Topological configuration

Topological Configurations.—It may be noticed from the preceding that there is a certain relation between limit cycles and singular points. 

Theory of Bifurcations.—In the preceding sections we have reviewed a few more important points of the existing topological methods in the theory of oscillation, assuming that the topological configuration or the phase portrait remains fixed. 

Suppose we have a certain topological configuration, say, SUS in our previous notation this means that the singular point is stable and the nearest cycle is unstable. The bifurcation of the first kind can be represented by the scheme ... 

Helicity is defined as a qualitative measurement of how a topological configuration is linked, knotted, or twisted. If A is the vector potential of electromagnetism, the quantity then defines magnetic helicity. In order to obtain a nonnull value for K, the condition A (V A) /- 0 must be verified in the volume... 

Tratch, S.S., Devdariani, R.Q. and Zefirov, N.S. (1990a). Combinatorial Models and Algorithms in Chemistry. Topological-Configurational Analogs of the Wiener Index. Zh.Org.Khim., 26, 921-932. 

FIG. 8 Rate of decay of the bond autocorrelation function (b(t)) for cyclic (ring) chains of 100 sites, using local moves and topological configurational bias moves, as a function of CPU time. 

A preliminary run may therefore be necessary to determine appropriate temperature spacings between replicas. Parallel tempering in temperature has been used to study simple glasses and isolated biological molecules [40-43]. It has also been used in conjunction with topological configurational bias to study linear and cyclic alkanes [37]. 

In principle, assignment of integral numbers to O and Ti atoms can be arbitrary, which results in many possible sequences for one equivalent topological configuration of (Ti02) cluster. In order to reduce the number of redundant sequences, we always assign smaller numbers to O atoms with smaller coordination number. Hence, for two O atoms 0(/) and 0 j), if the coordination number of 0(r) is smaller than that of 0(j), then / cluster structure and its sequence. ... 

Thirdly, the vibration frequency of flie H-O bond for water molecules confined in the plasma-treated silica surfaces is around 3,200 cm (bulk feature) and that confined in the untreated silica is around 3,400 cm (skin feature with higher [17], which indicates that hydrophilic interface produces no surface effect. The air gap between the water and the hydrophobic substrate [16] makes the contacting skin more like free. However, water confined in hydrophilic nanopores [50, 51] or wetting with hydrophilic topologic configurations [52] exhibits supercooling effect, or melts at lower temperature than usual because of the lacking of broken bonds. 

It is essential to segment the 0 H-0 bond into the stronger H-O bond and the weaker 0 H non-bond. Such a master—slave-segmented, flexible, polarizable, and fluctuating H-bond represents all 0-0 interactions in solid and liquid phases irrespective of topologic configurations or phase structures. 

Note that in the n-dimensional case, where n > 4, other topological configurations of may be realized. Such saddle-node bifurcations will definitely lead the system out of the class of systems with simple dynamics. For example, it is shown in [139, 152] that a hyperbolic attractor of the Smale-Williams type may appear just after the disappearance of a saddle-node periodic orbit. ... 

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