Scaling nesting similarity measures

IX. A FUZZY SCALING-NESTING SIMILARITY MEASURE AND THE FUZZY SCALING-NESTING DISSIMILARITY METRIC... 

X. FUZZY MEASURES OF CHIRALITY AND SYMMETRY DEFICIENCY, FUZZY SYMMETRY GROUPS, AND FUZZY SYMMORPHY GROUPS BASED ON THE FUZZY SCALING-NESTING SIMILARITY MEASURE... 

The fuzzy scaling-nesting similarity measure fs g and the fuzzy scaling-nesting dissimilarity metric dfJ,A,B) provide such alternatives. The entire treatment used for the introduction and definition of fuzzy measures of symmetry deficiency, fuzzy chirality, fuzzy symmetry, and fuzzy... 

The same treatment can be applied in the derivation of fuzzy symmor-phy groups based on FSNSM. In Section VI, fuzzy symmorphy and fuzzy symmorphy groups were developed based on the fuzzy Hausdorff-type similarity measure Sg. The steps of the entire derivation can be repeated for FSNSM fs. A valid description of fuzzy symmorphy is obtained if the fuzzy Flausdorff-type similarity measure Sg is replaced with the fuzzy scaling-nesting similarity measure fs in each equation and inequality of Section VI that involves the fuzzy Hausdorff-type similarity measure Sg. 

The fuzzy scaling-nesting similarity measure is simpler to visualize than variants of the Flausdorff metric hence, FSNSM is advantageous in chemical applications of electron density similarity analysis, where chemical intuition and visualization play a prominent role. 

Similarly, the fuzzy scaling-nesting similarity measures serve as the basis of measures of approximate symmetry, with respect to the fuzzy average. 

Semi-Similarity Measures and Scaling-Nesting Similarity Measures (SNSM)... 

Figure 6.5 A two-dimensional example of the Scaling-Nesting Similarity Measure (SNSM), applied to the planar objects A and B. A scaling of S(ba)=0.80 is sufficient for fitting object B within A, whereas a scaling of s(ab)=0-68 is needed for fitting object A within B. Note that, according to these one-sided measures (semi-similarity measures), A is less similar to B than B is to A. A symmetric SNSM of s b=0.74 is obtained as the average of the above one-sided scaling-nesting similarity measures.

A definition of the unconstrained scaling-nesting similarity measures is given below the constrained measures can be derived easily from these by applying the appropriate constraint while determining the maximum scaling factor. 

Scaling-nesting similarity measures (SNSM) are obtained by determining a maximum scaling factor that allows one molecular body to fit within another. For bodies initially scaled to unit volume, if the shapes are identical, then the scaling... 

---