Polynomial coefficients

If the isotropic coefficient is specified to be unity, a is just the total (integrated) cross-section. In Appendix A, an alternative quantum mechanical expression for this cross-section is obtained in the electric dipole approximation. By comparing the two expressions, it can be seen that the Legendre polynomial coefficients in Eq. (11) may be obtained from the inner summation terms in Eq. (A.15). Hence, the Legendre polynomial coefficients are... 

Symmetry Properties of Legendre Polynomial Coefficients in the Photoelectron Angular Distribution... 

Equation (17) is the heart of the excess acidity method for the determination of unknown p bh+ values in strongly acidic media. Without going into detail (which is tedious) polynomial coefficients have been calculated that enable the calculation of X for 0-99.5 wt% H2SO4 and 0-80 wt% HCIO4. These are used with equation (18) and are given in Table 1. The form of equation (18) was... 

Table 1 Polynomial coefficients giving X as a function of wt% acid at 25°C for aqueous sulfuric and perchloric acid mixtures." ...

Polynomial coefficient Gives X for aq. H2SO4 Gives X for aq. HCIO4... 

Subsequently this computer method was investigated in more detail,36 and it was found that it was not necessary to be as elaborate as equation (18). The polynomial coefficients given in Table 2 for HC1 and HC104 are used with the much simpler equation (19) ... 

The Xq scale for HCIO4 that can be obtained from Table 2 is derived using H0 indicators only (primary aromatic amines), rather than the broad mix of indicators of different type used in deriving X. Values of X calculated from these polynomial coefficients are given for H2SO4 in Tables 3 and 4, for HCIO4 in Tables 5 and 6 (with V0), and for HC1 in Tables 7 and 8, as a function of wt% acid (odd-numbered tables) and of the acid molarity (even-numbered tables). 

Thus it is not necessary to derive X scales at different temperatures. This has, nevertheless, been done,43 although the scales (now called Me) or polynomial coefficients enabling their calculation are not given.43 Values of log Ch+ for HC1... 

Tables 3 and 4 contain values of the log water activity and log sulfuric acid activity in molarity units. These can be obtained at any temperature by using the polynomial coefficients supplied by Zeleznik,45 which are based on all of the preexisting thermodynamic data obtained for this medium. The numbers were converted to the molarity scale using the conversion formula given in Robinson and Stokes 46 Molarity-based water activities are given for HCIO4 in Tables 5 and 6. These are calculated from data obtained at 25°C by Pearce and Nelson,17...

Insertion of the ambient temperatures in Eq. 1 results R0 in and a. Since the ambient temperature is calculated from a polynomial funtion (eq. 2) it is necessary to determine the polynomial coefficients by a least square fit regression over the ambient temperature range 0-50°C. Together with S0,. Si, b, and c, these values are written by the calibration computer to the non volatile memory of the microcontroller. As a calibration check two additional blackbody readings are performed at a third ambient temperature (see Fig. 3.48). 

Here, a, b, and c are polynomial coefficients that vary with temperature ... 

In react, dataset isotope.dat contains polynomial coefficients that define temperature functions for the fractionation factors of species, minerals, and gases. The factors describe fractionation relative to a reference species chosen for each element. The reference species for oxygen and hydrogen is solvent water, H2O. CO2 and H2S, in either aqueous or gaseous form, serve as reference species for carbon and sulfur. 

Let us lump together the m observations y (i = 1,..., m) into a vector y, the polynomial coefficients otj (/=0,...,n — 1) into a vector jc of unknowns, and define the (/—l)th power of the ith observable (uj> 1 as the current term atJ of the matrix Am, . We now apply the usual method. Polynomials of high degrees tend to generate nearly singular matrices A which result in excessive fluctuations. 

It is tempting to write a routine such as SavGo l bad. m, to perform the Savitzky-Golay filtering, but we will show its numerical weakness. F is built up by the appropriate range of x-values and used to calculate the polynomial coefficients as a=F y( i-n i+n), see e.g. equation (4.31). 

The modulus data were fitted with a second degree polynomial equation, and these functions were used in the calculations of the thermal stresses from Equations 1 through 3. The polynomial coefficients and the correlation coefficient for each sample are given in Table III. 

Initial screens can be distinguished between methods that are used to determine what factors are most important, and follow-up screens that allow optimization and improvement of crystal quality (Table 14.1). In experimental design, this is known as the Box-Wilson strategy (Box et al., 1978). The first group of screens is generally based on a so-called factorial plan which determines the polynomial coefficients of a function with k variables (factors) fitted to the response surface. It can be shown that the number of necessary experiments n increases with 2 if all interactions are taken into account. Instead of running an unrealistic, large number of initial experiments, the full factorial matrix can... 

Figure 1 shows an example of a chromatogram of alkylbenzenes approximated with Chebyschev series. The information present in the chromatogram is reflected in the values of the polynomial coefficients. However, the terms do not usually have a direct relationship with conventional analytical parameters. 

A typical application is given by Debets et al. A quality criterion for the characterization of separation in a chromatogram is modified by using Hermite polynomial coefficients in order to enhance the performance. The quality criterion can be used in... 

Fig. 2. Response surfaces of a separation quality criterion from chromatograms of sulfanilamide, sulfacetamide, sulfadiazine, sulfisomidine and sulfathiazole, with eluents consisting of water, methanol and acetonitrile, (a) with, and (b) without Hennite polynomial coefficients.

M60 Newton interpolations computation of polynomial coefficients and interpolated values 6000 6054... 

In both cases the resulting system of equations is tridiagonal and can be easily solved by the special method presented in Section l.S. Once the m values are known, equations (4.21) can be easily rearranged to obtain the polynomial coefficients. Computing the function value and the derivatives at any point x is then straightforward, whereas integration is facilitated by the relationship... 

The polynomial coefficients for these fits are given in Table 12.2. There is a very weak dependence of A -k, B -k, and C k on l k, which is often ignored. 

Making recourse to the saturation property of the design and substituting in succession the coordinates of experimental points 1 through 15 into the polynomial of Eq. (3.67), we determine the polynomial coefficients ... 

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