Non-Linear Functions

R. J. Kelly and R. E. Wheeling, A Digital Computer Program fior Optimifing Non-linear Functions, Mobil Oil Corp., Research Dept., Central Research... 

The "add-to-memory" signal averaging method currently available to us distorts fluorescence intensity versus time plots when the fluorescence intensity is a non-linear function of incident laser energy and the laser energy varies from shot to shot. For this reason we have not attempted detailed kinetic modelling of the observed fluorescence intensity decay curves recorded at high 532 nm laser fluence. 

Where k is a non-linear function of T, some mean value, k, will apply, where ... 

By comparing equations 1-4 and 9-11, one can see a one-to-one correspondence between terms in the mathematical description and the numerical program. The flexibility of TOPAZ derives from the fact that the parameters of equations 9-11 (p, Cp, kx, ky, qg, P, y) can be linear or non-linear functions of spatial coordinates, time, and temperature. 

Another approach is to transform the original columns of a measurement table by suitable non-linear functions in order to obtain some desirable effect. This is... 

The acceptance is generally a non-linear function of perceived intensity. A simple example is the salt level in a soup which clearly has a level of maximum acceptability between too weak and too salty a taste. 

Powell, M.J. D., "A Method of Minimizing a Sum of Squares of Non-linear Functions without Calculating Derivatives", Computer J., 7,303-307 (1965). 

Kaiser, K.L.E. (1983) A non-linear function for the calculation of partition coefficients of aromatic compounds with multiple chlorine substitution. Chemosphere 12, 1159-1165. 

The relationship between spot sales price and spot sales quantity can be modeled as a linear function within the feasible minimum and maximum quantities defined by the management of the company. Of course, the price-quantity relationship could also be modeled using a non-linear function depending on the actual price-quantity-bids the company receives. In this work the linear function showed a sufficient statistical fit based on the real data provided by the industry case. 

For a review on linear approximation for non-linear functions see Kallrath (2002b), p. 125. 

The last term on the right-hand side of (1.28) is the chemical source term. As will be seen in Chapter 5, the chemical source term is often a complex, non-linear function of the scalar fields , and thus solutions to (1.28) are very different than those for the z nm-scalar transport equation wherein S is null. 

The chemical-source-term closures described in this chapter do not require that the chemical source term be derived from a set of elementary reactions as done here. Indeed, closure difficulties will result whenever S() is a non-linear function of . 

On the other hand, the expected values of all non-linear functions of velocity and/or scalar gradients require two-point information, and thus cannot be found from /u,4>(V, 0 x, t). For example, the scalar dissipation rate involves the square of the scalar gradients ... 

The same note of caution applies to strongly non-linear functions j as in the case of a single random variable. 

When linear regression does not yield a good correlation, application of a non-linear function may be feasible (see Chapter 10). The parameter estimates for higher-order or polynomial equations may prove to be more difficult to interpret than for a linear relationship. Nevertheless, this approach may be preferable to using lower-order levels of correlation (B or C) for evaluating the relationship between dissolution and absorption data. 

In many instances non-linear functions can be linearised and in this way a non-linear, iterative fitting procedure can be reduced to an explicit linear fit. A typical example is the exponential decay of the intensity of the emission of a radioactive sample. We use the data already used for Figure 4-4, produced by the function Data Decay. m. 

Non-linear curves may be treated using Equation 9 directly, using the techniques of non-linear least squares, when appropriate. (Note that a non-linear calibration curve does not necessarily imply non-linear least squares. The latter is necessary only if the problem is non-linear in the estimated parameters (16). For example, y = a+bx+cx and y = a+bx are both non-linear functions, but only the latter is non-linear in the parameters.)... 

Assuming the value of r to be fixed, Eq. (24) is linear with respect to c and w. The constrained optimal values of these two parameters, denoted as Cl and Wx, are therefore easily determined using standard linear correlation formulae. Since Ci and Wi depend on r, this procedure defines the non-linear function Qi(r) = Q(ci(r), Wi(r), r). 

The retention time tj is a non-linear function of the ring size n and thus allows the identification of new species of the homologous series S by interpolation. Furthermore, the logarithm of the capacity factor k = (t — (with = death time of the chromatographic system) is a linear function of the ring size, n, although two such functions are obtained see Fig. 7... 

The non-linearity of the force-displacement relationship between a tip and a sample is sketched in Fig. 13.3(a). The force is a non-linear function of the tip-sample displacement, and depends on whether approach or retraction is under way. If the tip and the sample are well separated, at the far right on the graph, there is negligible interaction. During the approach, from right to left on the graph, initially there is an attractive interaction. At closer approach the force becomes repulsive. When reversing the displacement, the tip and the surface adhere until the contact is broken at a certain pull-off distance, which... 

In the vicinity of the stationary point n non-linear functions Fl in (2.1.1) could be expanded into the Taylor series... 

All local concentrations C of particles entering the non-linear functions F in equation (2.1.40) are taken at the same space points, in other words, the chemical reaction is treated as a local one. Taking into account that for extended systems we shouldn t consider distances greater than the distinctive microscopic scale Ao, the choice of equation (2.1.40) means that inside infinitesimal volumes vo particles are well mixed and their reaction could be described by the phenomenological reaction rates earlier used for systems with complete reactant mixing. This means that Ao value must exceed such distinctive scales of the reaction as contact recombination radius, effective radius of a dynamical interaction and the particle hop length, which imposes quite natural limits on the choice of volumes v0 used for averaging. 

It is possible to determine the characteristics of more complex non-linear functions and to derive their corresponding describing functions by adding and subtracting the describing functions of simpler functions. For example, the characteristic of... 

Nile Red was recently introduced as a solvatochromic dye for studying supercritical fluids (10). Although not ideal, Nile Red does dissolve in both nonpolar and polar fluids and does not lose its color in the presence of acids, like some previously used dyes. Major criticisms of Nile Red include the fact that it measures several different aspects of "polarity" simultaneously (polarizability and acidity (15)) yet it is insensitive to bases (10). However, in chromatography other single dimension polarity scales, like P, are routinely used. Measurements with Nile Red and other dyes indicate that the solvent strength of binary supercritical fluids is often a non-linear function of composition (10-14). For example, small... 

Figure 1. Solvent strength measured with two different solvatochromic dyes. On both scales solvent strength is a non-linear function of composition. Note that E q is more sensitive to hydrogen bonding than Ej. ...

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