Topological representation, definition

The meaning of a complex formation constant is not as clear-cut as one can wish. The behavior of aqua ions which may be either definite species such as Ni(H60)6" or vague representations of the standard solute in dilute solution has already been discussed. One then must discuss the question When is a ligand bound to a central atom Certainly, this problem does not occur for most complexes of multidentate ligands, but weak complexes are not in an easy position. Prue (89) correctly emphasized that without any specific attraction between two molecules, for purely topological reasons, one would find a complexity constant of 0.2 liter/mole from contact charge-transfer spectra in typical conditions. If one finds a smaller formation constant, the two molecules repel each other... 

Atom-type designations in atom-pairs have constitutional, topological, and electronic character - atoms of the same type share atomic identity, the same number of non-hydrogen bonding partners, and the same number of bonding n electrons (Fig. 13.1-4(b)). Because of this representation, molecules tend to have many, fewer than the theoretical maximum possible, atom-pairs (1/2 [n (n—l)]2 for a molecule with n atoms), both by virtue of having multiple atoms of the same type, and because the order of the two atoms appearance within an atom-pair is not important (Fig. 13.1-4(c)). A very significant aspect of the definition of atom-types is that Carhart et al. provided both a distance metric and a normalized similarity score for molecules based on the atom-pair definition (Fig. 13.1-4(d)). Formally, such a provision is a requirement for any metric descriptor space intended to afford a basis for comparison of molecular... 

With this definition, we can consider the structure representations of molecules in Section 7.2 as topological descriptors. At present, more than 5000 molecular descriptors can be computed [5]. They can be categorized by their data types (Table 7.11) or by their dimensionality (Table 7.12). 

Example 1.23. — Let T be a site with precanonical topology i.e. such that any representable presheaf is a sheaf Assume that there exists a family of coverings pi U, pt of the final object of T such that for any U in T the intersection of images of //< n(U, U,) in pt—Hotrdf, pt) is empty (such a family can be found for example in the site associated with any profinite group which is not finite). Consider the simplicial sheaves J, - = C(U,- pt) (see definition prior to Lemma 1.15) and let Ex he z. resolution fimctor on E SfafT). We claim that the canonical morphism Y[Ex i is not a weak equivalence. Indeed, by Lemma 1.15 each of 3 -s is weakly equivalent to the final object and therefore H Ex.% is weakly equivalent to the final object as well. On the other hand our condition on U, s implies that the product is empty. 

Wire frames with unified topology and geometry use the same entities as solid models but without face and surface definitions. Note that these wire frames are not the same representations as the wire frames in the early era of geometric modeling. When a wireframe model defines a shape unambiguously, a surface or solid model can be replaced by the simple wire frame for economical modeling. If necessary, a verified wire frame can be completed into a surface or solid. 

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