Vertex classification

The vertex classification for the DRP of Fig. 57a is shown in Fig. 51 e, and the tiling charges for the DRP are shown in Fig. 57/. Again, the appearance of the vertex and tiling charge plots is quite similar to those of the dense WCA liquid, with a similar tendency for polygons to form local... 

According to the above classification, the structures of LiNb(Ta)F6 and Li2Nb(Ta)OF5 should be composed of lithium cations and isolated octahedral complex ions, Nb(Ta)F6 or Nb(Ta)OF52, respectively. It is known, however, that the structure of these compounds consists only of octahedrons linked via their vertexes in the first case, and via their sides in the second case. The same behavior is observed in compounds containing bi- and trivalent metals. 

MOLCONN-Z EduSoft, LC www.eslc.vabiotech.com/molconn/ Molecular connectivity, molecular connectivity difference, and kappa shape indices, E-state indices, atom-type and group-type E-state indices, topological equivalence classification of atoms, other topological indices, counts of subgraphs paths, rings, clusters, etc. vertex eccentricities... 

The above introduced classification scheme of synthons will be of great importance for the elaboration of effective heuristics to reduce the enormous number of synthons from the family Sr (A), where A is a given vertex set. Forbidden synthons are removed from the family "(A). 

We now categorize the basis atomic orbitals in any, roughly spherical, cluster according to the number of nodal planes they possess that contain the radius vector. Hence, s atomic orbitals are classified as cr-type cluster orbitals, and so are p and d 2 orbitals, where the local axes at each vertex are chosen with the X and y directions tangential to the surface of the sphere so that z points outwards along the radius vector. Radially directed hybrids, such as sp are also a-type under this classification. Examples will be illustrated below. 7r-type cluster orbitals have one intrinsic nodal plane containing the radius vector, for example, p , Py, and dy in the same axis system. A y and Aj 2 y2 functions contain two such nodal planes, and are known as 5 orbitals. ... 

In addition, it is sometimes useful to relate the total valence electron count in boranes to the structural type. In closo boranes, the total number of valence electron pairs is equal to the sum of the number of vertices in the polyhedron (each vertex has a boron-hydrogen bonding pair) and the number of framework bond pairs. For example, in there are 26 valence electrons, or 13 pairs (= 2n + 1, as mentioned previously). Six of these pairs are involved in bonding to the hydrogens (one per boron), and seven pairs are involved in framework bonding. The polyhedron of the closo structure is the parent polyhedron for the other structural types. Table 15-8 summarizes electron counts and classifications for several examples of boranes. 

Our experience with KG classification has revealed that requirements (i) and (ii) can be fulfilled by making use of the concept of the so-called stipergropA. Each supergraph vertex represents a cycle in the initial KG while a pair of the supergraph vertices is connected... 

In what follows we discuss these criteria and present the main results. The number of vertices in a KG, JV, is not important for the classification, but it is convenient to introduce this criterion into the coding procedure immediately after the notation for the number of routes. The number of KG edges, E (the mechanism s elementary steps), is not regarded as a criterion because it is determined uniquely by the Horiuti rule M = E — J, where J is the number of linearly independent intermediates. In the case of non-catalytic reactions the number N includes the vertex with the so-called zero reagent. 

A linear discriminant function can be found using a linear programming approach (48,49). The objective function to be optimized consists of the fraction of the training set correctly classified. If two vertices have the same classification ability, then the vertex with the smaller sum of distances to misclassified points is taken as better. 

In the analysis of the topology of such nets we can look for a classification scheme that will allow one to uniquely assign equal nets (up to isomorphism) derived from totally different crystal structures. Remember that the topology is not influenced by the metrical properties of the structure (angles, distances), so that a 4-connected diamondoid net is such, even if highly distorted (the geometry around the nodes could be far from tetrahedral), and also, obviously, is not dependent on the chemical nature of each node/vertex. 

Rule 1 - The first simplex is determined performing a number of experiments equal to the number of factors plus one. The size, position and orientation of the initial simplex are chosen by the researcher, taking into accoimt his experience and available information about the system under study (Biuton and Nickless, 1987). In Fig. 8.1a, the first simplex is defined by the A, B and C vertexes. Performing experiments at the conditions indicated by these vertexes and comparing the results, we verify that they correspond, respectively, to the worst, second worst and the best of the three observed responses. You can easily verify this by observing the location of the simplex relative to the contour curves of the response surface. This classification is necessary so that we can define the location of the second simplex, done according to rule 2. 

A shape index, k, could be made equal to either of these symmetry/redundancy indices in an effort to encode information about this shape attribute. Computation of these zero-order indices depends upon proper classification of the atom (vertex) groups. For simple molecules, the chemist readily accomplishes this task by eye. For complex molecules and for automated computation use is made of computer programs... 

Figure 1.5 shows mainly physically valuable types of three-component azeotropic mixtures deduced by Gurikov (1958) by means of systematic apphcation of Eq. (1.12). In Fig. 1.5, one and the same structure cover a certain type of mixture and an antipodal type in which stable nodes are replaced by unstable ones and vice versa (i.e., the direction of residue curves is opposite). Besides that, the separatrixes are shown by the straight lines. Let s note that the later classifications of three-component mixture types (Matsuyama Nishimura, 1977 Doherty Caldarola, 1985) contain considerably greater number of types, but many of these types are not different in principle because these classifications assume light, medium, and heavy volatile components to be the fixed vertexes of the concentration triangle. 

We see that the concept of valence states of the vertex from 1-vertex S-graphs may be used as a criterion for the classification of these S-graphs as forbidden. unstable, and stable 1-vertex S-graphs. Hence, the family is equal to a union of these three disjoint subfamilies composed of the respective S-graphs,... 

Following the above introduced classification of S-graphs composed of more than one vertex, we decompose the family 3Fp (p 2) into three disjoint subfamilies and 3Pp containing forbidden, unstable, and stable S-graphs, respectively ... 

---