Linearity results

As can be seen from Figure 4, LBVs for these components are not constant across the ranges of composition. An iateraction model has been proposed (60) which assumes that the lack of linearity results from the iateraction of pairs of components. An approach which focuses on the difference between the weighted linear average of the components and the actual octane number of the blend (bonus or debit) has also been developed (61). The iadependent variables ia this type of model are statistical functions (averages, variances, etc) of blend properties such as octane, olefins, aromatics, and sulfur. The general statistical problem has been analyzed (62) and the two approaches have been shown to be theoretically similar though computationally different. 

The effect of a controlled device may not be proportional to its movement. In particular, the shape of valve plugs and the angle of opening of dampers will not give a linear result, and the signal from the controller must take this into account [10, 62]. 

The graphic demonstration thereof is self-explanatory. The linear(resultant) vector defines a fluctuating electromagnetic field. Each of the rotating vectors describes a helix around the time axis. 

With T) = 1300 °C, Too = 25 °C and 7Fj0 = 1, we have two equations in two unknowns m" and nip for the extinction conditions. However, under only suppression of the burning rate, Equation (9.80) applies, giving a nearly linear result in terms of q" and m" as well as 02,oo. The nonlinearity of the blocking effect can be ignored as a first approximation, and the experimental results can be matched to this theory. By subtracting Equation (9.81a) from Equation (9.80), we express the critical mass loss flux as... 

Initial values for a non-linear fit of Eq. (1) can be achieved by linearizations. Most conventional linearizations result from the transformation of the Michaelis-Menten equation, and are plotted according to ... 

Linearity is evaluated by appropriate statistical methods such as the calculation of a regression line by the method of least squares. The linearity results should include the correlation coefficient, y-intercept, slope of the regression line, and residual sum of squares as well as a plot of the data. Also, it is helpful to include an analysis of the deviation of the actual data points for the regression line to evaluate the degree of linearity. 

Some reactions are characterized by straight-line plots of TAS versus having a slope of approximately one, where this linearity results from compensatory, or off-setting, changes of AH and TAS. For this reason, the change in the Gibbs energy of activation, AG = AH - TAS, is a better description of the variation in the reaction than either AH or alone . See also Isokinetic Relationship... 

In practice, the real signal is never ideal. Systematic and random errors often occur due to, e.g. unresolved or badly resolved peaks, non-linearity (resulting in concentration-dependent peak shapes), noise and drift. 

Statistical approach. The linearity results can be subjected to statistical analysis (e.g., use of statistical analysis in an Excel spreadsheet). The p-value of the -intercept can be used to determine if the intercept is statistically significant. In general, when the p-value is less than 0.05, the... 

Intrinsic Accuracy. Intrinsic accuracy indicates the bias caused by sample matrix and sample preparation. In this approach, a stock solution is prepared by using known quantities of related substance and drug substance. The stock solution is further diluted to obtained solutions of lower concentrations. These solutions are used to generate linearity results. In addition, these linearity solutions of different concentrations are spiked into placebo. The spiked solutions are prepared according to the procedure for sample analysis. The resulting solutions, prepared from the spiked solution, are then analyzed. If the same stock solution is used for both linearity and accuracy and all of these solutions are analyzed on the same HPLC run, the response of linearity (without spike into matrix) and accuracy (with spike into matrix) can be compared directly. Any differences in response indicate the bias caused by matrix interference or sample preparation. To determine the intrinsic accuracy at each concentration level, one can compare the peak area of accuracy (with matrix) with that of linearity (without matrix) at the same concentration (Figure 3.11). This is the simplest approach, and one would expect close to 100% accuracy at all concentration levels. 

Injection parameters. If the injection reproducibility or linearity results are problematic, ensure that the sample vial cap is put on correctly. Sometimes, if the cap is put on incorrectly, the vial cannot be pressurized and injection either fails or is irreproducible. Also, check to make sure that no air bubbles are present in the sample vial. If air is injected into the capillary, poor results will be obtained. 

Let us notice that the eigenvalues Aa in equation (2.29) are considered constant here and henceforth. The same applies to ipa. However, the introduced dissipative matrices are, generally speaking, functions of invariants papa or of mean values (papa). The latter are functions of the velocity gradients, the expansion of which begins with a second-order term. It will be necessary to take this into account when discussing the non-linear results of the calculations. 

In the pharmaceutical industry, typically one will examine the previously defined API either as mixture of polymorphic forms or as a mixture of crystalline and amorphous phases (both having a simple linear intensity proportionality to concentration). Alternatively one may encounter a mixture of hydration or solvation states, in which cases the intensity would not necessarily be directly proportional to concentration. Figure 12.4 demonstrates the deviation from linearity resulting from differences in... 

Table 14.2 Concentrations of Cu, Pb, Zn, and Cd and linearization results for dissolved (<0.2 im) samples collected at three sites in the Narragansett Bay in June 1994.

Figure 4 Typical viscosity response of a polysaccharide polyanion and a neutral molecule to concentration, showing electroviscosity in a dilute dispersion of the polyanion (negative slope segment) and linearity resulting from interactions and cancellation of electroviscosity (positive slope). P represents the polyanion and P° represents its neutral counterpart.

In contrast, Eq. (44) for the constant charge case fails badly for Kh< 1, particularly when Ka < 5, as shown in Fig. 3b. Even for Ka —10, the Derjaguin result errs by 10% at Kh % 0.5, and quickly becomes more inaccurate as Kh 0. However, it does provide an accurate approximation to the linear result when the conditions mentioned above hold, namely when 1 /Ka Kh Ka. In assessing the merits of the linear Derjaguin approxi-... 

For small potentials, Eq. (56) simplifies to the linear result given by Eq. (24). 

Perhaps the first comparison that should be made is that between the potential near an isolated, charged plate as predicted by the linear and nonlinear Poisson-Boltzmann equations. The linear result is given by... 

For a homogeneous, low-viscosity fluid the probability P(p,M) is essentially a delta function centered at

and and the linear result is recovered. However, when Vyqr is much greater than one, there should be a thermodynamic distribution of density and modulus for a region of size 2ir/q. 

TABLE 9-14. Linearity Results (Assay and Content Uniformity)... 

TABLE 9-18. Linearity Results for Related Substances METHOD VALIDATION... 

If Cq is known as a function of the capillary number and the surfactant properties, the functional form of the frequency and bubble volume can be approximated from the linear results. However, a model for Cq in constricted angular tubes does not exist. If one assumes that snap off occurs as soon as the thread becomes axisymmetric, then the base state thread radius is approximately the half width of the channel at the point snap off occurs. The experimental observations of Arriola and Ni along with the theoretical predictions of Ransohoff and Radke indicate that snap off takes place very near the constriction neck. Therefore, the radius of the bubbles formed should be slightly larger than the half width of the constriction neck. In fact, approximating Cq by the constriction half width, one observes from equations 14 and 15, that the snap off frequency and bubble volume are independent of the liquid flow rate once the critical liquid flow rate has been exceeded. Ni measured the dependence of snap off on the bubble velocity, the velocity of... 

Figure 1.11 gives the scaled potential distribution y(r) around a positively charged spherical particle of radius a with yo = 2 in a symmetrical electrolyte solution of valence z for several values of xa. Solid lines are the exact solutions to Eq. (1.110) and dashed lines are the Debye-Hiickel linearized results (Eq. (1.72)). Note that Eq. (1.122) is in excellent agreement with the exact results. Figure 1.12 shows the plot of the equipotential lines around a sphere with jo = 2 at ka = 1 calculated from Eq. (1.121). Figures 1.13 and 1.14, respectively, are the density plots of counterions (anions) (n (r) = exp(+y(r))) and coions (cations) ( (r) = MCxp(—y(r))) around the sphere calculated from Eq. (1.121). 

Linear results or infinite value for steady state viscosity. 

Unbounded transient viscosity at high rates. Linear results or infinite value for steady state viscosity. 

Essig and Caplan [15] have made the point that, since a priori the physical constraint is arbitrary, it may be chosen such that v varies linearly with /is-/tp. After having shown that linear flow-force relationships may have great advantages for biological systems. Stuck [16] suggested that the latter may have evolved in such a manner that linearity resulted. Below we shall examine the case in which classical... 

Ciurczak and associates " reported a NIR method of determination of particle size of pure, granular substances. The method is based on theories of reflected light, in which reflectance increases as the particle size decreases. The reference method was a low-angle laser light-scattering (LALS) particle sizer (Malvern). The researchers found linear results for particles above 85 mm, but less accurate results for smaller particles. 

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