Symmorphy group

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. 

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. 

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. 

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. 

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... 

One such approach, the symmorphy group approach [43,108], is based on the extension of the family of point symmetry operators to a much richer family of operations which preserve the general morphology of objects. (Note that the term "symmorphy" is used in a different sense in the crystallography literature, with reference to the symmorphic space groups of crystallography, also called semi-direct... 

The family of all symmorphy equivalence classes of a given object p(r) form the symmorphy group hp of the object p(r). This group hp is formally defined [43,108] as the quotient group of group gp with respect to the symmorphy equivalence nip. The product of two symmorphy classes Tj and T2 in the symmorphy group hp is defined as the class T3,... 

Most symmorphy groups hp are rather complicated and their direct use for molecular shape characterization and shape similarity analysis is not a trivial task. Some simplifications are possible using a technique based on the Brouwer fixed point theorem, as described in reference [43]. 

The family of all symmorphy transformations of the given object p(r) form a subgroup gp of the group G of all homeomorphic transformations of the 3D space. This subgroup gp is, in fact, defined by the shape properties of the 3D object, and it provides a complete characterization of its shape, in our case, the shape of the molecular charge density function p(r). 

However, the group gp is much too complicated for practical purposes of molecular shape characterization. Fortunately, the behavior of transformations t of family gp far away from the object p(r) is of little importance, and one can introduce some simplifications. Let us assume that the 3D function considered [e.g., an approximate electron density function p(r)], becomes identically zero outside a sphere S of a sufficiently large radius. As long as two symmorphy transformations tj and t2 have the same effect within this sphere, the differences between these transformations have no relevance to the shape of p(r), even if they have different effects in some domains outside the sphere. All such transformations t of equivalent effects within the relevant part of the 3D space can be collected into equivalence classes. In the symmorphy approach to the analysis of molecular shape, these classes are taken as the actual tools of shape characterization. 

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