Neighbor relations

For fluid membranes, in which neighbor relations are not maintained, the free energy of a membrane is often written in the form [27,30]... 

Ecologically, female meadow voles are territorial, know their neighbors, and are more tolerant of each other. They exemplify the dear enemy concept familiar neighbors reduce aggression toward one another because they pose less threat to each other than newcomers without a territoiy, who might compete for territory, mates, or resources. Males are dispersal prone, and neighbor relations are more ephemeral. Each male s home range overlaps with those of several females (Ferkin, 1988). 

The concept of chemistry space pervades, either explicitly or implicitly, much of the literature in chemoinformatics. As is discussed in Subheading 3., chemistry spaces are induced by various similarity measures. The different similarity measures do not, however, give rise to topologically equivalent chemistry spaces—nearest-neighbor relations are generally not preserved among chemistry spaces induced by different similarity measures. The consequences of this are manifold. An especially egregious consequence is that the results of similarity searches based on different similarity measures can differ substantially. And there is no easy solution to this problem. 

Fig. 12. Sequence spaces for the sequence/structure shown in the right column, (a) Sequence for ssRNA of length 3 with monomers G and A only, (b) Part ofthe sequence space for the 1,4-benzodiazepin-2-one derivative library constructed in Ref. 156. For clarity, not all neighbor relations are shown with lines in this figure. Points which differ at one site are called 1-mutant neighbors. For example, the points marked o are all 1-mutant neighbors of the point marked .

Fig. 7. Geometric structure of the (0001) oriented single crystal surface of trigonal V2O3. for a surface termination A BA. Vanadium and oxygen centers are shown as light large and dark small balls, respectively, with connecting sticks indicating nearest neighbor relations.

Consider a MIDCO G(a) and a choice for the curvature parameter b, and assume that the shape domains of relative convexity of G(a) have been determined. By using an appropriate neighbor relation to describe the mutual arrangements of the domains along the MIDCO surface G(a), the corresponding shape matrix s(a,b) and the associated shape graph g (a,b) can be defined [109,110,158,193]. 

Two D j domains are considered N-neighbors if they have a common boundary line. For a more precise description, the N-neighbor relation between two Djj i shape domains is defined in terms of their closures clos(D i). In accord with the definitions given in Chapter 3, the closure clos(Dji j) of a domain D, contains all the points of D j as well as all of its boundary points. The formal definition of the N-neighbor relation is given below ... 

This neighbor relation is similar to the "symmetric strong neighbor relation" between some potential surface catchment regions of reaction topology, used in the analysis of reaction mechanisms [106,343-345]. 

In most cases, nonzero N-neighbor relations are found for the (Dq , Di p) and (Dij, D2,i") types of pairs of D domains. Note that each D j domain is a maximum connected component of the set A of all points of G(a) with index (i, consequently, a nonzero N-neighbor relation is impossible between two D shape domains of the same index p. 

A nonzero N-neighbor relation between a Dq and a D2 domain is possible... 

One of the most useful shape codes is based on shape matrices. As we have seen in Chapter 5, the N-neighbor relation N(D j, D j ) of various curvature domains and, given by Equation (5.8), leads to a shape matrix... 

To design a graph, one must define the objects in terms of vertices and edges. The simplest approach is to view all the domains Dc(a, K) into which the surface G a, K) is partitioned (cf. Eq. [26]) as vertices. In this case, the edges are defined by the neighbor relations among domains. ... 

Based on the above neighbor relation, and by direct analogy with the model of ref.[20], we define the point symmetry graph g(M,sym) of the nuclear configuration space M of the given stoichiometric family of chemical species by the following relations ... 

By replacing the point symmetry domains Gy, and Gj.j. of equations (8)-(10) with the D y and D. j. domains of the eigenvalue sign distribution of local Hessian matrices within the nuclear configuration space M, and by replacing the symbol sym with the symbol hess, one obtains the analogous neighbor relation... 

As for the other partitionings mentioned above, these neighbor relations and graphs can be restricted to various subsets S, such as relaxed cross sections, and in particular, to individual catchment regions C(X,i) of the nuclear configuration space M. This approach leads to the local shape domain graphs g(S,x) and g(C(X,i),x), respectively. 

The same approach is applicable for the comparison of molecular surfaces belonging to two different molecules. If the neighbor relations among the corresponding domains on the two molecular surfaces are the same, then the shapes of the two contours are similar within the above context. " ... 

In most cases the following types of pairs of domains have nonzerp N-neighbor relations ... 

In exceptional cases a nonzero N-neighbor relation between a Do and a Dz domain is also possible. A point r of the molecular surface G(a) may simultaneously belong to a Do,/ domain and to the closure of a Dz,i domain. 

The qj in Eqs. (64) and (65) are the same as in Hiickel s theory. We know that the are the components of the normalized eigenvectors of the structure matrix which is defined by the neighboring relations of the carbon atoms only. Therefore we find the charge densities of a w-electron system are—also in the given expansion of Hiickel s theory—determined only by the structure of the molecule. 

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