Third-order polynomial function

For self-associating protein systems, third-order polynomial functions provided a good fit over the accessible range The data on AG° must show the direction of the chemical change, toward the minimum in the Gibbs function. If this proves true, the equation can be applied in the standard or nonstandard state. For protein unfolding or DNA unwinding, nonlinear models are needed Consistent with Occam s razor, the simplest description is used to describe the system, and complexity is increased only if warranted by the experimental results. 

In order to have a linear correlation between KJS and concentration over a wide range of concentrations, the KJS value is corrected with a third-order polynomial function, so that/(A/S) is proportional to the analyte concentration or enzyme activity ... 

With an apphed DC voltage, the current Idc of IPMC actuator does not vanish at the steady state, which indicates the existence of polymer resistance. Such resistance, however, is not linear [Bonomo et al. (2007)]. In this work, we use a third-order polynomial function Y(V) to empirically approximate Idc-... 

Instead of doing the calculations based on the apparent molar volumes of the solute (and the corresponding approximation of attributing all volumetric changes of the solution to changes in the volume of the solute), one can also do the calculations based on the estimation of partial molar volumes for both components. This is in fact a much better option since it does not contain the aforementioned approximation. In order to do so, the molar volumes of the solutions as a function of the solute mole fractions were fitted to third-order polynomial functions, and the corresponding derivatives were ealeulated in order to yield (by extrapolation to the pure component compositions) the eorresponding partial molar volumes. 

To incorporate anharmonicity effects in a description of a diatomic, a third-order polynomial function is a starting point for the potential, and we can express this generally as... 

Various compilations of densities for organic compounds have been published. The early Landolt-Bomstein compilation [23-ano] contained recommended values at specific temperatures. International Critical Tables [28-ano-l] provided recommended densities at 0 °C and values of constants for either a second or third order polynomial equation to represent densities as a function of temperature. This compilation also gave the range of validity of the equation and the limits of uncertainty, references used in the evaluation and those not considered. This compilation is one of the most comprehensive ever published. Timmermans [50-tim, 65-tim], Dreisbach [55-die, 59-die, 61-dre] and Landolt-Bomstein [71-ano] published additional compilations, primarily of experimental data. These compilations contained experimental data along with reference sources but no estimates of uncertainty for the data nor recommended values. 

Table 11.9 Third-order polynomials relating rednced partition functions to inverse of squared absolute temperature x = 10 X (from Clayton and Kieffer, 1991).

The result illustrated in Figure 6.7 shows a satisfactory fit and the next groups of parameters included in the refinement were a more complex background (third order polynomial instead of a linear function) and peak asymmetry. The fit further improves and when the calculated and observed... 

Liquid crystals can also be used since the temperature-dependent residual static dipole-dipole coupling caused by the anisotropic orientational order produces, in turn, a temperature-dependent static splitting of the proton spectrum. The first study used a 5% v/v solution of benzene in the room temperature mesophase ZLI-1132 liquid crystal in a 5 mm NMR tube. Over 60 lines were present in the spectrum (66 at 360 MHz and 64 at 600 MHz). The authors measured the frequency separation of the outermost lines as a function of temperature at 360 MHz, and fitted this to a third-order polynomial, giving ... 

A procedure similar to that used to solve equation (66) can be applied to solve equation (110). Again, the solution is given by the Jacobi elliptic functions. The third-order polynomial under the square root on the r.h.s. of (110) has the roots... 

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 ... 

Spline functions using piecewise third-order polynomials can be used to fit each interval between standards. The entire function and its first and second derivatives are continuous ... 

We have seen that the most reasonable choice for the function v(a,x) that approximates the unknown function y(x) is a polynomial. We have also seen that it is usually suitable to split the interval in elements. Finally, we demonstrated that to obtain the (P - l)-order polynomials valid inside each element, P support points are necessary for the first and second strategies. In the case of third strategy, we need the value of the variables and their first derivatives at the extremes of each element and the value of the variables in a certain number of internal points of the element they are 0 for a third-order polynomial, 1 for fourth-order polynomial, 2 for fifth-order polynomial, and 3 for sixth-order polynomial. These points should be opportunely selected. 

The relationships between response variables, y, and factors, x, are represented through mathematical expressions that take the form of correlations or regression equations. These linear or nonlinear mathematical expressions are often expressed as a polynomial, logarithmic, exponential, or trigonometric function. For example, the variation of the specific heats of gases, Cp, has been expressed as a third-order polynomial with respect to temperature ... 

Tables 5.7-5.9 report the voltage as a function of temperature and third-order polynomials are generally sufficiently accurate to calculate the...

The total energy was then fitted to a third order polynomial as a function of die strain (e), from which Young s modulus was derived (3J) (Table II). 

This transformation provides non-dimensional HSI values, where H is normalized with 2n and I is normalized with the maximum pixel count for an individual sensor of the digital camera (255 for each 8-bit sensor). Furthermore, the minimum and maximum hue and saturation values, respectively, detected in temperatures imder the red start temperature of the TLC material (25 °C) were used as a threshold level imder which the negative radian value of H would be used in order to obtain a better fit to a continuous third-degree polynomial function. 

Fig. 2.7 Experimental data of Fig. 2.6 represented in form of polynomial function (2.49) (points), and fit by third order polynomial equation (line)...

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