Linear Dependence and Inconsistency

In Chapter 3 we discussed linear dependence and inconsistency in the case of two equations. We will not discuss completely the questions of consistency and independence for sets of more than two equations, but we will make the following comments, which apply to sets of linear inhomogeneous equations ... 

However, not only the contact lines themselves are beneath all the points but their slope is also too small to explain the true concentration dependence. Perhaps these difficulties stimulated Stevens to propose the original finite-sink approximation instead of the existing theory [103,250]. The inconsistency of this model is seen from the very fact that it does not discriminate between the Stem-Volmer and the steady-state constants, k and k = lim, >rXj k(t), and ascribes to the latter the concentration dependence, which is not inherent to k in principle. At the same time the model predicts the linear dependence of the results in coordinates 1/k versus c1/3, and this linearity was confirmed experimentally a number of times [6,248,250,251], According to their finite-sink model [248], the straight lines representing this dependence for different temperatures should intersect at a common point where the static quenching limit is unambiguously located. ... 

Although the dependence on the gas velocity appears to be reasonably explained by the above described model approaches, the experimental data available in the literature are giving even in this respect an inconsistent picture Merrick and Highley (1974), Arena et al. (1983), and Pis et al. (1991) also found the linear dependence on the excess gas velocity (m — Mmf) to be valid. As an example. Fig. 15 shows the results of Pis et al. (1991), which were obtained in a fluidized bed column of 0.14 m in diameter. The distributor had orifices of 1 mm in diameter on a 5 mm square pitch. Unfortunately, no distinction was made between the measured attrition rate and the influence... 

We can derive the same inference from Tables 30.2, 30.4 and 30.5, specifying current ratings for different cross-sections. The current-carrying capacity varies with the cross-section not in a linear but in an inconsistent way depending upon the cross-section and the number of conductors used in parallel. It is not possible to define accurately the current rating of a conductor through a mathematical expression. This can be established only by laboratory tests. 

We must stress, however, that the Black-Halperin analysis has been conducted for only a single substance, namely, amorphous silica, and systematic studies on other materials should be done. The discovered numerical inconsistency may well turn out to be within the deviations of the heat capacity and conductivity from the strict linear and quadratic laws, repsectively. Finally, a controllable kinetic treatment of a time-dependent experiment would be necessary. 

This model is incorrect because the linear thermal expansivity for both components in the isotropic and oriented state is assumed to be the same. The concequences of this assumption are quite different for Px and For px, it does not play any essential role, because in the isotropic and oriented state perpendicular to the draw axis the thermal expansivity is determined solely by intermolecular interactions. For P, this suggestion may lead to a principal inconsistency. This conclusion is evident from comparison of the calculated and experimental -dependences of P and P for PE and PP according to Eqs. (107) and (108). For Px, the agreement between the model calculation and the experiment is quite satisfactory for all draw ratios. On the other hand, Eq. (108) does not describe the X-dependence of P( at all. This equation does not yield negative values of P even in case of a limited orientation of crystallites (fc = 1) because it is based on the suggestion that Pam is always positive and pam > Pcr. ... 

High-level DAE software (e.g., Dassl) makes a time-step selection based on an estimate of the local truncation error, which depends on the difference between a predictor and a corrector step [13,46]. If the difference is too great, the time step is reduced. In the limit of At 0, the predictor is just the initial condition. For the simple linear problem illustrated here, the corrector will always converge to the correct solution y2 = 1, independent of the time step. However, if the initial condition is y2 1, then there is simply no time step for which the predictor and corrector values will be sufficiently close, and the error estimate will always fail. Based on this simple problem, it may seem like a straightforward task to build software that identifies and avoids the problem, and there is current research on the subject [13], The problem is that in highly nonlinear, coupled, problems the inconsistent initial conditions can be extremely difficult to identify and fix in a general way. 

One may see that in Eq. (2.18) O0/3> depends linearly on the concentration of the quencher, which is inconsistent with the experiment (see Fig. 2.15). Therefore, in further description of experimental data we use Eq. (2.19), which corresponds to the case when the recombination sites are not identical to the adsorption sites. In this case, A and Kads are the varied fitting parameters. Physical meaning of parameter A corresponds to the ratio between the probability of electron transfer to the quencher molecules and the total probability to disappear upon recombination. 

Some of the molecular theories of multicomponent diffusion in mixtures led to expressions for mass flow of the Maxwell-Stefan form, and predicted mass flow dependent on the velocity gradients in the system. Such dependencies are not allowed in linear nonequilibrium thermodynamics. Mass flow contains concentration rather than activity as driving forces. In order to overcome this inconsistency, we must start with Jaumann s entropy balance equation... 

The similar order of magnitude of the reactivities of methyl-substituted 1,3-dienes (Table 4) which depended on the number but not on the position of the substituent was strong evidence that allyl cations serve as reaction intermediates in these reactions. The rate decrease with increase in the ring size of the cycloalkadienes was attributed to the increased deviation of the jr-system from planarity. The reactivities of 1,3-dienes deviated markedly from the roughly linear relationship between the rates of proton and carbenium ion additions to alkenes. These deviations were ascribed to abnonnally low reactivity of the conjugated jr-systems. although this interpretation was inconsistent with the similar behavior of alkenes and dienes in the structure-reactivity relationship for hydration . 

It has recently been shown [ 12] that time-dependent or linear-response theory based on local exchange and correlation potentials is inconsistent in the pure exchange limit with the time-dependent Hartree-Fock theory (TDHF) of Dirac [13] and with the random-phase approximation (RPA) [14] including exchange. The DFT-based exchange-response kernel [15] is inconsistent with the structure of the second-quantized Hamiltonian. 

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