts vx. [I], the concentration of the inhibitor) will be nonlinear depending on the relative magnitude of the [I], [If,. .., and [If terms in the rate expression. 

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