Error function properties

In this chapter, the elution curve equation and the plate theory will be used to explain some specific features of a chromatogram, certain chromatographic operating procedures, and some specific column properties. Some of the subjects treated will be second-order effects and, therefore, the mathematics will be more complex and some of the physical systems more involved. Firstly, it will be necessary to express certain mathematical concepts, such as the elution curve equation, in an alternative form. For example, the Poisson equation for the elution curve will be put into the simpler Gaussian or Error function form. 

In recent years some theoretical results have seemed to defeat the basic principle of induction that no mathematical proofs on the validity of the model can be derived. More specifically, the universal approximation property has been proved for different sets of basis functions (Homik et al, 1989, for sigmoids Hartman et al, 1990, for Gaussians) in order to justify the bias of NN developers to these types of basis functions. This property basically establishes that, for every function, there exists a NN model that exhibits arbitrarily small generalization error. This property, however, should not be erroneously interpreted as a guarantee for small generalization error. Even though there might exist a NN that could... 

In view of the fact that recent parameterisations make use of reference data from high-level calculations, the corresponding error functions used to develop these methods can in principle involve any given property that can be calculated. Thus, in addition to structural information, the error function can involve atomic charges and spin densities, the value for the wavefunction, ionisation potentials and the relative energies of different structures within the reference database [26, 32], Detailed information concerning the actual wavefunction can be extremely useful for... 

Here, erfcx is the error function complement of x and ierfc is its inverse. The physical properties are represented by a, the thermal diffusivity, which is equal to K/pCp, where k is the thermal conductivity, p is the density and Cp, the specific heat capacity at constant pressure. The surface temperature during this irradiation, Ts, at x = 0, is therefore... 

Note that s is a dummy variable the value of the integral depends only on the value of the upper limit. Tables of the error function are available and values can be calculated from power series [Dwight (1961), Kreyszig (1988)]. The error function has the properties erf(0) = 0 and erf(°°) = 1. Equation 10.31 can be written in terms of the error function as... 

Substituting the boundary conditions, using the properties of the error function given above, shows that —A = B = u0. The velocity field is therefore... 

Using the property of the error function listed in Appendix 8B... 

The solutions of a diffusion equation under the transient case (non-steady state) are often some special functions. The values of these functions, much like the exponential function or the trigonometric functions, cannot be calculated simply with a piece of paper and a pencil, not even with a calculator, but have to be calculated with a simple computer program (such as a spreadsheet program, but see later comments for practical help). Nevertheless, the values of these functions have been tabulated, and are now easily available with a spreadsheet program. The properties of these functions have been studied in great detail, again much like the exponential function and the trigonometric functions. One such function encountered often in one-dimensional diffusion problems is the error function, erf(z). The error function erf(z) is defined by... 

In addition, the properties for the two sections of the burner are smoothed across the interface using an error function in order to avoid a discontinuity. 

The measurements of the local properties of two-phase systems during cultivation indicate that radial profiles of ds are fairly uniform. Also, their longitudinal variations are fairly moderate, except in the neighborhood of the aerator (1, 4). The same holds true for the spacial variations of the local relative gas holdups. At low superficial gas velocities the specific interfacial area, a, is fairly uniform also At high superficial gas velocities (turbulent or heterogeneous flow range) the radial profile of a has a shape of an error function, with its maximum in the column center (5). The behavior of these parameters near the aerator depends on the aerator itself and on the medium character. 

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