Homeomorphic structures

The topological space Spec A is not a sufficiently complicated geometrical object to capture the full structure of A, since the topology is so weak. Indeed, for a field k, all the spaces Spec k[X, Y]/ f (X, Y) for irreducible/are homeomorphic. Consequently one tries to add more structure while still keeping a geometric flavor. 

Identification of all the molecular structures compatible with (r,R,t), that is construction of a set (]j> of all distinct equivalence classes of <7(R, t)>. Two molecular graphs belong to the same equivalence class Qj if and only if they are homeomorphic [27,28]. The set Qj is finite, provided that the system under consideration consists of a finite number of particles (nuclei and electrons). 

One finds that the structure diagrams obtained for F(r X) and p(r X) for the HjO system are homeomorphic in the sense that both exhibit an identical partitioning of the control space yielding the same sets of structures, both stable and, as illustrated in Fig. 3.13, unstable. In addition to finding the same sets of structures for both fields, it has been found that their mechanisms of structural change are also the same. Thus, the bifurcation and conflict... 

Homeomorphs of the two-point bicycle show up in an interesting collection of structures, ranging from the simple cage-type bicyclophane (73) [93, 94] to the extended homeomorphs by Lehn et al. [95], Vogtle et al. [96], and Moore et al. [97]. Other molecular representations are Moore and Bedard s flat molecular turnstile (74) [98], Hart and Vinod s cuppedophanes (75) [99-102], and Okazaki et al. s bowl-shaped bicyclic cyclophane (76) [103,104]. 

In the simplest approach, a lattice can be wrapped on the surface of a particle homeomorphic to a sphere [64]. Consider the connectivity and attendant domain structure defined by the five Platonic solids [45]. If d denotes the dimensionality of the surface of the particle, N the number of... 

The mathematical term topology involves geometrical properties which remain invariant given continuous deformation in space [2, 3]. Two structures are said to be topological equivalent or isotopic if they can be interconverted into one another by continuous deformation. If not they are called topologically distinct or not isotopic. One of these topological invariants is connectivity. Structures with identical connectivity are termed homeomorphic. Transferring these terms to molecular structure results in three types of isomers ... 

The structures XI to XIII or XIV and XV are topological stereoisomers since they have identical connectivity (homeomorphic), but no continuous deformation will allow them to interconvert (not isotopic). Furthermore, structures XII and XIII are topological enantiomers and the knots XII or XIII and the unknotted ring XI are topological diastereomers. 

The complexity of hierarchical structures is related symbatically to the number of types of building entities (building blocks) which are not homeomorphic with respect to one another. 

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