Reciprocal Extremal Problems

Here x and y are the lengths of the edges of the rectangle. Problems of this type frequently appear in mathematical physics, in particular in the more advanced calculus of variation. We will address this set of somehow related problems as reciprocal extremal problems. 

However, the theorem of reciprocity is a wave optical argument that does not consider intensities where we easily find the differences. For example, if one thinks of a STEM as an inverted HRTEM one would not detect any intensity in an image since it is an inherent property of a point detector to collect no intensity. On the other extreme side, the ability to form an intense and focused probe is a valuable ability that boosts local spectroscopy. Obviously, the best choice of tools cannot be a matter of exclusion but must relate to the problem at hand that needs solving . 

In 1934, Dean Burk (1904-88) and Hans Lineweaver (1907- ), at the United States Department of Agriculture, proposed an extremely simple mathematical solution to this experimental problem. By simply plotting the reciprocals of... 

Although pump casing failures are likely to be extremely serious, they are also quite rare. Causes of casing failure include overpressure, mechanical stress, and reaction with the process materials. If the internal rotating or reciprocating element fails, then there is likely to be a major process upset. However, such a failure will probably not create safety problems because the casing will remain intact. Casing failures are normally very serious because ... 

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