Error function variations

A useful trial variational function is the eigenfunction of the operator L for the parabolic barrier which has the form of an error function. The variational parameters are the location of the barrier top and the barrier frequency. The parabolic barrierpotential corresponds to an infinite barrier height. The derivation of finite barrier corrections for cubic and quartic potentials may be found in Refs. 44,45,100. Finite barrier corrections for two dimensional systems have been derived with the aid of the Rayleigh quotient in Ref 101. Thus far though, the... 

Other Related Methods.—Baerends and Ros have developed a method suitable for large molecules in which the LCAO form of the wavefunction is combined with the use of the Xa approximation for the exchange potential. The method makes use of the discrete variational method originally proposed by Ellis and Painter.138 The one-electron orbitals are expanded in the usual LCAO form and the mean error function is minimized. 

A neural network is typically trained by variations of gradient descent-based algorithms, trying to minimize an error function [77]. It is important that additional validation data be left untouched during ANN training, so as to have an objective measure of the model s generalization ability [78],... 

Concentration, analytic solution Concentration, numerical solution Coefficient of variation Dimensionless diffusion coefficient Molecular diffusion coefficient, L /t Expectation operator Error function Shale fraction... 

An error functional Ajj is defined, related to approximate solutions of Eqs. (2), which is minimized with respect to variations of the coefficients a, of Eq. (4), on a discrete set of points fit with weight u>( ) in three-dimensional space ... 

To demonstrate the systematics of how an oscillatory time variation in the reflectivity is possible during dissolution, we assume that the surface is characterized by occupation factors described by an error function profile (Fig. 28A). These occupation factors can be thought of as blocks of orthoclase, so that occupation factors <1 represent a partially filled layer that is locally stoichiometric. The position of the error function moves continuously as the surface dissolves. For simplicity, we first assume that the interface width does not change during dissolution. The reflectivity is calculated as a function of the error function position. At the anti-Bragg condition, neighboring terraces are out of phase. The phase factor for each layer, n, varies as ( <9 ) = (-1) , so that interfacial structure factor becomes ... 

The Gaussian error function correctly describes the variations to be expected in a population of values of infinite size, an obviously unattainable level of replication in practice. With the general guidelines embodied in this function, a body of statistical rules applicable to small sets of data has been developed. 

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. 

Figure 7 The variation of tile interfacial -width as a function of annealing time, t, obtained from the discontinuous error functional fi m of the interfacial profile model used to fit a selection of the rrflecthity data obtained for the OSt-dPSj system (reproduced from reference [14]).

The relative errors or variational coefficients of cumulative distribution functions increase from zero at the maximum particle size to high values of more than a hundred percent at the minimum size class. 

The principle of the internal standard is based on the calculation of relative values, which are determined within the same analysis. One or more additional substances are introduced as a fixed reference parameter, the concentration of which is kept constant in the standard solutions and is always added to the analysis sample at the same concentration. For the calculation, the peak area values (or peak heights) of the substance being analysed relative to the peak area (or height) of the internal standard are used. In this way, potential volume errors and variations in the function of the instrument are compensated for and quantitative determinations of the highest precision are achieved (see also ISO 5725-6,1994). Standard deviations of less than 5% can be achieved with internal standardization. 

The tabulated values of error function were used in the work reported here. However, it is believed that the electrical analog can save time in developing a required solution where numerous initial and boundary conditions are to be tried, where these conditions are to vary with time, or where a variation of thermal properties is to be considered. 

Handbook of Heterogenous Kinetics A. 10.1.2. Variations of error function... 

Figure A.10.1. Variations of error function within range (0,co)...

Figure 9.1 shows a simple example of a data set (which has previously been considered) to illustrate some of the factors to be considered in this chapter. Shown is a data set with random errors but which has some obvious nonlinear functional variation of the Y variable with flie value of an assumed independent parameter, X in this case. The solid line curve represents a simple proposed functional relationship between the independent variable and flie dependent variable as indicated in the figure. In this chapter the emphasis will be primarily on estimating values of a set of parameters in the fitting model equation and secondarily on the characteristics of the random noise in the data. In the previous chapter, the primary emphasis was on the characteristics of flie random variations about some mean value. For the data set in Figure 9.1 it can be seen that the model curve is about half the time above the data points and about half the time below the data points as would be... 

There are problems to be considered and avoided when using Hquid-in-glass thermometers. One type of these is pressure errors. The change in height of the mercury column is a function of the volume of the bulb compared to the volume of the capillary. An external pressure (positive or negative) which tends to alter the bulb volume causes an error of indication, which may be small for normal barometric pressure variations but large when, for example, using the thermometer in an autoclave or pressure vessel. 

Assume Z and R to be constant for the expander. This assumption can lead to some minor errors in the calculation. However, if Pj is assumed constant then the variation for Z can be rolled into the new function of R, and the total error becomes even smaller. Now, Equation 7-4 can be simplified to ... 

To evaluate the required condenser area, point values of the group UAT as a function of qc must be determined by a trial and error solution of equation 9.181. Integration of a plot of qc against 1/17AT will then give the required condenser area. This method takes into account point variations in temperature difference, overall coefficient and mass velocities and consequently produces a reasonably accurate value for the surface area required. 

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