An Alternative Formulation

An exact solution of the Maxwell-Stefan equations may be obtained in a somewhat different way as shown below. Equations 1.2.12 are written in n — 1 dimensional matrix form 

In so far as [B] (or, more generally, [D] ) depends on the composition of the mixture and as the composition is, in turn, a function of position 17, we may regard [T ] as a function of 77. Thus, Eq. 8.3.50 is a first-order matrix differential equation of order n — 1 with a variable coefficient matrix [ (17)]. This equation may be solved by the method of repeated substitution as shown in Appendix B.2. The solution is 

Equations 8.3.52-8.3.57 were presented as an exact solution of the Maxwell-Stefan equations for diffusion in ideal gas mixtures by Burghardt (1984). Equations 8.3.52-8.3.57 are somewhat less useful than Eqs. 8.3.15-8.3.24 because we need to know the composition profiles in order to evaluate the matrizant. Even if the profiles are known, the computation of the fluxes from either of Eqs. 8.3.62 or 8.3.63 is not straightforward and not recommended. It is with the development in Section 8.4 in mind that we have included these results here. 

In 1964 Toor and Stewart and Prober independently put forward a general approach to the solution of multicomponent diffusion problems. Their method, which was discussed in detail in Chapter 5, relies on the assumption of constancy of the Fick matrix [D] along the diffusion path. The so-called Tinearized theory of Toor, Stewart, and Prober is not limited to describing steady-state, one-dimensional diffusion in ideal gas mixtures (as we have already demonstrated in Chapter 5) however, for this particular situation Eq. 5.3.5, with [P] given by Eq. 4.2.2, simplifies to 

We emphasize the difference between the Toor, Stewart, and Prober approach and the exact method considered above by using the subscript av thus, [Pavl- For practical purposes, this means that [Pavl has to be evaluated by employing suitable average mole fractions, av iri the definition of the (Eqs. 2.1.21 and 2.1.22). The arithmetic average mole fraction av = + F/sX normally is recommended for calculation of (Stewart 

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The possibility of a nongradient component in the integrand introduces some difficulty and an alternative formulation has been proposed [273,283]. [At positions that are close to a ci, the alternative approximates well to the angle shown above in Eq. (73).]... 

An alternative formulation of QM/MM is the energy subtraction method. In this method, calculations are done on various regions of the molecule with... 

The point of the weighting function can be illustrated by considering the two extremes of complete scrambling (for which ) = 1), and complete routing (forwhich f() = 1/Fp). If we choose to make f(F) = F ,we have constructed a convenient function that can be adjusted to take intermediate values (for this particular DIFF), where CO may take any values between 0 (where f(F) = 1 scrambling) and -1 (where f(F) = 1/F direct routing). Later (Section 4) we use an alternative formulation, in which... 

Also, an alternative formulation of equation (17) can be conceived if one wants to distinguish between ground state, monoexcitations, biexcitations,. .. and so on. Such a possibility is symbolized in the following Cl wavefunction expression for n electrons, constructed as to include Slater determinants up to the p-th (pp) unoccupied ones l9klk=i,ni Then, the Cl wavefunction is written in this case as the linear combination ... 

One reason for the relatively large RMS deviations, compared to the active sites of MMO and RNR, is that the active-site residues are not coordinated to the selenium (see Figure 2-8). The lack of a structural anchor leads to a relatively unstable active-site geometry. An alternative formulation is that the presence of a metal center with strong ligand interactions is one reason the active-site model works comparatively well for many metal enzymes. 

An alternative formulation is that given two finite code sets, the Post Correspondence Problem inquires whether any word is coded alike in both codes. 

Some remarkable chemistry is observed when silenes react with heteroatom systems, in particular carbonyl compounds (]>C=0) and imines Q>C=N—R). The reaction with ketones was first described by Sommer (203), who postulated formation of an intermediate siloxetane which could not be observed and hence was considered to be unstable even at room temperature, decomposing spontaneously to a silanone (normally isolated as its trimer and other oligomers) and the observed alkene [Eq. (14)]. Many efforts have been made to demonstrate the existence of the siloxetane, but it is only very recently that claims have been advanced for the isolation of this species. In one case (86) an alternative formulation for the product obtained has been advanced (204). In a second case (121) involving reaction of a highly hindered silene with cyclopentadienones,... 

It is not known if the proposed 9 complex observed in nonaqueous solvents, is formed in water. An alternative formulation would have the hydroxo ligand still coordinated to iron and this hypothetical species 10 could also be deprotonated to give a ra zs-dioxo species. In fact, our mechanistic theory for the pH dependent rate of interaction Fem-TAML complexes with H202 is... 

An alternative formulation is based on the concept of immediate batch precedence. In contrast to the previous model, allocation and sequencing decisions are divided into two different sets of binary variables. This idea is described in the work presented by Mendez et al. [30], where a single-stage batch plant with multiple equipment in parallel is assumed. Relevant work following this direction can also be found in Gupta and Karimi [31]. Key variables are defined as follows ... 

As in the steady-state case, the implementation of the chi-square test is quite simple, but has its limitations. One can use a more sophisticated technique such as the Generalized Likelihood Ratio (GLR) test. An alternative formulation of the chi-square test is to consider the components of jk separately (this may be useful for failure isolation information). In this case we compute the innovation of the ith measurement as... 

An alternative formulation of the Fukui functions based on the mutual polarization of the reagents can be constructed as follows [24,32]. In a chemical reaction, the description of a local electron transfer does not need to involve any actual change... 

An alternative formulation is based on a rather than on q. It starts with the expression of q as a function of a and introduction in equation (6.229). After integration, taking into account the appropriate boundary conditions, one obtains, at the first wave, in = /A( I — ay=i), with... 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

An alternative formulation of the phase-transfer DCC concept was reported in 2008 by the Sanders group [75]. In this case, thiol monomers were dissolved in water on either side of a U-tube containing chloroform (Fig. 1.23). After allowing the system to reach equilibrium, monomer distribution was identical in both aqueous solutions, and mixed species (e.g., 51) were observed in the chloroform layer. 

With the basic problem in standard form it is easy to switch to an alternate formulation by scaling the matrices P and Q. By scaling P by Hp/ o and scaling Q by l/Ao Ip, we obtain... 

In the original formulation of the R12 method [23], which is often referred to as "the standard approximation" or SA, a large uncontracted OBS was needed as the RI basis, strongly limiting the applicability of the R12 method to relatively small systems. Klopper and Samson [26] later pioneered the use of a separate RI basis set and considerably improved the applicability and accuracy of MP2-R12. Valeev developed an alternative formulation of the RI procedure [27] known as the complementary auxiliary basis set (CABS) approach, which had smaller RI errors. Ten-no, on the other hand, proposed the use of a numerical quadrature for the same purpose [28]. Kedzuch et al. later used the CABS approach to develop a particularly elegant and practical formalism for the MP2-R12 energies [30]. 

As remarked in Approach 1, a potential complication with Expectation Model I lies in computing a suitable range of values for the operational risk factor 0j. Therefore, an alternative formulation of minimizing variance while adding a target profit constraint is employed for Expectation Model II ... 

We formulated quantum mechanics in Chapter 1 in terms of operators and their eigenfunctions. There exists an alternative formulation (Section 2.3) in terms of matrices and their eigenvectors, which is sometimes more convenient than the operator formulation. Matrix algebra was developed by Cayley in the mid-nineteenth century. Scientists remained unaware of matrices until 1925, when quantum mechanics was born. Today there is hardly a branch of science from physics to psychology that does not use matrices. 

This relationship may be inserted back into the general case of eqn. (40) to produce an alternative formulation of the Butler—Volmer relationship. 

The hypothetical reaction mechanism shown in Fig. 10 is a variation of Scheme (3), and is consistent with the redox and pH dependence of the EPR-detectable nickel species (65). Hydrogen is known to undergo heterolytic cleavage (81) it is proposed that this is an intramolecular reaction, leading to the formation of a nickel (II) hydride and a protonated base in the enzyme (Step 1 in Fig. 10). The Ni-C species is postulated to be a protonated Ni(I) species. An alternative formulation for this state would be a dihydrogen complex, Ni(III) H2, as suggested by Crabtree (104). Ultimately the exact mechanism can only be determined by kinetic measurements. 

Fig. 10. Hypothetical reaction cycle for D. gigas hydrogenase, based on the EPR and redox properties of the nickel (Table II). Only the nickel center and one [4Fe-4S] cluster are shown. Step 1 enzyme, in the activated conformation and Ni(II) oxidation state, causes heterolytic cleavage of H2 to produce a Ni(II) hydride and a proton which might be associated with a ligand to the nickel or another base in the vicinity of the metal site. Step 2 intramolecular electron transfer to the iron-sulfur cluster produces a protonated Ni(I) site (giving the Ni-C signal). An alternative formulation of this species would be Ni(III) - H2. Step 3 reoxidation of the iron-sulfur cluster and release of a proton. Step 4 reoxidation of Ni and release of the other proton.

An alternative formulation for the torus-computing algorithm is to solve for an invariant circle along with a nonlinear change of coordinates that makes the action of the stroboscopic map conjugate to a rigid rotation on the circle. This is equivalent to the parameterization... 

This structure should be contrasted with an alternative formulation of this species as a complex containing one linear and one bent nitrosyl which is suggested in (110) for the Ru analog. The dinitrogen dioxide ligand differs... 

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