Polynomial degree

To make the pair potentials flexible but fast to evaluate we chose the U-y rik) = W f qik) as low degree polynomials in... 

This third-degree polynomial has the value zero with two eigenvalues ... 

FIG. 7-2 Linear analysis of catalytic rate equations, a), (h) Sucrose hydrolysis with an enzyme, r = 1curve-fitted with a fourth-degree polynomial and differentiated for r — (—dC/dt). Integrated equation,... 

The correlation for C° of the ideal gas at low pressure is a third degree polynomial, which is a function of temperature. 

Interpolation with nonequally spaced data may be accomplished by the use of Lagrange Polynomials, defined as a set of n degree polynomials such that each one, P.(x) (j = 0, 1,. . n), passes through zero at each of the data points except... 

Fig. 11-26 Decade-averaged data of Northern hemisphere tree ring records from 1750-1979 and 7th-degree polynomial fit of the data. The vertical extension of blocks represents 95% confidence limits of the mean. The open circles give the change of —0.65% in atmospheric CO2 observed from 1956 to 1978 by Keeling et al. (1979). (Adapted from Peng et al, 1983.)...

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. 

A seventh degree polynomial fit to the theoretical curves has been used (columns A and B). In column A use is made of the variational MR SD-Cl energies while in column B estimated full-CI energies 111] are utilised... 

The fraction undissolved data until the critical time can be least-square fitted to a third degree polynomial in time as dictated by Eq. (29). The moments of distribution ij, p2, and p3 can be evaluated from Eqs. (30) through (32), with three equations used to solve for three unknowns. These values may be used as first estimates in a nonlinear least-squares fit program, and the curve will, hence, reveal the best values of both shape factor, size distribution, and A -value. 

Equation 56-27 contains scaled coefficients for the zeroth through third derivative convolution functions, using a third degree polynomial fitting function. The first row of equation 56-27 contains the coefficients for smoothing, the second row contains the coefficients for the first derivative, and so forth. 

The same degree polynomial, for a lower-order derivative. 

For this reason, the reader will find another very interesting exercise to compute the sums of the squares of the coefficients for several of the sets of coefficients, to extend these results to both higher order derivatives and higher degree polynomials, to ascertain their effect on the variance of the computed derivative for extended versions of these tables. Hopkins [8] has performed some of these computations, and has also coined the term RSSK/Norm for the 2((coefl7Normalization factor)2) in the S-G tables. Since here we pre-divide the coefficients by the normalization factors, and we are not taking the square roots, we use the simpler term SSK (sum squared coefficients) for our equivalent quantity. Hopkins in the same paper has also demonstrated how the two-point... 

The reason why the authors do not sort the DCT data before fitting the DCT polynomial is as follows. If the data get sorted then it becomes monotonically increasing or decreasing (according to the sorting). Hence the data can be very well modeled with low degree polynomial with very small mean square error. Thus, when one gets back to the DCT coefficients from the polynomial, the modified data (and also the data that are not modified) are not disturbed much and consequently the watermark is not removed. 

Method (b). The (C,t) data are fitted by a fourth degree polynomial on the second graph, from which the derivative is... 

The other plots are made with the software TABLECURVE. The special function F2 used there is a log-normal relation and F3 is a sine-wave function. Usually a ratio of low degree polynomials also provides a good fit to bell-shaped curves here five constants are needed. The Gamma distribution needs only one constant, but the fit is not as good as some of the other curves. The peak, especially, is missed. 

Final substitution of v will give the sought for result, which will be a ratio of fourth and third degree polynomials in r. 

Cubic interpolation to find the minimum of f(x) is based on approximating the objective function by a third-degree polynomial within the interval of interest and then determining the associated stationary point of the polynomial... 

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

Figure 4-24. In all panels the true data are represented by the line marker. The top panel displays the noisy ( ) data the middle panel shows the result of a 4th degree polynomial fitted through 11 noisy data points ( ) and the bottom panel, the result of a 2nd degree smoothing through 21 noisy data points ( ).

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. 

Any gth-degree polynomial of the number operators for these R orbitals... 

Warning. In step 1, if you use a computer to fit a polynomial to the data it could lead to disaster. For example, consider fitting a sixth-degree polynomial to the seven data points, or an (n - 1) degree polynomial to n points. 

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