Topological dimensionality

Combining the building unit and overall topology dimensionalities a classification scheme. 

For convenience, we shall classify the molecular models according to their topological dimensionality, p. A molecular conformation defined by the set of nuclear position vectors is a zero-dimensional (OD) model. A one-dimensional (ID) model corresponds to a molecular skeleton, defined by the set of nuclear positions and their connectivity (bond) matrix. Contour surfaces of one-particle molecular properties such as electron density or electrostatic potential are topologically two-dimensional (2D) models embedded in three dimensions. Finally, we find a true three-dimensional (3D) model whenever an entire one-electron property over all space is involved. This model can be regarded as the continuum of all 2D isoproperty surfaces. The difference among the models is summarized in Figure 1. We shall deal with pD models in this work (p = 0,1, 2, 3). Each of them requires a different type of shape descriptor. 

Dynamic methods such as gas relative permeability combined with aspects from percolation theory can be employed to obtain structural information on the porous network topology (dimensionality and connectivity) as well as on the pore shape. Model membranes with well defined structure formed by compaction of non-porous spherical particles, are ideal for testing the different characterization techniques. One should bear in mind though, that dynamic methods should be used as a complement of, rather than as an alternative to, the equilibrium methods. Furthermore, capillary network models are not always appropriate for the derivation of structural information, and for membranes made by compaction of spheres the use of advanced pore modeling tools based on random sphere packing geometry is required. 

Figure 6. Two-dimensional (top) and 3D (bottom) representations of a peaked (a) and sloped (b) conical intersection topology. There are two directions that lift the degeneracy the GD and the DC. The top figures have energy plotted against the DC while the bottom figures represent the energy plotted in the space of hoth the GD and DC vectors. At a peaked intersection, as shown at the bottom of (a), the probability of recrossing the conical intersection should be small whereas in the case of a sloped intersection [bottom of ( )l, this possibility should be high. [Reproduced from [84] courtesy of Elsevier Publishers.]...

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. 

Several research groups have built models using theoretical desaiptors calculated only from the molecular structure. This approach has been proven to be particularly successful for the prediction of solubility without the need for descriptors of experimental data. Thus, it is also suitable for virtual data screening and library design. The descriptors include 2D (two-dimensional, or topological) descriptors, and 3D (three-dimensional, or geometric) descriptors, as well as electronic descriptors. 

In the parabolic model the equations for caustics are simply Q+ = Q, and Q- = <2-- The periodic orbits inside the well are not described by (4.46), but they run along the borders of the rectangle formed by caustics. It is these trajectories that correspond to topologically irreducible contours on a two-dimensional torus [Arnold 1978] and lead to the quantization condition (4.47). 

Figure S.2 Schematic and topological diagrams of an up-and-down fi barrel. The eight p strands are all antiparallel to each other and are connected by hairpin loops. Beta strands that are adjacent in the amino acid sequence are also adjacent in the three-dimensional structure of up-and-down barrels.

There is a second family of small lipid-binding proteins, the P2 family, which include among others cellular retinol- and fatty acid-binding proteins as well as a protein, P2, from myelin in the peripheral nervous system. However, members of this second family have ten antiparallel p strands in their barrels compared with the eight strands found in the barrels of the RBP superfamily. Members of the P2 family show no amino acid sequence homology to members of the RBP superfamily. Nevertheless, their three-dimensional structures have similar architecture and topology, being up-and-down P barrels. 

Blood-brain barrier permeation of 7, among other drugs, was predicted from its three-dimensional molecular structure by a computational method (0OJMC2204). The combination of molecular topological methods using 137 quinolones, including 7 provided an excellent tool for the design of new... 

Figure 2.9 shows a typical Gc topology, computed for a one-dimensional lattice consisting of four viu tices and evolving according to totalistic rule T2. 

A partial analogy between the dynamics of CA and the behaviors of continuous dynamical systems may be obtained by exploiting a fundamental property of CA systems namely, continuity in the Cantor-set Topology. We recall from section 2.2.1 that the collection of all one-dimensional configurations, or the CA phase space, r = where E = 0,1,..., fc 9 cr and Z is the set of integers by which each site of the lattice is indexed, is a compact metric space homeomorphic to the Cantor set under the metric... 

Fig. 5.3 Natural topologies C,p for three one-dimensional additive rules ( , s sum modulo q). For an arbitrary acting on a given lattice C, is always (Bq aneigMoTsY...

Kaplunovsky and Weinstein [kaplu85j develop a field-theoretic formalism that treats the topology and dimension of the spacetime continuum as dynamically generated variables. Dimensionality is introduced out of the characteristic behavior of the energy spectrum of a system of a large number of coupled oscillators. 

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