Sufficiency condition search

In this framework, optimizing the division scheme of the potential seems to allow us to approach such uniqueness [65]. By searching through the function space, one hopes to find a unique representation B = B(j ) after renormalization. Such endeavors are beyond the scope of research in the field of SCIETs. The logic is as follows If the closure is exact, it must necessary satisfy all known thermodynamic and structural conditions. The reverse question, the sufficiency condition, requires more care. If such a functionality provides, a posteriori, accurate results for all properties, though not proof of unique functionality, it is favorably disposed to it. 

We end this chapter by pointing out that the variation of an objective functional will provide us with important clues about its optimum, similar to what a differential does for an objective function. The second variation will provide some auxiliary conditions and help in the search for optimal solutions. The necessary and sufficient conditions for the optimum of an objective functional will be the topic of the next chapter. As expected, those conditions will use the concepts we have developed here. 

In this chapter, we wish to touch on a number of automated AR construction schemes. At the time of writing, research in AR theory has witnessed a shift toward the development of numerical AR constmction algorithms, with less emphasis placed on general AR theory. These developments have arisen primarily out of a practical need to determine candidate regions for complex, higher dimensional problems, which are not easily computed by hand, but which are still important for practical problems of interest. AR construction methods provide a numerical basis wherefrom theoretical predictions may be compared with in the search for a sufficiency condition. Inasmuch as how... 

A sufficient condition for having Poissonian stationary distribution in a certain class of birth and death processes was given by Whittle (1968). However, his assumptions are slightly different from those of chemical reactions, therefore the search for precisely defined assumptions or for certain classes of reactions is necessary. 

On one rather technical point, the optimum theory does differ from classical calculus of variations. This is in regard to the Weierstrass- or end condition, which is not satisfied by the rigorous optimum control theory. The reason for this nicety in the sufficient conditions for an optimum (and the mathematics will not be pursued further) is that optimum control theory is based on the utilitarian search for a greatest or a least value of a functional rather than a maximum or a minimum. These latter terms have precise mathematical meaning in relation to their turning values, whereas all we may require in application is the least value of a cost functional. The example of a finite straight line may make this point clearer, for such a line has neither maximum nor minimum and yet will have, at its ends, greatest and least values. 

These searches based on adjacent C , atoms are very fast in operation because they make use of the faa that a necessary, but not sufficient, condition for two sets of atoms to have the same structure is that the sum of the interatomic distances is the same. This observation allows rapid rejection of nonhits that can then be checked in more detail if required. ... 

A consistent data model is a necessary but not sufficient condition for a distributed system that provides data access methods acceptable to users. It is also essential to provide a coherent way to search for and update information in multiple, diverse databases. This has implications for applications running in part at the workstation GUI, in that users want a single interface to diverse information. 

A sufficient condition for convergence of Eq. 1.29) to the root x is dial g fjc) < I for all X in the search interval. Fig. 1,2b shows the case when this condition is not valid and the method diverges. This analytical test is often difficult in practice. In a computer program it is easier to determine whether x - jTjI < jjCj - jc, and, therefore, the. successive x values converge. The advantage of this method is that it can be started with only a single point, without the need for calculating the derivative of the function. 

Katz et fl/.[l] searched the literature for data that could be used to identify the pertinent dispersion equation for a packed column in liquid chromatography. As a result of the search, no data was found that had been measured with the necessary accuracy and precision and under the sufficiently diverse solute/mobile phase conditions required to meet the second criteria given above. It became obvious that a... 

The chemical process industries are competitive, and the information that is published on commercial processes is restricted. The articles on particular processes published in the technical literature and in textbooks invariably give only a superficial account of the chemistry and unit operations used. They lack the detailed information needed on reaction kinetics, process conditions, equipment parameters, and physical properties needed for process design. The information that can be found in the general literature is, however, useful in the early stages of a project, when searching for possible process routes. It is often sufficient for a flow-sheet of the process to be drawn up and a rough estimate of the capital and production costs made. 

There were a number of practical and theoretical issues to be addressed. A key scientific question was whether fixation in formalin modified antigens in a reversible or irreversible manner. To be more specific, was there any theoretical or prior scientific evidence that the effects of formalin fixation on proteins could be reversed, and if reversed, was the structure of protein restored to a sufficient degree for recovery of antigenicity With these key questions in mind, one of the authors (Shi) spent many days and nights in 1988 searching the chemical literature under somewhat adverse conditions, with a second job as an apprentice in a supermarket, and prior to the increased efficiency of such searches that is afforded today by the Internet and online databases. The answer was finally found in a series of studies of the chemical... 

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