Higher-order equations, rewriting

Because Pdb(oP,o, Pe,oi) is a higher-order polynomial of the Boltzmann factor, Equation 14.2 cannot have the same structure as Equation 14.1 that is, rewriting Equation 14.2 as... 

An iterative method for developing higher order approximations to the solution to Equation 1 can be devised by using some of the ideas of singular perturbation theory (15). To display the systematics of the procedure let us rewrite Equation 7 as... 

If the mass transfer rate is small, then the velocity profile is unaltered by diffusion and the above series expression for velocity can be applied in the convective-diffusion equation. Here, Sc being large, we can assume that convection dominates over diffusion. We assume that at close to the wall, p is small and higher-order terms are neglected. We rewrite equations (4.96)... 

This valence model Hamiltonian contains high-order many-electron operators due to the presence of products of projection operators and would be clearly much too complicated for practical calculations. Moreover, no computational savings would result compared to a standard all-electron treatment, since the derivation given so far merely corresponds to a rewriting of the Fock equation for a valence orbital in a quite complicated form (eqns. 41 and 43). Actual reductions of the computational effort can be achieved only at the price of approximations, i.e., by the actual elimination of the core electron system and the simulation of its influence on the valence electrons by introducing a simplified valence-only model Hamiltonian containing the pseudopotential If higher than two-electron operators in eqn. 45 are omitted this corresponds to the formal substitutions ... 

Qualitatively we can do the same for the polymeric fractal, if one extends the meaning of s to some variable space of dimensions in order to take the higher dimensional connectivity into account. Thus the variable R(s) covers the entire fractal object. Then the values of q are no longer onedimensional quantities, they are defined as ds dimensional variables. Rewriting the equation for the viscosity for a polymeric fractal and replacing the sum by an integral ... 

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