Topological neighbor

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

If the polymers under consideration are regarded as model networks in which/ chains connect a given junction point with/nearest topological neighbor junction points, one can assume a certain ordering in the network structure [134]. If now the junction is placed into a cell of a crystalline... 

The possibility of chain rearrangement so that topological neighboring junctions move in concert to resist chain deformation is implicit in the structure of the theory. Both minor and major chain rearrangements can be expected depending on the extent of chain entanglement. 

Within the domain defined crudely by the locations of these closest topological neighbors (four in number in a tetra-functional network) as many as fifty or more other junctions may reside. The situation is depicted in Fig. 3. The paths (not shown) that connect spatial neighbors, indicated by X in Ihe figure, to the central junction may include many chains we shall be content to let these connections remain obscure. It suffices to observe that elastomeric networks are copiously interpenetrated structures, as these considerations clearly show. They bear little resemblance to a lattice-or even to a disordered lattice - often Invoked Incorrectly as a suitable analog. 

Fig. 4. (A) A central tetrafunctional junction surrounded by four other topologically neighboring junctions and a number of spatially neighboring junctions. (B) Schematic drawing of a slip link, with its possible motions along the network chains specified by the distances a, and its locking into position as a cross-link.

The Kohonen network or self-organizing map (SOM) was developed by Teuvo Kohonen [11]. It can be used to classify a set of input vectors according to their similarity. The result of such a network is usually a two-dimensional map. Thus, the Kohonen network is a method for projecting objects from a multidimensional space into a two-dimensional space. This projection keeps the topology of the multidimensional space, i.e., points which are close to one another in the multidimensional space are neighbors in the two-dimensional space as well. An advantage of this method is that the results of such a mapping can easily be visualized. 

Fig. 3.45 Time evolution of rule T12 on (a) r — 2 lattice, (b,c) intermediate lattices, defined by populating an r=2 lattice with a fraction p of vertices that have 6 nearest-neighbors, with p6 0.15, pc 0.30, and (d) r = 3. We see that the class-3 behavior on the pure range-r graphs in (a) and (b) can become effectively class-2 on certain intermediate (or hybrid) topologies.

Fig. 8,20 First five iterations of an SDCA system starting from a 4-neighbor Euclidean lattice seeded with a single non-zero site at the center. The global transition rule F consists of totalistic-value and restricted totalistic topology rules C — (26,69648,32904)[3 3[ (see text for rule definitions). Solid sites have <7=1.

Figure 7, Schematic diagram of the topology of communication when the DFFD algorithm is employed. As in Fig. 5, each processor is labled [i,j], referring to the particular chunk of [R, r ] for which it is responsible. However, communication is only between nearest neighbors that is, processor (ij) sends and receives data only from (i 1,/) and (ij 1).

The first step in the DG calculations is the generation of the holonomic distance matrix for aU pairwise atom distances of a molecule [121]. Holonomic constraints are expressed in terms of equations which restrict the atom coordinates of a molecule. For example, hydrogen atoms bound to neighboring carbon atoms have a maximum distance of 3.1 A. As a result, parts of the coordinates become interdependent and the degrees of freedom of the molecular system are confined. The acquisition of these distance restraints is based on the topology of a model structure with an arbitrary, but energetically optimized conformation. 

Points on the zero-flux surfaces that are saddle points in the density are passes or pales. Should the critical point be located on a path between bonded atoms along which the density is a maximum with respect to lateral displacement, it is known as a pass. Nuclei behave topologically as peaks and all of the gradient paths of the density in the neighborhood of a particular peak terminate at that peak. Thus, the peaks act as attractors in the gradient vector field of the density. Passes are located between neighboring attractors which are linked by a unique pair of trajectories associated with the passes. Cao et al. [11] pointed out that it is through the attractor behavior of nuclei that distinct atomic forms are created in the density. In the theory of molecular structure, therefore, peaks and passes play a crucial role. 

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