Topological classification

E. G. Mazur.S, Graphic Representation of the Periodic System during One Hundred Years, University of Alabama Press. Alabama, 1974. An exhaustive topological classification of over 700 forms of the periodic table. 

The simplest example of topological classification is the knotting of individual strands, i.e., self-entanglement. However, the effects attributed to entanglement in non-crosslinked polymers are clearly intermolecular. The simplest such case is that of pair-wise classification without self-entanglement. Consider the following three examples of strand pairs ... 

Edwards (300) was able to make some progress on the problem by showing that topological classification can serve only to raise the modulus. As a simple example, consider all strand pairs in a network which have, within some small tolerance, a specified set of internal junction coordinates. Suppose there are B such pairs, and that the strands of each pair, labeled 1 and 2, have co1 and co2 distinguishable configurations each as free strands, and fractions (gt)0 and (g2)o respectively which have the end-to-end distances specified by the equilibrium junction coordinates. If the crosslinks were formed in the system at equilibrium, then the total number of configurations for each strand of the pair is cj1(g1)0 and o)2(02h> and the number available to the pair is (o1(o2(gl)0(g2)0. 

For a surface, in which a finite 3-fullerene can be embedded, the number x is, therefore, a non-negative integer. Let us use the topological classification of finite closed maps in Theorem 1.2.1 and recall the expression of x in Theorem 1.2.2 ... 

Fig. 10 Topological classification of linear polymeric polyradicals, including a scheme showing the importance of regiospecific connectivity.

The most thorough topological classification of the dynamic behavior of the copolymerization systems suppose to make them out by their kinds, each of them is determined by the types of all SPs as well as by manifolds separating their basins (regions of their attraction). Every kind is characterized by the type of its phase portrait. The case of the binary copolymerization obviously is trivial since each of the types (5.8) consists of only one genus which in its turn includes a single kind. 

The system of genus 1 is regarded to be an exclusive one since it only has the stable inner azeotrope when the coefficient oq is positive (see Table 5.1). Therefore the four kinds of the system of genus 1 correspond to the four combinations of signs of the oq and p values. If, as it is usually admitted under the topological classification of the dynamic systems, one doesn t distinguish for the sake of... 

The topological classification of an individual extended motif can be accomplished according to generally accepted criteria used in a variety of structural contexts that are here described. First, however, we want briefly to mention the pioneering work of the researchers who introduced the basic concepts for the rationalization, enumeration and classification of the extended crystal structures, using different approaches. 

Polyad PfWOT Topology classification Topology Examples (PDB code)... 

A sensor is a device capable of converting a physical or chemical quantity into readable information. In addition to a topological classification (local or distributed, intrinsic or extrinsic), a more basic classification can be made according to the optical parameter affected by the external factor intensity, phase, wavelength, and polarization. 

This result follows from the general Theorem 2.1.3 on complete topological classification and on canonical representation of constant-energy surfaces of integrable systems (on four-dimensional manifolds of the three simplest types). In particular, this yields a simple classification of all nonsingular constant-energy surfaces for integrable systems. 

Theorem 2.1.2. (Fomenko. The topological classification theorem FOR three-dimensional CONSTANT-ENERGY SURFACES OF INTEGRABLE... 

First of all we observe that the network (6.26) contains an A2-element as integrated circuit for the autocatalytic production of X. In addition to this autocatalytic loop there is a second loop formed by the reaction R2. From the topological point of view, there is a strikingly close resemblance between the Brusselator in (6.26) and our autocatalytic excitation model in (6.7). Despite this topological resemblance, the two networks behave fundamentally differently, namely multiple steady states in (6.7) and a limit cycle in (6.26). This shows that a purely network topological classification of types of models will find its limits or is at least a pretentious program. 

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