Quadratic polynomial

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

The measured relationships between piezoelectric polarization and strain for x-cut quartz and z-cut lithium niobate are found to be well fit by a quadratic relation as shown in Fig. 4.4. In both materials a significant nonlinear piezoelectric effect is indicated. The effect in lithium niobate is particularly notable because the measurements are limited to much smaller strains than those to which quartz can be subjected. The quadratic polynomial fits are used to determine the second- and third-order piezoelectric constants and are summarized in Table 4.1. Elastic constants determined in these investigations were shown in Chap. 2. 

Testing the adequacy of a model with respect to its complexity by visually checking for trends in the residuals, e.g., is a linear regression sufficient, or is a quadratic polynomial necessary ... 

The experiments were carried out in random order and the responses analyzed with the program X-STAT(11) which runs on an IBM PC computer. The model was the standard quadratic polynomial, and the coefficients were determined by a linear least-squares regression. 

The resulting data of the Box-Behnken design were used to formulate a statistically significant empirical model capable of relating the extent of sugar 3deld to the four factors. A commonly used empirical model for response surface analysis is a quadratic polynomial of the type... 

Although the fit does not capture the onset of 02(w > 26) near 234 nm, it does about as well as can be done using a quadratic polynomial. 

Figure 66-3 A quadratic polynomial can provide a better fit to a nonlinear function over a given region than a straight line can in this case the second derivative of a Normal absorbance band.

The improvement in the fit from the quadratic polynomial applied to the nonlinear data indicated that the square term was indeed an important factor in fitting that data. In fact, including the quadratic term gives well-nigh a perfect fit to that data set, limited only by the computer truncation precision. The coefficient obtained for the quadratic term is comparable in magnitude to the one for linear term, as we might expect from the amount of curvature of the line we see in Anscombe s plot [7], The coefficient of the quadratic term for the normal data, on the other hand, is much smaller than for the linear term. 

The basis for this calculation of the amount of nonlinearity is illustrated in Figure 67-1. In Figure 67-la we see a set of data showing some nonlinearity between the test results and the actual values. If a straight line and a quadratic polynomial are both fit to the data, then the difference between the predicted values from the two curves give a measure of the amount of nonlinearity. Figure 67-la shows data subject to both random error and nonlinearity, and the different ways linear and quadratic polynomials fit the data. 

Table 15.1 contains osmotic pressure data calculated from the work of Browning and Ferry [3] for solutions of polyvinyl acetate in methyl ethyl ketone at 10°C. Plot H/vv against w, fit the data to a quadratic polynomial, and calculate the number-average molar mass from the intercept with the n/w axis. 

To find L 2 it is necessary to calculate (dAH/dn2) T,p at various molalities of HCl. The data in Figure 18.5 have been fitted to a quadratic polynomial by the method of least squares. The equation obtained is... 

The base groups of the Anderson-Beyer-Watson-Yoneda method are listed in Table 22.1. They are modified by appropriate substitutions to yield the desired molecule. Thus, aliphatic hydrocarbons can be built up from methane by repeated substitutions of methyl groups for hydrogen atoms. Other compounds are formed by substitution of functional groups for CH groups. All values in the tables are in units of J mol or J moP as appropriate. The heat capacity constants are similar to those discussed in Chapter 4 but for a quadratic polynomial in T/1000. 

Polynomial regression is frequently used, and for a set of data that is slightly curved, a small quadratic term could correct for the small deviation from a linear equation. However, polynomials are not necessarily monotonic, and a quadratic function can have a maximum or minimum when the coefficients Co and C do not have the same sign and dy/dx = 0. Let us consider a quadratic polynomial regression for the boiling points of the normal paraffin. The regression shows... 

These results are not shown.) The latter two are already smaller than the leading terms (AifaAL = 0001, 0223, and 2023) by nearly two orders of magnitude. These and the higher terms can safely be neglected. The A coefficients may be represented by quadratic polynomials in the bond distances, r and r2. From these, the radial transition matrix elements, Eq. 4.20, are obtained with the help of the H-H matrix elements, v j v2j2 Qn v j v 2f2) [425],... 

Pleshanov (P4) extends the integral heat balance method to bodies symmetric in one, two, or three dimensions, using a quadratic polynomial for the approximate temperature function. Solutions are obtained in terms of modified Bessel functions which agree well with numerical finite-difference calculations. 

The responses of nisin and lactic acid were correlated by nonlinear regression using the following full quadratic polynomial model ... 

FIGURE 10.17 Strong filtering of noisy data using quadratic polynomials. Filtering was done with a five-point smoothing window. The true signal is shown as a dotted trace. 

Fig. 4.6 Temperature-compensated creep plot for tension and compression on a glass-ceramic. Data are compensated by an Arrhenius correction to 700°C. Data from Ref. 56 were replotted. The curves are least square fits of quadratic polynomials to the data. An increase in the stress exponent of the creep data is clearly indicated.

The design used is a function of the model proposed. Thus, if it is expected that the important responses vary relatively little over the domain, a first-order polynomial will be selected. This will also be the case if the experimenter wishes to perform rather a few experiments at first to check initial assumptions. He may then change to a second-order (quadratic) polynomial model. Second-order polynomials are those most commonly used for response surface modeling and process optimization for up to five variables. 

The curves in Figs. 2, 3, and 4 were obtained by a quadratic polynomial interpolation. 

---