Gauss error function

TABLE 7.12 Practical Modifications of the Gauss Error-Function Velocity ... 

Application of the Gauss error-function equation for velocity profile in the form proposed by Shepelev (Table 7.12) in Eq. (7.39) results in the following formula for the centerline velocity in Zone 3 of the compact jet ... 

Air velocity in each jet cross-section, described using the Reichardt Gauss error-function profile... 

The equation for the centerline temperature differential in Zone 3 of the compact jet derived" from Eq. (7.61) using the Gauss error-function temperature profile (Table 7.14) is... 

We consider a wave packet of finite width. For the sake of simplicity we represent its amplitude at any moment by a Gauss error function (such as actually occurs in the ground state of the harmonic oscillator, Appendix XXXII, p. 343) ... 

This is based on the arbitrary assumption, discussed further below, that the distribution of combining energies (which are proportional to log K) in a normal antibody population obeys a Gauss error function. Ko, the average K value, is that represented by the peak of the distribution curve. 

Studies described in detail in Chapter 10 have indicated that many, if not most, specifically purified antibody populations contain significant amounts of homogeneous subpopulations. It is therefore evident that the Gauss error function must provide an inaccurate picture of the actual distribution of K values. Attempts have therefore been made to analyze experimental data in terms of the real distributions (79,80). 

The method is of particular value for demonstrating the presence of a bimodal distribution, with peaks of concentrations of molecules of high and low affinity. (The Gauss error function assumes a single peak with decreasing concentration of molecules on each side.) Werblin and Siskind used their computational techniques to demonstrate the continued presence of low affinity antibody, up to one year after initial immunization, coexisting with antibody of high affinity present after the first few weeks. As with other techniques, the lower limit of affinity detectable is dependent on the antibody concentration. 

This is just a constant times a Gauss error function or sinq)le gaussian-type function. This wavefunction is sketched in Fig. 3-2 and is obviously symmetric. 

Error function relates to the normal density of probability (pdf)/x(x) of Gauss. fx(x) is given by... 

The number of reflection intensities measured in a crystallographic experiment is large, and commonly exceeds the number of parameters to be determined. It was first realized by Hughes (1941) that such an overdetermination is ideally suited for the application of the least-squares methods of Gauss (see, e.g., Whittaker and Robinson 1967), in which an error function S, defined as the sum of the squares of discrepancies between observation and calculation, is minimized by adjustment of the parameters of the observational equations. As least-squares methods are computationally convenient, they have largely replaced Fourier techniques in crystal structure refinement. 

Given the n observations, our aim is to obtain the best estimates X of the m unknown parameters to be determined. Gauss proposed the minimization of the sum of the squares of the discrepancies, defining the error function S, which, after assignment of a weight w, to each of the observations, is given by... 

Values of the error function and its first derivative (the Gauss distribution) are tabulated1. 

The function e a< x z ), is called a Gauss error curve, having been used by Gauss to describe distribution functions similar to our /( ) in the theory of errors. It equals unity when x = x0, and falls off on both sides of this point symmetrically, being reduced to 1 /e when... 

This is Gauss s function, whose value is getting near to value 1, when the difference is growing. Parameter cr is a mean-root-square-error from normal sorting. 

The choice of the specific orthogonal polynomial is determined by the convergence. If the signal to be approximated is a bell-shaped function, it is evident to use a polynomial derived from the Gauss function, i.e. one of the so-called classical polynomials, the Hermite polynomial. Widely used is the Chebyschev polynomial one of the special features of this polynomial is that the error will be spread evenly over the whole interval. 

Computational the most demanding task is locating the minimum of the function (3.89) at step (ii). Since the Gauss-Newtan-Marquardt algorithm is a robust and efficient way of solving the nonlinear least squares problem discussed in Section 3.3, we would like to extend it to error—in-variables models. First we show, however, that this extension is not obvious, and the apparently simplest approach does not work. 

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