Error function values

Rms difference vs. refined structure (A) Distance error function value... 

TABLE 4.3 Look-up Table of Error Function Values... 

Two kinetic models, namely, pseudo-first-order and pseudo-second-order, were used to investigate the adsorption process of methyl orange, methyl blue and safranine T onto synthesised and commercial zeolite. Kinetic parameters along with correlation coefficient for the pseudo-second-order kinetic model are listed in Table A.3. The calculated correlation coefficient is closer to unity for the pseudo-second-order kinetic model than the pseudo-first-order kinetic model. Therefore, the sorption reaction can be approximated more favourably by the pseudo-second-order kinetic model for methyl orange, methyl blue and safranine T onto synthesised and commercial zeolite. MPSD error function values as shown in Table A.3 are also considerably lower for the pseudo-second-order kinetic model, reinforcing the applicability of the pseudo-second-order kinetic model. It may be seen that the initial sorption rate (h) continuously increased with increase in Cq. This is due to the increase in driving force due to the increase in Q. 

Table 4.3 Values of kA for several error function values ...

Thus, we solve a two-point boundary value problem instead of a partial differential equation. When the diffiisivity is constant, the solution is the error function, a tabulated function. 

Fig. 5.2. Computed percentage error (absolute value) for the He-N2 (j, = 0) system using potential function HFD1. The state to state inelastic cross-sections are compared at several collision energies as a function of A j transitions. The B value for N2 is taken to be about 2 [207],...

The variable y in the expression under the integral sign is an auxiliary variable the value of the integral depends only on the limits of integration (i.e., on the value of u). The numerical values of the error function vary from zero for m = 0 to an upper limit of unity for m —(this value is practically attained already for u 2). Plots of functions erf(n) and erfc(n) are shown in Fig. 11.2. 

If the assumption of normality is grossly violated, ML estimates of the parameters can only be obtained using the error-in-variables" method where besides the parameters, we also estimate the true (error-free) value of the measured variables. In particular, assuming that Ey i is known, the parameters are obtained by minimizing the following objective function... 

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... 

Table 7.2 gives tabulated values of the error function and related functions in the solution of other semi-infinite conduction problems. For example, the more general boundary condition analogous to that of Equation (7.27), including a surface heat loss,... 

Other more convenient equations for f(cr) that do not require access to values of the error function are in problem P5.02.08. 

The preceding results are applied to develop a strategy that allows us to isolate the source of gross errors from a set of constraints and measurements. Different least squares estimation problems are resolved by adding one equation at a time to the set of process constraints. After each incorporation, the least square objective function value is calculated and compared with the critical value. 

Then the values of the test statistic for all combinations are compared with the critical value. The presence of gross errors correspond to the combinations with the low objective function value (ofv). Detailed algorithms for Stages 1 and 2 are included in Appendix B. 

The objective function value is equal for different combinations of gross errors... 

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... 

Again, this solution converges very slowly for small extents of loss, i.e., for small values of Stt/d1. In this case, the solution expressed as an error function series should be used (Appendix 8B)... 

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