Stepwise refinement

In the nonrelativistic limit, only the (LL LL) class of multi-centre integrals is needed this class is still the most important in relativistic calculations. In terms of the conventional expansion in powers of a, a DHF calculation at this level includes all the one-centre relativistic effects, screening by the Coulomb interaction, and some of the spin-other-orbit effects. The (LL 55) and (55 55) 

Include one-centre integrals of all classes (these are so cheap to make there is little point in excluding them) together with the (LL LL) integrals which would be needed for the equivalent nonrelativistic Hartree-Fock calculation. 

Add multi-centre integrals of the (55 SS) class, along with the multicentre Breit interaction integrals if desired. 

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A fundamental property of the Fourier transform is that of superposition. The usefulness of the Fourier method lies in the fact that one can separate a function into additive components, treat each one separately, and then build up the full result by summing the individual results. It is a beautiful and explicit example of the stepwise refinement of complex problems. In stepwise refinement, one successfully tackles the most difficult tasks and solves problems far beyond the mind s momentary grasp by dividing the problem into its ultimately simple pieces. The full solution is then obtained by reassembling the solved pieces. 

Components can themselves be treated as systems and further subdivided into components, and so on. The level of granulation need not be exhaustive a common-sense approach shonld prevail. Usnally only one or two stepwise refinements are appropriate. 

Complex Refinement Structures. Flowsheets are refined across various hierarchical abstraction levels and depicted in specific representation formats. In the context of the IMPROVE project, we are mainly interested in two of the three commonly used flowsheet variants the abstract flow diagram (AFD) and the process flow diagram (PFD). Thus, the flowsheet editor supports the full hierarchy of complex stepwise refinement operations from AFD to PFD and assures a set of consistency constrains in order to preserve their semantic correctness. 

Aljer and Devienne [5] consider the use of a formal specification language as the foundation of real validation process. They propose architecture based upon stepwise refinement of a formal model to achieve controllable implementation. Partitioning, fault tolerance, and system management are seen as particular cases of refinement in order to conceptualize systems correct by proven construction. The methodology based on the refinement paradigm is described. To prove this approach, the B-HDL tool based on a combination of VHDL and B method formal language has been developed. 

Wirtb N., "Program Develo Hnent I Stepwise Refinement", Comm ACM, vol 14,1971, pp 221-227. 

The Constraint Processing Logic is responsible for the stepwise refinement of the design parameters. The Automatic Conflict Solver takes a conflict between two or more constraints as an input and generates a list of possible solution ordered by a... 

Turkevich who established the first reproducible standard procedure for the preparation of metal colloids [44] also proposed a mechanism for the stepwise formation of nanoclusters based on nucleation, growth, and agglomeration [45,46]. This model, refined by data from modern analydical techniques and results from thermodynamic and kinetic studies, is in essence stiU valid today (Figure 2) [82]. 

Although newer technologies are always under development, the basic kraft chemical recovery process has not been fundamentally changed since the issue of its patent in 1884. The stepwise progression of chemical reactions has been refined for example, black liquor gasification processes are now in use in an experimental phase. The precise details of the chemical processes at work in the chemical recovery process can be found in Smook s Handbook.12 The kraft chemical recovery process consists of the following general steps ... 

Because the whole idea of a tiered approach of the kind outlined above is in its initial stages, it will have to be validated and discussed further and will in all probability need to be refined afterwards. The aim here is to introduce the idea of a stepwise approach to the assessment of the risk to re-entry workers. The outlined procedure should be used to calculate the dermal re-entry exposure for real examples of rather dermally toxic compounds in order to gain experience with the recommended procedure. 

Concerted [3 + 2] cycloaddition was proposed as the mechanism for dihydroxylation originally by Boesken195 and Criegee166 and reiterated in a refined manner by Corey.172 196 197 Another mechanism, stepwise [2 + 2] cycloaddition, has been proposed by Sharpless (Scheme 50).198-200... 

Such studies may yield useful information, but only when more refined values of the ionization potential of HgR are available. For purposes of discussion, it will be assumed that the stepwise mechanism is correct. 

The outcome of each simulation was used to compare the ability of each stepwise sequence to predict EU classifications and to reduce and refine the use of animals, with the corresponding ability of the EPISKIN test, when used as a stand-alone alternative method. 

The results of this study and of a previous study (Worth et al., 1998) show that stepwise approaches to hazard classification, in which alternative methods are applied before animal tests, provide a promising means of reducing and refining the use of animals. In these approaches, fewer animal experiments need to be conducted, and of those chemicals tested in vivo, the majority are found to be non-toxic. 

Thus, the level of sophistication which one may consider for the application of rubber-like elasticity theory to epoxy networks may depend on the application. For highly crosslinked systems (M < 1,000), a quantitative dependence of the rubbery modulus on network chain length has recently been demonstrated , but the relevance of higher order refinements in elasticity theory is questionable. Less densely crosslinked epoxies, however, are potentially suitable for testing modern elasticity theories because they form via near quantitative stepwise reactions. Detailed investigations of such networks have been reported by Dusek and coworkers in recent studies ... 

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