Topological aspects

In molecular physics, the topological aspect has met its analogue in the Jahn-Teller effect [47,157] and, indeed, in any situation where a degeneracy of electronic states is encountered. The phase change was discussed from various viewpoints in [144,158-161] and [163]. 

Mezey, P.G. (1990) Three-dimensional topological aspects of molecular similarity. In Concepts and Applications ofMolecular Similarity, Johnson, M.A. and Maggiora, G.M. (Eds.), Wiley, New York. 

In contrast to common ANNs, Kohonen networks produce self-organized topological feature maps (Kohonen [1982, 1984]). The basic idea of Kohonen mapping is that information in data usually contains not only an algebraic but also a topological aspect. These double aspect is shown schematically in Fig. 8.25 where the data and the structure of them are composed. 

The basic aspects of the structural systematic and the chemical bonding are discussed as well as ambiguous topological aspects and selected physical properties (magnetism, conductivity) of the respective alloys. 

Kholodenko AL (1996) Reptation theory geometrical and topological aspects. Macromol Theory Simul 5 1031-1064... 

The glass transition temperature Tg is one of the most important structural and technical characteristics of amorphous solids. The correlations of Tg of linear polymers with their chemical composition, molecular weight, rigidity and symmetry of chains, as well as some other characteristics of macromolecules are well documented 57,58) Thg information on networks is much poorer. At present, for networks there exists mainly one parameter in structure-Tg correlations. It is the concentration of crosslinks — a parameter which is very insufficient, since in networks there are chemical crosslinks of different functionality (connectivity) which are distinguished by their molecular mobility. This means that the topological aspect of the network structure should be taken into account in the Tg analysis. Another difficulty connected with Tg determination of polymers lies in vitrification occurring during polymer formation (Sect. 6). 

In conclusion, in ID materials the topological aspects of the FS are clear in principle. However, an experimental technique to map out the FS in detail is just developing. In addition, theoretical understanding of the observed features in transport measurements is far from being complete in contrast to the well elaborated Lifshitz-Kosevich theory for 3D metals. Nevertheless, the principal simple quasi-ID form of the FS allows for an enormously rich variety of experimentally observable physical phenomena which were discussed partially in Sect. 2.2.3. These have stimulated a large amount of theoretical effort to obtain a better understanding of these effects in relation to the electronic band structure. 

Although it was that simple to derive the rule of three functions in biochemistry in general, this modification cannot be calculated as straightforwardly because the reduction of autocatalytic order due to oligomerization depends sensitively on (bio-)chemical network topology. In addition to topological aspects of feedback in the network, the exact kind (manner) of dimerization which interconnects identical metal ions needs to be known, likewise the way by which the substrate interacts with this multimetal center (for example, the Mo dimerization effect is also seen with Mo-dependent nitrogenases but and the other reduc-... 

In Chapter III we amply illustrated the relationships between Green s functions, continued fractions, and moments here we need only summarize a few relevant results and focus on some peculiar aspects (such as topological aspects) inherent in the structure of the Hamiltonians on a lattice. 

P. G. Mezey, Three-Dimensional Topological Aspects of Molecular Similarity, in Concepts and Application of Molecular Similarity (M. A. Johnson and G. M. Maggiora, eds.), pp. 321-368. Wiley, New York, 1990. 

Balaban, A.T., Niculescu-Duvaz, I. and Simon, Z. (1987b). Topological Aspects in QSAR for Biologically Active Molecules. Acta Pharm.Jugosl.,37,7-36. 

Narumi, H. and Hosoya, H. (1985). Topological Index and Thermodynamics Properties. III. Classification of Various Topological Aspects of Properties of Acyclic Saturated Hydrocarbons. Bull.Chem.Soc.Jap., 58,1778-1786. 

Up to now we just looked at the topological aspect of the problem and have understood that in systems with many degrees of freedom, KAM tori cannot block the global motion of the system in the entire state space (energy surface). Then how can the system move beyond the KAM tori ... 

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