Absolute value linear

Absolute value linear equations I ax + b I = c, with two solutions... 

Much can be predicted from these parameters. Linear correlations have been estabhshed between the R value and the drawabhity parameter, LDR. Thus higher R values are associated with better deep drawabhity. AR, on the other hand, predicts the height and location of ears. Eor example, larger absolute values of AR predict higher ears during deep drawing cup shapes. 

The absolute value of a proportional error depends upon the amount of the constituent. Thus a proportional error may arise from an impurity in a standard substance, which leads to an incorrect value for the molarity of a standard solution. Other proportional errors may not vary linearly with the amount of the constituent, but will at least exhibit an increase with the amount of constituent present. One example is the ignition of aluminium oxide at 1200°C the aluminium oxide is anhydrous and virtually non-hygroscopic ignition of various weights at an appreciably lower temperature will show a proportional type of error. 

At first sight, equation 9.1 implies that the relationship between Q and AT is linear. Whereas this is approximately so over limited ranges of temperature difference for which U is nearly constant, in practice U may well be influenced both by the temperature difference and by the absolute value of the temperatures. 

It can be seen from Fig. 15.2 that in semilogarithmic plots of AE vs. log/, the polarization characteristics are linear [i.e., obey the Tafel equation (6.3)]. Slopes b practically coincide for most metals and have values of 0.11 to 0.13 V. However, the absolute values of polarization recorded for a given current density (CD) vary within... 

In order to determine the activation volume for the HS LS conversion AI l. Mossbauer spectra were recorded at 260 K for different pressures. A linear regression of Inknc versus pressure p following Eq. (55) gives A FJl = — 22 + 3 A. Thus AF is negative and its absolute value is less than AF, the... 

A comparahve analysis of coefficients and descriptors clarifies the relationship between lipophilicity and hydrophobicity (Y in Eq. 4 is the molar volume which assesses the solute s capacity to elicit nonpolar interactions (i.e. hydrophobic forces) which, as also clearly stated in the International Union of Pure and Applied Chemistry definitions [3] are not synonyms but, when only neutral species are concerned, may be considered as interchangeable. In the majority of partitioning systems, the lipophilicity is chiefly due to the hydrophobicity, as is clearly indicated by the finding that the product of numerical values of the descriptors V and of the coefficient v is larger in absolute value than the corresponding product of other couples of descriptors/coefficients [9]. This explains the very common linear rela-... 

There was a clear linear correlation between (a + (l)-values and the differences in activation enthalpies. The absolute value of the activation enthalpy difference went through a minimum, meaning that at medium angle sum there was no distinct preference of one or the other o-QM type, whereas at extreme angle sums —either very large or very small ones—one of the o-QM types is largely preferred over the other one. 

With respect to the segmental diffusion, the characteristic frequencies of the cyclic systems vary with Q3 as in the case of the linear chains in dilute solution (see Sect. 5.1.2). The absolute values are independent of the topology of the polymers and their molecular masses, and thus exhibit the same deviations from the theoretical predictions that have just been pointed out for dilute solutions of linear homopolymers. 

The linear Mjj versus Wjj e plots imply that the number of poly(MVE) chains produced per unit initiator, N (cf. eq. 2), is constant during the polymerization. The absolute values of N were greater than unity (2.50 and 2.79 at [p-DCC]0 = 1.0 and 2.0 mM, respectively), suggesting that chain transfer to monomer may have occurred during the early stages of the polymerization. 

One must note that probability alone can only detect alikeness in special cases, thus cause-effect cannot be directly determined - only estimated. If linear regression is to be used for comparison of X and Y, one must assess whether the five assumptions for use of regression apply. As a refresher, recall that the assumptions required for the application of linear regression for comparisons of X and Y include the following (1) the errors (variations) are independent of the magnitudes of X or Y, (2) the error distributions for both X and Y are known to be normally distributed (Gaussian), (3) the mean and variance of Y depend solely upon the absolute value of X, (4) the mean of each Y distribution is a straight-line function of X, and (5) the variance of X is zero, while the variance of Y is exactly the same for all values of X. 

The reaction enthalpy of the hydride transfer step usually has a low absolute value. Whether hydride transfer is exo- or endothermic depends on the stability (evidenced by the heat of formation) of the involved carbenium ions. Branched carbenium ions are more stable than linear ones. Longer carbenium ions are more stable than shorter ones. Replacement of a long-chain carbenium ion by... 

The trust region problem is to choose Ax to minimize PI in (8.54) subject to the trust region bounds (8.55) and (8.56). As discussed in Section (8.4), this piecewise linear problem can be transformed into an LP by introducing deviation variables p, and The absolute value terms become (p, + ,) and their arguments are set equal to Pi — nt. The equivalent LP is... 

In the simplest case, that of time-independent behaviour, the shear stress depends only on the shear rate but not in the proportional manner of a Newtonian fluid. Various types of time-independent behaviour are shown in Figure 1.19(a), in which the shear stress is plotted against the shear rate on linear axes. The absolute values of shear stress and shear rate are plotted so that irrespective of the sign convention used the curves always lie in the first quadrant. 

In diatomic molecules, T2 = 0, and thus the expectation value of C vanishes. This is the reason why this operator was not considered in Chapter 2. However, for linear triatomic molecules, t2 = / / 0, and the expectation value of C does not vanish. We note, however, that D J is a pseudoscalar operator. Since the Hamiltonian is a scalar, one must take either the absolute value of C [i.e., IC(0(4 2))I or its square IC(0(412))I2. We consider here its square, and add to either the local or the normal Hamiltonians (4.51) or (4.56) a term /412IC(0(412))I2. We thus consider, for the local-mode limit,... 

The range of rjk is — 1 to +1 a value of +1 indicates a perfect linear relationship, a value of —1 indicates a perfect inverse linear relationship absolute values of approximately <0.3 indicate a poor or no linear relationship. The Pearson correlation coefficient is best suited for normally distributed variables however, it is very sensitive to outliers. This coefficient is the most used correlation measure as usual also throughout this book the term correlation coefficient will be used for the Pearson correlation coefficient. 

It is evident that in centrosymmetric non-linear media the induced polarization in the +E direction is equal (in absolute value) to the induced polarization in the opposite direction -E. However, in... 

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