Polynomial second-degree

For the systems investigated, the increase of / . with a expressed by a second degree polynomial according to Mandel [1]... 

Coefficients and standard deviation,s, of the least-squares second-degree polynomial representing the titration curve of PGA with different strong bases at 298 K. 

The application of the general method can be illustrated by the example shown in Pig. 3. The series of data points is fitted by a polynomial of second degree. Two points will be employed on either side of the point to be inter-... 

The performance statistics, the SEE and the correlation coefficient show that including the square term in the fitting function for Anscombe s nonlinear data set gives, as we noted above, essentially a perfect fit. It is clear that the values of the coefficients obtained are the ones he used to generate the data in the first place. The very large /-values of the coefficients are indicative of the fact that we are near to having only computer round-off error as operative in the difference between the data he provided and the values calculated from the polynomial that included the second-degree term. 

Let us solve this relationship for t. Switching to an equality sign, we write this relationship as a second-degree polynomial in N/r,... 

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. 

EXAFS data are processed to obtain radial structure functions (RSFs). First, the non-EXAFS components are subtracted from the data. Pre-edge absorption is removed using the Victoreen correction (International Tables for Crystallography, 1969) of the form AX + BX . The monotonic decrease of absorbance beyond the edge, called the photoelectric decay, is subtracted out after approximating it either by a second degree polynomial or a spline-function (Eccles, 1978). The normalized x(k) is then expressed as... 

Following application of the method of steepest ascents, it will usually be advisable to investigate the neighborhood of the optimum more carefully, fitting at least a second-degree polynomial to the response surface as described in Section VI B. 

Go/dZi can be calculated and should be independent of concentration if our procedure is correct. The results presented in Tables I and II show that this is indeed the case. Finally AG°t is obtained by analytical integration. The variation of dGo/dZi with Z is fitted to two second-degree polynomial functions, one for the region of the maximum (Figure 1) and one for the rest of the curve ... 

Trial and error shows that a= — 1 is a solution of this equation, which means that x - (— 1) = x + lisa factor of 3x3 — 2x + 1. It is now possible to express 3x3 — 2x + 1 in the form (x + l)(3x2 -3x —1). Note that the second degree polynomial in parentheses does in fact factorize further, but the resulting expression is not very simple in appearance. 

The characteristic equation for the above equation is the second-degree polynomial... 

Fig.12 Concentrations of TBE and BDE-209 as a function of year in a northern Lake Michigan sediment core. Second-degree polynomials have been fitted to these data as visualization aids. From Hoh et al. [60]...

Scheffe [5] suggested to describe mixture properties by reduced polynomials obtainable from Eq. (3.11), which is subject to the normalization condition of Eq. (3.2) for a sum of independent variables. We shall demonstrate below how, for instance, such a reduced second-degree polynomial is derived for a ternary system. The polynomial has the general form ... 

Then we arrive at the reduced second-degree polynomial in three variables ... 

Figure 3.9 3,n -lattices a) for a second-degree polynomial, b) for an incomplete-third-degree polynomial, c) for a third-degree polynomial, d) for a fourth-degree polynomial... 

Figure 5-1 Examples of Savitsky Golay smoothing. A second-degree polynomial was used for the left column and a third-degree polynomial was used for the right column with a 5-, 15-, and 25-point smooth.

It is assumed that the temperature, T, takes the form of a polynomial. In other words, second-degree... 

Figure 9 Time domain ESEEM ratio (a) and Fourier transform (b) generated by dividing time domain ESEEM data obtained for Fe(II)NO-TauD treated with aKG and Ci- H-taurine with data obtained imder identical conditions for Fe(II)NO -TauD treated with aKG and taurine. Data sets were normahzed to their maximum amplitudes prior to division. A second-degree polynomial was used to remove residual background decay. ESEEM data were collected imder conditions identical to those of Figure 8(a)...

The variation of the solubilities of most substances with temperature is fairly regular, and usually increases with temperature. When water is the solvent, breaks may occur in solubility curves because of formation of hydrates. Figure 16.1(a) shows such breaks, and they can be also discerned in Figures 16.2(b) and (c). Unbroken lines usually are well enough represented by second degree polynomials in temperature, but the Clapeyron-type equation with only two constants, Inx = A + B/T, is of good accuracy, as appears for some cases on Figure 16.1(b). 

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