Symmetry deficiency

Mezey, P. G. (1997) A proof of the metric properties of the symmetric scalingnesting dissimilarity measure and related symmetry deficiency measures. Int. J. Quantum Chem. 63, 105-109. 

Molecular chirality " and various other, more general symmetry deficiencies ... 

IV. FUZZY SYMMETRY DEFICIENCY MEASURES, FUZZY CHIRALITY MEASURES, AND FUZZY SYMMETRY GROUPS BASED ON THE MASS OF FUZZY SETS AND FUZZY HAUSDORFF-TYPE METRICS... 

A measure 5(/l,R,gg) of the symmetry deficiency of fuzzy set A in symmetry element R according to the fuzzy Hausdorff-type similarity measure s can be defined as... 

Alternative symmetry deficiency measures of fuzzy sets are defined following the treatment of symmetry deficiency of ordinary subsets of finite n-dimensional Euclidean spaces, introduced earlier. To this end, we shall use certain concepts derived as generalizations of concepts in crisp set theory. 

The measure m A) of a fuzzy set A is analogous to the concept of area viy) or, in general, the n-dimensional volume of an ordinary set T, where the formal mass-density of ordinary set Y can be interpreted as the constant membership function /Liy(x) = 1 for every point x of set Y. For fuzzy sets, however, the concept of volume does not appear particularly useful, and a formal total mass, taken as the integral of Eq. (100), appears more appropriate. In this context, the role of volume as applied for weighting purposes for ordinary sets can be played by the formal total mass m A) of fuzzy sets, and this function miA) will be used for scaling various symmetry deficiency measures derived for fuzzy sets. 

A fuzzy set B is called an R-deficient set if B has none of the point symmetry elements of family R. However, by analogy with the case of crisp sets, it takes only infinitesimal distortions to lose a given symmetry element. Consequently, unless further restrictions are applied, the total mass difference between a fuzzy set of a specified symmetry and another fuzzy set that does not have this symmetry can be infinitesimal. As a result, i -deficient fuzzy sets and fuzzy R sets can be almost identical. Nevertheless, the actual symmetry deficiencies of fuzzy continua, such as formal molecular bodies represented by fuzzy clouds of electron densities, can be defined in terms of the deviations from their maximal R subsets and minimal R supersets, defined in subsequent text. 

If B is a maximal mass fuzzy B subset of fuzzy set A, and C is a minimal mass fuzzy B superset of fuzzy set A, then the relations among these fuzzy sets define measures for symmetry deficiency. The relationships... 

Chirality, an important shape property of molecules, can be regarded as the lack of certain symmetry elements. Chirality measures are in fact measures of symmetry deficiency. These principles, originally used for crisp sets, also apply for fuzzy sets. Considering the case of three-dimensional chirality, the lacking point symmetry elements are reflection planes a and rotation-reflections 82 of even indices. Whereas the lacking symmetry elements can be of different nature in different dimensions, nevertheless, all the concepts, definitions, and procedures discussed in this section have straightforward generalizations for any finite dimension n. 

For a chiral fuzzy object A, the largest achiral fuzzy object that fits within A, as well as the smallest achiral fuzzy object that contains A, are of special importance. Following the method used for general symmetry deficiencies, one may compare the masses m of these fuzzy objects, and use these comparisons to evaluate the degree of the deviation of the fuzzy object A from achirality. To avoid pathological cases, we shall follow the restriction used for general symmetry deficiencies and consider only fuzzy objects A that have finite, nonzero mass and are nowhere infinitely thin. ... 

The fuzzy Hausdorff-type similarity measures can be used for a direct comparison of a fuzzy set A and its various R sets and / -deficient sets, providing alternative fuzzy symmetry deficiency measures. 

For example, the simplest of these fuzzy symmetry deficiency measures,... 

Various generalizations of syntopy are possible by replacing the configuration space distance by some alternative similarity measures. In fact, any of the similarity measures and symmetry deficiency measures may serve as a suitable parameter in the definition of fuzzy syntopy membership functions. 

VIII. CHIRALITY MEASURES AND SYMMETRY DEFICIENCY MEASURES FOR CONTINUA USING THE SNDSM METRIC... 

Besides chirality, more general symmetry deficiencies can be treated within a unified framework using the metric SNDSM dJ A,B). First we shall describe a general treatment applicable with any metric defined for shapes of finite continua, followed by comments on the special case of SNDSM d A,B). 

The preceding general treatment of approximate symmetry is especially compatible with the SNDSM metric. The quantity d iA,AiR,c)) is the corresponding SNDSM shape distance between the original set A and the fully / -symmetric set AiR,c) with respect to center c, whereas d iA, (/ )) is the SNDSM shape distance between the original set A and the fully / -symmetric set AiR) with respect to the center of mass of set A. The larger the value for d iA, AiR,c)) or d iA, AiR)), the higher the / -symmetry deficiency of set A with respect to the arbitrary center c of the center of mass, respectively. Several related results are discussed in the Appendix. 

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 measures of symmetry deficiency, fuzzy chirality, fuzzy symmetry, and fuzzy symmetry groups described in Section IV in terms of the fuzzy Hausdorff-type metrics provide new tools for the characterization and comparison of fuzzy objects. However, the computation of the fuzzy Hausdorff-type metric is often cumbersome and alternative formulations may have certain advantages. 

Alternative fuzzy symmetry deficiency measures are defined in terms of a maximal R subset B, maximal mass R subset B, minimal R superset C, and minimal mass R superset C, of fuzzy set A, discussed in Section IV. If fuzzy set D denotes any one of the R subsets B,B or R supersets C,C of fuzzy set A, D G B, B, C,C, then a fuzzy symmetry deficiency measure... 

XI. THE CENTER OF MASS OF A FUZZY SET, THE CENTER OF MOLECULAR ELECTRON DENSITY, AND FUZZY CENTRAL MEASURES OF SYMMETRY DEFICIENCY... 

Using the center of mass concept of fuzzy sets, the symmetry deficiency measure of finite continua, described in Section VIII, can be generalized for fuzzy sets. The chirality and more general symmetry deficiencies of fuzzy sets can be treated within a unified framework using the fuzzy metric FSNDSM diJ A,B). 

The dissimilarity of fuzzy sets A and A(R,c) provides a measure of the symmetry aspect R for set A with respect to center c. A large measure of dissimilarity implies a higher degree of symmetry deficiency of fuzzy set A, with respect to symmetry represented by element R. This symmetry deficiency can be described using either one of the fuzzy set dissimilarity metrics. For example, if the fuzzy metric FSNDSM df/A,B) is used, then one obtains the fuzzy symmetry deficiency measure jf JA) ... 

Alternative fuzzy symmetry deficiency measures (. ) of fuzzy set A with respect to reference point c and symmetry represented by element R are defined as... 

TTie dissimilarity of A and Af i p p provides a symmetry deficiency measure analogous to the ZPA continuous symmetry measure of discrete point sets. As a dissimilarity measure, both the SNDSM metric and the Hausdorff metric, or any other dissimilarity measure suitable for continua, are applicable. 

Using the Hausdorff metric for the dissimilarity of sets A and Af f p p, one obtains another generalization of the ZPA continuous symmetry measure of discrete point sets to crisp continuum sets, leading to a new symmetry deficiency measure /Jzpa( > that is a valid measure of... 

The actual value of this Hausdorff measure of symmetry deficiency also depends on the positioning P of R with respect to A as well as on the choice of an associated partitioning of A. 

Another measure for the symmetry deficiency of set A, independent of positioning and partitioning, is given by the infimum of ZPA " >taken over all the allowed positionings and partitionings. This measure, h p/ A, A f p), is defined as... 

These symmetry deficiency measures and others employing different general dissimilarity measures exploit the advantages of the elegance of... 

Fuzzy dissimilarity measures, such as the fuzzy FSNDSM metric fs(/l, B), and any one of the fuzzy Hausdorff-type dissimilarity metrics, for example, f(A,B), can be applied to the pair of set A and the folded-unfolded set Aff p p. These fuzzy dissimilarity measures generate fuzzy symmetry deficiency measures analogous to the ZPA continuous symmetry measure of discrete point sets. 

By taking the infimum for all the allowed choices of P, a symmetry deficiency measure of crisp or fuzzy set A is obtained that is independent of positioning and partitioning. The corresponding df I/l, /Iff.uf,/ )... 

Using any one of the versions of the fuzzy Hausdorff-type metrics for the dissimilarity of sets A and An f/fp, for example, the commitment weighted fuzzy Hausdorff-type dissimilarity metric f(A,B), one obtains another generalization of the ZPA continuous symmetry measure of discrete point sets to crisp or fuzzy sets. The corresponding symmetry deficiency measure f(A,A(f ( p p) provides a measure for the symmetry aspect R for crisp or fuzzy set A, with reference to the given positioning P of R with respect to A and to the choice of the associated partitioning of A. 

Following the principles of the ZPA approach, these symmetry deficiency measures are generalizations of the folding-unfolding approach, equally applicable to crisp continuum sets and fuzzy sets, for example, to entire electron density distributions of molecules and various molecular fragments representing fuzzy functional groups. 

In this book we shall place only limited emphasis on symmetry, since there is little use of symmetry in the shape characterization of more complicated molecules most of which have only trivial symmetry. The reader may find many excellent texts on molecular. symmetry in the literature (for a selection see references [73-79]). Note, however, that deviations from a given symmetry and various symmetry deficiency measures are important and more generally applicable tools for shape characterization. These latter subjects are discussed in Chapter 8. 

SYMMETRY AND APPROXIMATE SYMMETRY, SYMMETRY DEFICIENCY MEASURES, SYNTOPY, AND SYMMORPHY... 

The Quantification of Approximate Symmetry Symmetry Deficiency Measures... 

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