First degree equations slope

The curve drawn illustrates how the model fits measured data. The first derivative of Equation 30.4 allows calculating the slope at any strain. The same model can be used to fit any relative torque harmonic, for instance the 3rd one, T(3/l). Note that in using Equation 30.4 to model harmonics variation with strain, one may express the deformation (or strain) y either in degree angle or in percent. Obviously all parameters remain the same except C, whose value depends on the unit for y. The following equality applies for the conversion C(y,deg) = x C(y,%), where a = 0.125 rad. 

Moreover, it should be noticed that polymerization rates were determined from the maximum slope of the kinetic curves, namely at degrees of conversion between 20 and 40%. At that time, the large increase in viscosity of the photoresist may already have reduced the chain mobility, thus favoring radical isolation and first-order termination. It is therefore very likely that the intensity exponent of the photopolymerization rate equation will be less than 0.85 in the early stages and that it increases with conversion to reach almost unity in the solid network. Such a kinetic behavior was indeed observed for the photopolymerization of neat hexanedioldiacrylate (31). 

From a statistical viewpoint it is difficult to state a definite answer because several problems accumulate here. First, there is not an exact solution to the comparison of two population means, for which we have to estimate simultaneously (from the limited experimental data available) their average values and their associated variances, as is the case in laboratories. Second, the equations stated above were deduced for normal distributions but the slopes derived from a least-squares fit follow approximately a Student s distribution. We must bear in mind that although the theoretical slope and intercept of the population follow a normal distribution their estimators do not because the latter (along with the variance of the regression itself) must be estimated from (usually) a very reduced number of data and the number of degrees of freedom - dof- must be taken into account. In statistical terms an intermediate pivot statistic must be introduced to obtain an approximate Student s distribution. ... 

With such a large hydrodynamic section of the molecules (d = 30-40 A), the degree of stretching of the molecules p = Lid of the fractions studied is low from 7-5.5 to 1-0.7. In this region of p, the dependences of [nl and D on M for rod-shaped particles in the form of Mark-Kuhn equations are characterized by values of exponents a and b of less than 0.5. For this reason, Eqs. (3.17)-(3.19) for approximation of the dependences of [t]], D, and Sq on M (cf. Fig. 3.4, lines 1, 2, 3) should only be considered a first approximation. Lines 1, 2, and 3, plotted by the method of least squares with slopes of less than 0.5, more accurately convey the features of the hydrodynamic properties of short-chain molecules and quantitatively more precisely correspond to the experimental data. The relations... 

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