Undetermined term

The undetermined term is given by applying a mass balance for the liquid phase ... 

It is easily seen that the principal closure problem is maintained since three undetermined terms appear in the volume averaged equations (3.186). 

As in the volume averaged equations discussed in sect 3.4.1, three undetermined terms can be identified in the time averaged equations (3.259). The first term, —V (/3fc(pfcv V )rfc) denotes the covariance or correlation terms. The second term, Tkikj, accounts for the effects of interfacial stress. 

So far in the derivation of the averaged equations the basic concepts used are considered fairly rigorous, but in order to put (3.329) and (3.330) into directly usable forms several modeling approaches have emerged proposing quite different manipulations and approximations of the undetermined terms. 

The definitions contain undetermined terms such as frequently , over long periods of time , occasionally etc. Concrete numbers, which would make sense, are avoided. An orientation can be provided by the numbers indicated in Table 4.19, which can be applied analogously to zones 20-22. 

As expected, three groups of undetermined terms appear in the averaged Eq. (3.291). The first term, V ... 

Henee, for this non-degenerate root, one is able to solve for all of the vj elements in terms of one element that is yet undetermined. 

The applications of quantitative structure-reactivity analysis to cyclodextrin com-plexation and cyclodextrin catalysis, mostly from our laboratories, as well as the experimental and theoretical backgrounds of these approaches, are reviewed. These approaches enable us to separate several intermolecular interactions, acting simultaneously, from one another in terms of physicochemical parameters, to evaluate the extent to which each interaction contributes, and to predict thermodynamic stabilities and/or kinetic rate constants experimentally undetermined. Conclusions obtained are mostly consistent with those deduced from experimental measurements. 

In the presence of excess bromine, the first-order rate coefficient was 10.3 x 10" 5, but kinetic studies here were complicated due to rapid reaction of bromine with the 2,4,6-tribromophenol to give the intermediate (LXII), with SO3 replaced by Br) which slowly decomposed with a rate coefficient k2 to give two products. By analysis in terms of two consecutive first-order reactions, the values of k2 and k2 were determined as 9.2 x 10"5 and 3.75 x 10"4 and the latter rate being the faster means that two moles of bromine were consumed for every mole of sulphonate undergoing substitution in fact, more than two moles were consumed, the reason for this being undetermined. 

In Appendix A, we follow the derivation of Shi and Rabitz and carry out the functional variation of the objective functional [Eq. (1)] so as to obtain the equations that must be obeyed by the wave function (vl/(t)), the undetermined Lagrange multiplier (x(0)> the electric field (e(t)). Since the results discussed in Section IV.B focus on controlled excitation of H2, where molecular polarizability must be considered, the penalty term given by Eq. (3) is used and the equations that must be obeyed by these functions are (see Appendix A for a detailed derivation) ... 

The adiabatic approximation involves neglect of these off-diagonal terms, on the basis that En(Q) — Em(Q) (xm VQH xn). The diagonal elements (xn Vg x ) are undetermined by this argument, but the gradient of the normalization integral, (x x ) = 1, shows that... 

The gene product is termed cystic fibrosis transmembrane conductance regulator (CFTR), and it codes for a chloride ion channel. It may also carry out additional (as yet undetermined) functions. 

Derivation of the Boltzmann distribution function is based on statistical mechanical considerations and requires use of Stirling s approximation and Lagrange s method of undetermined multipliers to arrive at the basic equation, (N,/No) = (g/go)exp[-A Ae/]. The exponential term /3 defines the temperature scale of the Boltzmann function and can be shown to equal t/ksT. In classical mechanics, this distribution is defined by giving values for the coordinates and momenta for each particle in three-coordinate space and the lin-... 

The exact solution of the problem leads to the same expression with a proportionality constant between 3 and 5, depending on the definition of the thickness of the boundary layer. In the following sections, the preceding evaluation procedure is applied to a large number of problems, particularly to complex cases for which limiting solutions can be obtained. As already noted in the introduction, the terms in the transport equations will be replaced by their evaluating expressions multiplied by constants. The undetermined constants will then be determined from solutions available for some asymptotic cases. 

The space-charge current density in vacuo expressed by Eqs. (3) and (4) constitutes the essential part of the present extended theory. To specify the thus far undetermined velocity C, we follow the classical method of recasting Maxwell s equations into a four-dimensional representation. The divergence of Eq. (1) can, in combination with Eq. (4), be expressed in terms of a fourdimensional operator, where (j, 7 p) thus becomes a 4-vector. The potentials A and are derived from the sources j and p, which yield... 

St. John s wort raises other concerns, as well. First of all, no one should attempt to diagnose and treat their own depression. Then there is the fact that most St. John s wort studies lasted only a few weeks, so long-term effects remain undetermined. In the short term some people may experience side effects such as upset stomach, skin rash, or tiredness, although these are less common with St. John s wort than they are with prescrip-... 

Summarizing this section, we conclude that, from an experimental point of view, the ideal network theory expressed by Eq. (III-9) seems to be obeyed rather well only in the case of swollen networks. The determination of the elasticity parameters is complicated by the presence of a free enthalpy of dilution term. In spite of this, it is possible to conclude that 1/2 is the most likely value for B in Eq. (III-9). The factor A remains undetermined because experimentally the most one can obtain is A g o 2 8. Concerning this memory parameter, a dependence on the stage of dilution during crosslinking definitely exists, (Eq. Ill-19), but its dependence upon the nature of the diluent used in swelling studies needs further investigation. 

Besides the identified products, there were isolated several transformation products, the structure of which were not detd. Another undetermined problem was the fate of portion of Centr l(ca 50%) which was apparently depleted from the proplnts and could not be accounted for in terms of Centr derivs in the extract... 

The amplitude a, which remains undetermined at this stage is fixed by the solvability conditions imposed on the equations for the higher order terms of the perturbation expansion (5). We arrive in this way at the following set of bifurcation equations for the normalized amplitude (3 = ta. 

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