Symmorphy

VI. A THIRD FUZZY SYMMETRY APPROACH SYMMORPHY AND FUZZY SYMMORPHY GROUPS BASED ON FUZZY HAUSDORFF-TYPE METRICS... 

Another description of approximate symmetry, based on symmorphy groups, can also be generalized using the fuzzy Hausdorff-type distances. 

Another generalization of symmetry and a fundamentally different algebraic structure is obtained in the symmorphy approach. (The... 

This family of operators can be regarded as an extension of the family of point symmetry operators. Symmorphy is a particular extension of the point symmetry group concept of finite point sets, such as a collection of atomic nuclei, to the symmorphy group concept of a complete algebraic shape characterization of continua, such as the three-dimensional electron density cloud of a molecule. In fact, this extension can be generalized for fuzzy sets. 

First we shall discuss ordinary symmorphy, followed by an extension to fuzzy sets. 

These two observations serve as the basis for the generalization of symmetry to symmorphy. For a general point set configuration K, it is K that selects a subset of special reflections, rotations, and inversions from the set of infinitely many such operations of the space. The conditions for selection is the indistinguishability of the original and transformed configurations. 

In symmorphy, the preceding interpretation of symmetry and the selection principle are extended in two ways ... 

The shape of a continuum A can be characterized by a subfamily Gsph(/1) of Gpo , where the subfamily Gsp, (/4) contain all those homeomorphisms S from G on, that bring the transformed object Sv4 into an arrangement that is indistinguishable from A. Since for such homeomorphisms S the morphologies of the transformed object and the original object A are indistinguishable, S is called a symmorphy transformation of object A. 

The set Gjph( ) of all symmorphy transformations S of the object A is a subgroup of Gp, . Clearly, if neither S, and S, alters the appearance of the morphology of A, then the transformations S3, defined as transformation Sj followed by S2 and denoted as the formal product... 

In the special case of selecting the three-dimensional object as a molecular charge density function p(r), the analogies between point symmetry and symmorphy are rather clear. By analogy with the point symmetry of nuclear arrangements, the molecular charge density function pit) can provide a criterion for selecting the symmorphy transformations of p(r) from the infinite family G, of homeomorphisms of the three-dimensional space. A homeomorphism 5 is a symmorphy transformation... 

For the molecular charge density function p(r), all those homeomor-phisms S of family Ghom are symmorphy transformations for which... 

Some simplifications are possible if equivalence classes of symmorphy transformations can be defined where operations S from the same class transform the space occupied by the object A the same way and differ only in parts of the space where A is not present. Furthermore, using the Brouwer fixed point theorem, a subgroup structure of symmorphy groups Gjph( ) provides a more detailed characterization of molecular shape. These aspects will not be reviewed here. 

By analogy with fuzzy symmetry elements, a fuzzy set A is said to have the fuzzy symmorphy element S( (3) corresponding to the symmorphy operation S at the fuzzy level /3 of the fuzzy Hausdorff-type similarity measure Sg if and only if the fuzzy similarity measure Sg between Sv4 and A is greater than or equal to /3 ... 

If in relationship (124) the fuzzy similarity measure Sg is replaced by either one of the unsealed similarity measures tg and Zg or the scaled similarity measures s, tf, and Zf, then the fuzzy symmorphy element 5( )3) is specified in terms of the corresponding fuzzy Hausdorff-type similarity measure. 

In the limiting case of y3 = 1, similarity becomes indistinguishability. In this case, the fuzzy symmorphy element S ) becomes an ordinary symmorphy element corresponding to the symmorphy operation S. The maximum fuzzy level l3(A,S,Sg) at which the fuzzy symmorphy element S( j8) is present for the fuzzy set A is given as... 

A Juzzy symmorphy operator S( g) of fuzzy symmorphy element 5( 0 exhibited by fuzzy set A at the fuzzy level )8 (according to the fuzzy Hausdorff-type similarity measure Sg) is defined by its action on the given fuzzy set A ... 

Operator has the same role as operator in the case of fuzzy symmetry operators. To exploit the fuzzy indistinguishability of sets A and Szl at the fuzzy level )3, operator resets the values of fuzzy membership functions if and only if the fuzzy symmorphy element... 

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... 

We say that a fuzzy set A has the fuzzy symmorphy group at fuzzy level if A has the fuzzy symmorphy element Rip ) at the fuzzy... 

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

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 maximum fuzzy level p(A,S,fs) at which the fuzzy symmorphy element S( /3) is present for the fuzzy set A, that is. 

The measure (zl,S,fs) of the symmorphy deficiency of fuzzy set A in symmorphy element S according to the FSNSM fs is defined as... 

We say that a fuzzy set A has the )8 fuzzy symmorphy group at fuzzy level fi if A has the fuzzy symmorphy element 5( )3 ) at the fuzzy level /3 of the FSNSM fs g for each symmorphy operation S of the crisp symmorphy group sph- The fuzzy symmorphy roup G pjj(fs, ) 3t the fuzzy level P that is the supremum of the levels of all fuzzy... 

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

Imperfect symmetry can be viewed in the broader context of molecular distortions. In this book the emphasis is on descriptions and measures based on topological arguments [54,55,106,108,158,240,243,247,248,345,444,445], or on the related concepts of syntopy [252,394,395] and symmorphy [43,108] discussed briefly in subsequent parts of this chapter. For a variety of alternative approaches and important additional insight, the reader may consult references [58,446-449]. 

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