Contracts versus systems

Table 6.1 gives electrochemical data for representative linked M02 and Wj systems. The main virtue of examining related M02 and W2 " systems is that the size of ions and their solvation are essentially identical due to the lanthanide contraction, and so the differences in electrochemical data reflect electronic differences due to the 4d versus 5d orbitals and the relative energies of the respective Mj-fi orbitals. 

With respect to the above it is noteworthy that Kent et al. [107] performed their study using narrow polydispersity probe and matrix polymers. The insensitivity of Rg versus polymer concentration below C only occurs if the molar mass of the probe and background polymer are similar. If the matrix polymer is of much lower molar mass, it can freely penetrate the probe polystyrene molecules and act as a poor viscous solvent inside the probe coils. In that case a decrease in Rg can also be observed at polymer concentration below C [107], In real free-radical polymerizations, polymer molecules with a wide variety of molar masses will be present simultaneously and it can thus be expected that all macroradicals will experience coil contraction to some extent in dilute solutions (except for the very smallest macroradicals). The magnitude of this effect will thus be dependent upon the molecular weight distribution (MWD) of the polymer and thus also upon the systems polymerization history. 

The difference in the ketonizing ability of a 3-tricarbonyl system versus a p-dicarbonyl moiety may explain the regioselective preference for "normal" ring contraction. 

A homoclinic bifurcation is a composite construction. Its first stage is based on the local stability analysis for determining whether the equilibrirun state is a saddle or a saddle-focus, as well as what the first and second saddle values are, and so on. On top of that, one deals with the evolution of a -limit sets of separatrices as parameters of the system change. A special consideration should also be given to the dimension of the invariant manifolds of saddle periodic trajectories bifurcating from a homoclinic loop. It directly correlates with the ratio of the local expansion versus contraction near the saddle point, i.e. it depends on the signs of the saddle values. 

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