Fuzzy sets symmetry measure

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... 

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 fuzzy set A is said to have the fuzzy symmetry element R( jS) corresponding to the symmetry operation R at the fuzzy level )3 of the fuzzy Hausdorff-type similarity measure if and only if the fuzzy similarity measure measure R and A is greater than or equal to )3 ... 

Various fuzzy subsets B of 4 may have the fuzzy symmetry element B(/3) at different fuzzy levels /3 of the fuzzy Hausdorff-type similarity measure s. Here the concept of fuzzy subset is interpreted in the usual way for the fuzzy subset B of fuzzy set A the condition... 

Note that if a fuzzy set A has the fuzzy symmetry element / ()3) corresponding to the symmetry operator R at the fuzzy level j8 of the fuzzy Hausdorff-type similarity measure then the application of R on 4 generates a set R indistinguishable from set A at the fuzzy level (i. For the fuzzy set A the application of symmetry operator R of fuzzy symmetry element R(p ) present at the fuzzy level is completed by a formal recognition of the indistinguishability of set R/1 and set A at the given fuzzy level. This additional step, for which the sufficient and necessary condition is the presence of fuzzy symmetry element Ri ) at the fuzzy level /S, involves operator setting the membership functions of elements of the fuzzy set R/1 equal to those of fuzzy set A. 

A fuzzy set A has the / fuzzy symmetry group G Sg, /3 ) at fuzzy level if has the fuzzy symmetry element R( j8 ) at the fuzzy level j8 of the fuzzy Hausdorff similarity measure for each symmetry operation R of the crisp symmetry group G. The )3 fuzzy symmetry group G(s, at the fuzzy level /3 that is the supremum of the levels j8 of all )3 fuzzy symmetry groups G(Sg, (3 ) of A is the fuzzy symmetry group G s, of the fuzzy set A ... 

Analogous fuzzy symmetry groups Git13), G(z, )3), Gisf, ), G(tf,p), and GizpP) of fuzzy set A are defined for the unsealed similarity measures t and and scaled similarity measures Sp tf, and Zf, respectively. 

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. 

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. 

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. 

A fuzzy set generalization of nuclear point symmetry in terms of these two syntopy models is applicable to all nuclear arrangements. Using appropriate membership functions, syntopy provides a measure of symmetry resemblance of actual, general nuclear configurations to ideal, fully symmetric nuclear configurations. 

Assume that R is the ordinary symmetry operator corresponding to the fuzzy symmetry element Rip ) present for fuzzy set A according to the FSNSM fs g. A fuzzy symmetry operator R(fs) of fuzzy symmetry element Rip ) present for fuzzy set A at the fuzzy level p of the fuzzy similarity measure fs is defined by its action on the fuzzy set A ... 

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... 

THE CRISP AVERAGE OF FUZZY SETS, AND RELATED FUZZY SYMMETRY MEASURES... 

The fuzzy and crisp average concepts are the basis of further measures of approximate symmetry of a crisp or fuzzy set A. Consider a point c as the center for a (possibly only approximate) symmetry element R oi A. 

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. 

Whereas both the external f -deficiency measure SRyc(A) and the R-deficiency measure ARyB(A) are mathematically correct and valid tools for comparisons of the degrees of symmetry deficiencies and their correlations with unusual reactivities of functional groups, the actual physical existence of fuzzy sets involved in the determination of the internal -deficiency measure SRyB(A) nevertheless suggests that this measure is likely to provide more reliable correlations with variations in the chemical reactivity of functional groups. 

THE FUZZY SETS APPROACH TO THE MEASUREMENT OF SYMMETRY AND CHIRALITY... 

A measure of the degree of symmetry aspect R for fuzzy set A according to fuzzy Hausdorff-type similarity measure Sg is the maximum fuzzy level (A,R,jg) = sup g[o.ii -j (RA,A) > P) at which the fuzzy symmetry element R P) is present for the fuzzy set A. 

The CSM described in Section 2 is a symmetry deficiency measure which is applicable to both discrete sets and to continua. For crisp continuum sets and fuzzy sets, the crisp and fuzzy versions of the Hausdorff metric provide generalizations of the CSM approach. 

The infimum /(A, Aff, , . ./>) = inf/. /(A,Aff,uf, ,/>) taken over all the allowed positionings and partitionings P gives another symmetry deficiency measure. These symmetry deficiency measures are equally applicable to discrete sets, crisp continuum sets, and fuzzy sets, including nuclear distributions and fuzzy electron density distributions of molecules, molecular fragments, and functional groups. 

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