Partitioning fuzzy boundary

If the electron density partitioning results in subsystems without boundaries and with convergence properties which closely resemble the convergence properties of the complete system, then it is possible to avoid one of the conditions of the Holographic Electron Density Fragment Theorem , by generating fuzzy electron density fragments which do not have boundaries themselves, but then the actual subsystems considered cannot be confined to any finite domain D of the ordinary three-dimensional space E3. 

Figure 6.1 shows the stockholder decomposition of the theoretical deformation density of the cyanoacetylene molecule, H—Cs=C—C=N (Hirshfeld 1977b). The overlap density in the bonds is distributed between the bonded atoms. The assignment of part of the density near the hydrogen nucleus to the adjacent carbon atom manifests the difference between fuzzy and discrete boundary partitioning methods. 

Classical mathematics requires that a concept has a precise definition which partitions a class of objects into two classes (a) those that belong, and (b) those that do not belong. But the real world is filled with concepts such as many, old, slow, large. These do not have sharply defined boundaries, but still convey information. Fuzzy logic deals with the application of such knowledge. 

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