Separability assumption

Note that D and JF do not appear separately in (8.6) and (8.7) but only in the combination D + /JF/co, which we may interpret as the total electric displacement and assume that D in (8.5) is this quantity. In fact, we could have done this in previous chapters but refrained from doing so because the notion of conductivity is well established. However, it is not possible to determine from macroscopic experiments of the type discussed in this book if the imaginary part of the refractive index originates from free or bound charge currents. Thus, we need not make separate assumptions about the relations between D and E and between JF and E. 

Also presented in Figure 14 are the computer predictions of the breakthrough fronts based on three separate assumptions. 

Under the separability assumption, the primal problem for fixed y — yk takes the form... 

Now, we can state the algorithmic procedure for the vl-GBD under the separability assumption. 

Remark 1 The separability assumption results in condition C3 being satisfied. In fact, condition (73 can be satisfied with a weaker assumption than separability, and this was discussed in the previous sections. 

In the sequel, we will discuss the effect of the separability assumption on the primal subproblem, dual subproblem and the primal and Lagrange relaxation master problems. 

Remark 2 ConditionCl oftheGCD with the separability assumption becomes that... 

In conjugate systems, with both a- and re-type orbitals, the tr-re separability assumption is made. That is, it is assumed that the interaction between the a- and re-type electrons may be neglected. Parr, Ellison, and Lykos (1956) have shown that in such a case the total function, 0, may be written as the product... 

For a center-to-center separation z of 7.5 nm, one radius worth of surface-to-surface separation, already violating the large-separation assumption, the interaction is... 

Firstly, it has been shown that there may be many experimental problems in a direct determination of the experimental fimction. In shear, damping functions obtained from step strain and from step strain rate experiments do not match each other. This poses an important question on the validity of the separability assumption in the short time rai e. Significant departures from this factorization have already been observed in the case of narrow polystyrene fractions by Takahashi et al. [54]. These authors found that time-strain superposition of the linear and nonlinear relaxation moduli was only possible above a cert2un characteristic time. It is interesting to note that this is predicted by the Doi-Edwards theory [10] and according to this theory, this phenomena is attributed to an additional decrease of the modulus connected to a tube contraction process and time-strain separability may hold after this equilibration process has been completed. Other examples of non-separability were also reported by Einaga et al. [55] and more recently by Venerus et al. [56] for solutions. 

The Redfield equation, Eq. (10.155) has resulted from combining a weak system-bath coupling approximation, a timescale separation assumption, and the energy state representation. Equivalent time evolution equations valid under similar weak coupling and timescale separation conditions can be obtained in other representations. In particular, the position space representation cr(r, r ) and the phase space representation obtained from it by the Wigner transform... 

If we wish to see the delocalization, we can change the means of detection. For instance, we can trap the molecules in low-temperature beams. This minimizes the energy imparted to the molecule as we "touch" it. In these experimental setups, we can dispense with the separability assumption if conditions warrant. However, even in such beams, the assumption is most often justified. Because most rotational levels are separated by about 250 cm , we have to trap atoms for minutes at temperatures of milli-Kelvins before the separability assumption fails. ... 

The following discussion is most suitable for flat and spiral-wound separators. Assumptions mnde in the model development are summarized below,... 

In deriving the theory, three separate assumptions must be made. It must be asked how many links between molecules are present in a solution of a given concentration, what will be the structure of the network formed, and what will be the elastic modulus of such a network Three simple assumptions were made in this case. 

The theory outlined in Section II was based on the assumption that vibrational relaxation occurs on a time scale slow compared with the bath (translation and rotation) degrees of freedom. In this case, a Markov approximation (separation of time scales) can be made and the relaxation can be described through rate equations the resulting population decay is given by an exponential or a sum of exponentials. This time scale separation assumption is certainly valid for the small molecules that require a nanosecond or longer to relax, but in the picosecond or subpicosecond domain which applies to larger molecules non-Markovian effects may be present. In this section we outline the results of some theoretical studies of non-Markovian (nonexponential) relaxation. 

Under the Stokes flow and particle separation assumptions, the viscous force between two approaching particles should scale as fiaU, with n the Newtonian viscosity of the medium and U the approach velocity. With U ay, where y is the applied shear rate, the energy dissipation within the gap between the particles scales as fJi-a y. We have here assumed that the interaction frequency between the particles is of the order of y. This will be true so long as the particle concentration is not so high that we are close to the maximum packing fraction for which flow can occur, a point which is discussed in greater detail in the following section. 

This is referred to as the separability assumption in that the aerial image and the standing wave can be calculated independently and then multiplied together to yield the total intensity within the resist. ... 

A third assumption in RRKM theory is that the special coordinates, dq and dp, are perpendicular to all other coordinates, and that they can therefore be separated from the other coordinates. This separability is most valid at the saddle point and at very low energies where the vibrational modes of the transition state can be treated as normal modes. However, as the energy is increased, there will be coupling between the reaction coordinate and the rest of the modes, especially for the case of a complicated reaction, for example, HCl loss om C2H5CI+. The separability assumption is one that... 

Several points require consideration on identification of AH,(CH2) - T,(GVGIP)AS,(CH2) as -AGha(CH2). The points include the separability assumption of Equations (5.3) and (5.4), the relevance of the model protein to such identification, and the choice of reference state in order that the nonlinearity of hydrophobic-induced pKa shifts be included. From the data of Butler, the separability is reasonable for a simple CH group, but examination of the calculated result is required to be satisfied whether or not extension to more complex substituents is warranted. As the inverse temperature transition of (GVGVP) has been experimentally shown to involve no Raman detectable changes in secondary structure, the elastic-contractile model proteins of focus here reasonably represent the best known model available for such an effort. It should be noted, however, that NMR studies on the temperature and solvent dependence of peptide NH and... 

If the various degrees of freedom are separable (assumption 1 above), then the partition function can be factored into the product of partial partition functions ... 

Turner (Ref. [156]) argues that the zeroth law is a consequence of the first and second laws and therefore is not a separate assumption in the axiomatic framework of thermodynamics. The term law for this principle is also questioned by Redlich (Ref. [138]). 

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