Cubic Functions

A simple inspection shows that cubic functions (splines) shown graphically in Figure 2.5 satisfy the above conditions. 

The ideal condition would be when the rotor output is a cubic function of wind speed. But in practice this may not be so. It is found to be linear or a near quadratic (square) function of the wind speed, as shown in Figure 6.64... 

However when the SF cross-section is analyzed, its composite nature still results in a twofold increase in rigidity, compared to an equivalent amount of solid plastic, since rigidity is a cubic function of wall thickness. This increased rigidity allows large structural products to be designed with only minimal distortion and deflection when stressed within the recommended values for a particular foamable plastic. 

As has been discussed in the preceding sections, it is expected that the surfactant monolayer exhibits "N"-shape nonlinearity in its dynamic tt-T characteristics. Thus, we would like to discuss the kinetics, assuming that G(Z0 is a cubic function. 

If the first derivatives of f(x) are available, only two points are needed, and the cubic function can be fitted to the two pairs of the slope and function values. These four pieces of information can be uniquely related to the four coefficients in the cubic equation, which can be optimized for predicting the new, nearly optimal data point. If (xx,fvf ) and (x2,/2,/ 2) are available, then the optimum 3c is... 

The nature of any degradation relationship will determine whether the data should be transformed for linear regression analysis. Usually, the relationship can be represented by a linear, quadratic, or cubic function on an arithmetic or logarithmic scale. Statistical methods should be employed to test the goodness of fit of the data from all batches and combined batches (where appropriate) to the assumed degradation line or curve. 

Algebraic expressions for terms M and C were derived using Dewar s PMO method (for C in a version similar to the co-technique [57] in order to calculate carbocation stabilization energies). The size factor S is simply a cubic function of the number of carbon atoms [97], The three independent variables of the model were assumed to be linearly related to the experimental Iball indices (vide supra). By multilinear regression analysis (sample size = 26) an equation was derived for calculating Iball indices from the three theoretical parameters. The correlation coefficient for the linear relation between calculated and experimental Iball indices is r = 0.961. 

On an interval [Xy i, of a variable x, there exists one cubic function 5, (x) and only one such that ... 

Veg and Veg are the wavenumbers of the electronic transitions for absorption and fluorescence in the gas phase, respectively the other terms are as in Eq. (6-2). Eq. (6-5a) has been widely used for the determination of dipole moment changes from the solvent dependence of spectra. The main source of error is the Hmited accuracy of the estimated value for the interaction radius uw of the solute molecule since Av is a cubic function of Uw. 

In van der Pol s day, the active element was a vacuum tube today it would be a semiconductor device. It acts like an ordinary resistor when / is high, but like a negative resistor (energy source) when / is low. Its current-voltage characteristic V = / I resembles a cubic function, as discussed below. 

All these equations can be rearranged into a cubic function of specific volume. The form of the Redlich-Kwong and Redlich-Kwong-Soave equation of state is... 

Make a cubic B-spline pass through all the data points [x(z),))(/), i = 1,..., zm], A cubic spline is a cubic function of position, defined on small regions between data points. It is constructed so the function and its first and second derivatives are continuous from one region to another. It usually makes a nice smooth curve through the points. The following commands create Figure B.3. 

Data regressions based on the law of mass action are generally adequate for most situations. However, this model only retains validity in liquid-phase reactions at equilibrium without cooperativity. Reactions that involve solid-phase, multiple cooperative binding, and not reaching equilibrium, deviate from the model. Therefore, empirical equations that are not based on the law of mass action have been used for curve fitting also. Among these, polynomial (205) and spline functions are often used (206-209). Polynomial regression can be a second-order (parabolic) or third-order (cubic) function ... 

Often a polynomial may be substituted as a function if the true model is unknown. For example, a quadratic model may be substituted for an Emax model in a pharmacodynamic analysis or a quadratic or cubic function may be used in place of a power function... 

Fig. 15.6. Typical dynamics of model (15.7) in the oscillatory regime, a) Phase portrait in the (/i, 7V)-phase plane showing the limit cycle (solid line) and the isoclines (dotted lines). Note, that the prey isocline is a cubic function, b) Time series of prey (solid line) and predator (dashed line) show recurrent population outbreaks.

A functional series is one way of representing a function. Such a series consists of terms, each one of which is a basis function times a coefficient. A power series uses powers of the independent variable as basis functions and represents a function as a sum of the appropriate linear function, quadratic function, cubic function, etc. We discussed Taylor series, which contain powers of x — h, where h is a constant, and also Maclaurin series, which are Taylor series with h =0. Taylor series can represent a function of x only in a region of convergence centered on h and reaching no further than the closest point at which the function is not analytic. We found the general formula for determining the coefficients of a power series. 

An undesirable side-effect of an expansion that includes just a quadratic and a cubic term (as is employed in MM2) is that, far from the reference value, the cubic function passes through a maximum. This can lead to a catastrophic lengthening of bonds (Figure 4.6). One way to accommodate this problem is to use the cubic contribution only when the structure is sufficiently close to its equilibrium geometry and is well inside the true potential well. MM3 also includes a quartic term this eliminates the inversion problem and leads to an even better description of the Morse curve. 

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