INQUIRY other features

It may be fairly said that the field of non-Kekule compounds has expanded from a few curious and esoteric molecules to an active domain of inquiry and a potential source of materials for practical application. Theory and experiment have interacted fruitfully in these developments, but some serious limitations remain. For example, for many years, experimentalists have operated under the imperative of structural simplification. The ideal is to construct molecules that embody the features of theoretical interest but are as free as possible of complicating impedimenta such as extra substituents or other structural elements left over from the synthesis itself and not readily removed. 

In this section we describe some of the essential features of fractal functions starting from the simple dynamical processes described by functions that are fractal (such as the Weierstrass function) and that are continuous everywhere but are nowhere differentiable. This idea of nondifferentiability leads to the introduction of the elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. We find that the relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, the changes in time of phenomena that are best described by fractal functions are probably best described by fractional equations of motion, as well. In any event, this latter perspective is the one we developed elsewhere [52] and discuss herein. Others have also made inquiries along these lines [70] ... 

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