Fuzzy symmetry

IV. FUZZY SYMMETRY DEFICIENCY MEASURES, FUZZY CHIRALITY MEASURES, AND FUZZY SYMMETRY GROUPS BASED ON THE MASS OF FUZZY SETS AND FUZZY HAUSDORFF-TYPE METRICS... [Pg.155]

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 ... [Pg.156]

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... [Pg.156]

The maximum fuzzy level p(A,R,Sg) at which the fuzzy symmetry element Rip) is present for the fuzzy set A, that is,... [Pg.156]

A fuzzy symmetry operator R(ij) of fuzzy symmetry element R()3 ) present for fuzzy set A at the fuzzy level j3 of the fuzzy Hausdorff-type... [Pg.156]

Here R is the ordinary symmetry operator corresponding to the fuzzy symmetry element R(p ). [Pg.157]

The role of operator can be described as follows. To take full advantage of the fuzzy indistinguishability of sets A and R 4 at the fuzzy level j8, operator j, resets the values of fuzzy membership functions whenever this fuzzy indistinguishability is implied by the presence of the fuzzy symmetry element R( /3) at the fuzzy level /3. [Pg.157]

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. [Pg.157]

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 ... [Pg.158]

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. [Pg.158]

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. [Pg.162]

For example, the simplest of these fuzzy symmetry deficiency measures,... [Pg.163]

V. ANOTHER FUZZY SYMMETRY APPROACH SYNTOPY AND SYNTOPY GROUPS... [Pg.164]

Syntopy and syntopy groups were introduced in an early approach to a fuzzy set representation of approximate symmetry, where imperfect symmetry is regarded as fuzzy symmetry. Whereas any symmetry is a discrete property within a metric space, it is natural to consider a fuzzy set approach for a continuous extension of the discrete symmetry concept to quasisymmetric objects, such as some almost symmetric molecular structures. The syntopy approaches take into account the nonlocalized, quantum-mechanical, fuzzy nature of nuclear arrangements of molecules. [Pg.164]

By analogy with fuzzy symmetry operations, if a fuzzy set A has the three fuzzy symmorphy elements 5( /3), 5 ( /3) and S"( ) at fuzzy level and if the product... [Pg.171]

X. FUZZY MEASURES OF CHIRALITY AND SYMMETRY DEFICIENCY, FUZZY SYMMETRY GROUPS, AND FUZZY SYMMORPHY GROUPS BASED ON THE FUZZY SCALING-NESTING SIMILARITY MEASURE... [Pg.179]

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. [Pg.179]

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 ... [Pg.180]

We say that a fuzzy set A has the p fuzzy symmetry group G(fs, P ) at fuzzy level P if A has the fuzzy symmetry element Ri p ) at the fuzzy level P of the FSNSM fs g for each symmetry operation R of the crisp symmetry group G. [Pg.180]

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... [Pg.181]

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) ... [Pg.185]

Alternative fuzzy symmetry deficiency measures (. ) of fuzzy set A with respect to reference point c and symmetry represented by element R are defined as... [Pg.185]

THE CRISP AVERAGE OF FUZZY SETS, AND RELATED FUZZY SYMMETRY MEASURES... [Pg.185]

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. [Pg.194]

Imperfect symmetry can be regarded as fuzzy symmetry. The theory of fuzzy sets has found applications in many fields of engineering and natural sciences (see, e.g., references [386-393]), in particular, for the description of fuzzy molecular arrangements [103,106,251]. It is natural to consider fuzzy sets for a continuous extension of the discrete point symmetry concept to quasi-symmetric molecular structures [252,394,395]. [Pg.195]

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