Linearity assumption

It is usually advisable to plot the observed pairs of y versus r, to support the linearity assumption and to detect potential outhers. Suspected outliers can be omitted from the least-squares Tit and then subsequently tested on the basis of the least-squares fit. 

Equations 3.1.47 and 3.1.50 express the relation between system concentrations and the fraction conversion for variable-volume systems that satisfy the linearity assumption of equation 3.1.40. This assumption is a reasonably unre-strictive one that is valid for all practical purposes in isothermal constant pressure systems in... 

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

P(j+i)j for / = 1— 1 to be independent of Sc. This is the assumption employed in the SR model, but it can be validated (and modified) using DNS data for the scalar spectrum and the scalar-scalar transfer function. The linearity assumption discussed earlier implies that the rate constants will be unchanged (for the same Reynolds and Schmidt numbers) when they are computed using the scalar-covariance transfer spectrum. 

SC 1-5 Uncertainty in Risk Estimates SC 1-6 Basis for the Linearity Assumption SC 1-7 Information Needed to Make Radiation Protection Recommendations for Travel Beyond Low-Earth Orbit SC 9 Structural Shielding Design and Evaluation for Medical Use of X Rays and Gamma Rays of Energies Up to 10 MeV SC 46 Operational Radiation Safety... 

Multiplying Eq. (44) by fg, we obtain Rule 3 as a special case of the linearity assumption, Rule 2 ... 

To convert an optical signal into a concentration prediction, a linear relationship between the raw signal and the concentration is not necessary. Beer s law for absorption spectroscopy, for instance, models transmitted light as a decaying exponential function of concentration. In the case of Raman spectroscopy of biofluids, however, the measured signal often obeys two convenient linearity conditions without any need for preprocessing. The first condition is that any measured spectrum S of a sample from a certain population (say, of blood samples from a hospital) is a linear superposition of a finite number of pure basis spectra Pi that characterize that population. One of these basis spectra is presumably the pure spectrum Pa of the chemical of interest, A. The second linearity assumption is that the amount of Pa present in the net spectrum S is linearly proportional to the concentration ca of that chemical. In formulaic terms, the assumptions take the mathematical form... 

Equation (16.1) is an approximation it assumes that the measured spectral contribution from Pa is independent of the overall composition of the sample. In reality, the bulk absorption and scattering properties of different samples will cause variations in how much Raman signal is collected. In the case of urine and blood serum specimens, the variation is often negligible. For whole blood and turbid tissue specimens, the influence of bulk optical properties is more important. Methods of correcting for their effects and returning to the simple linearity assumptions of (16.1) are mentioned briefly in the next section. 

Therefore, any C present must be removed down to a mole fraction of 0.198 if the linear assumption is appropriate. To be conservative, it is assumad that a residual mole fraction of 0.100 is an acceptable concentration of C. The relative volatility of C to R can be calculated from vapor pressure values. 

Because the temperature profile is assumed to be linear (assumption 5 above), the heat transfer rate, which is given by Fourier s law, is (he same at all points across the film. It follows therefore that the heat transfer rate to the wall from the liquid film is given by ... 

The model linearity assumption, Eq. (154), is not critical. Badgwell (1997) has shown how these ideas can be readily extended to the case of stable nonlinear plants. 

Note, however, that Eq. (4.21) loses its validity in the case of strong multiplicative noise, as it is based on a linearization assumption. The remarks above have to be related to a purely qualitative level of interpretation. 

In quantum mechanics, the most general time evolution must be linear. That implies that having two solutions p i t) and p2 (f) of the evolution equation, we can find another solution by writing p(t) = a pi(t) + b As the density matrices are represented by ensembles of systems, it must be possible to consider the ensemble combined from px(t) and p2(t). The linearity assumption is satisfied if we have a time evolution of the form... 

Despite the challenges that nonlinear extrapolation for carcinogens offers in terms of regulatory implementation, it is important that the best available science guide risk assessment practices. Default assumptions designed to be health-protective, such as low-dose linear assumption, are useful when no other alternatives are available, but public health initiatives will be best served when grounded in biologically based approaches. 

This analytical solution review is tractable only for very limited assumptions, such as homogeneity and linearly elastic behavior (not to mention excluding variations that are time- or temperature-dependent). The first deviation that must be examined is the elastic linearity assumption for polishing pads. Polymers, in general, show behavior that lies between that of an elastic solid and a viscous fluid. The term viscoelastic has been applied to this behavior. 

The parallelism of the linear plots indicates one value for the activation energy, 104.8 kcal mole , and an enhanced rate for decomposition in the presence of increased amounts of Oj. A plot of the observed rate coefficients at a temperature of 5000° K versus the mole fraction of oxygen supports the linearity assumption for Xq, < 0.10. The straight line drawn in Fig. 15 represents the linearity assumption. The deviations of the points at higher mole fractions which are outside the limits of experimental error can most probably be accounted for by the increased contributions of vibration-vibration processes which would lower the rate coefficients for > 0.10. 

A summary of the decomposition results is shown in Fig. 16. The rate coefficients are bounded by the upper [61] and lower [13] broken lines for Oi Ar mixtures. Least squares analysis yielded values for fe, and of eqn. (B) with M and O2 respectively as collision partners. The fact that fea did not depend upon the identity of M lends additional support to the linearity assumption. The value of 2 was deduced from an analysis of the laser beam profile following the dip and was found to agree within a factor of two with other reports [61—63]. The ratio of 2/ 3, a measure of the atom efficiency, is of the order of 5—7. 

An increasing activation energy with decreasing temperature was noted in mixtures with mole fractions of HCl equal to 0.10 and 0.20 [118]. The linearity assumption was used to calculate values of ftnci Ar and... 

The independence of the atom and molecule catalysed rate coefficients with regard to different inert gas environments (the linearity assumption) and the validity of the equilibrium relation (K = fe, /fe, ) has been tested for [40], Oj [11], and HCl [118] by the laser-beam... 

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