Geometric progressive growth

NMR has accomplished a growth in a geometrical progression since the early sixties and virtually developed into extremely potential analytical tools not only useful for elucidation of complex structural determinations but also equally beneficial in the assay of pharmaceutical substances. 

In step-growth polymerization of bifunctional monomers, the mole fractions of successive polymers (with increasing number of structural units) are in a declining geometrical progression. 

A mole-fraction distribution that is a declining geometrical progression is called a Schulz-Flory distribution or most probable distribution and is quite common [29,30]. As later examples will show, it can arise from other mechanisms as well and can therefore not be taken as evidence for step growth. 

The degree of polymerization is seen to show the same dependence on the geometric-progression factor as in step-growth polymerization of bifunctional monomers (eqn 10.16) and free-radical polymerization with chain breaking by disproportionation or terminating chain transfer (eqn 10.42). 

The geometric models of solid state reactions are based upon the processes of nucleation and growth of product nuclei by interface advance. These processes are discussed individually in the next section, followed by a description of the ways in which these contributions are combined to give rate equations for the overall progress of reaction. 

We introduce the drift component briefly as follows. For an asset such as an ordinary share, which is expected to rise over time (at least in line with assumed growth in inflation), the drift can be modelled as a geometric growth progression. If the price process had no noise , the change in price of the stock over the... 

As mixing progresses, the interfacial area per unit volume increases and the striation thickness decreases. If there is a distribution of striation thickness then not only the mean but also the variance should be taken into consideration for the quality of mixing calculations. In Section 6.3.1 the interfacial area growth, or equivalently the striation thickness reduction, is calculated from geometrical arguments. 

Geometrical analysis of this interface gave a roughness exponent of 0.62. Analysis of the interface for samples of various Al concentration which were crystallized at various temperatures is in progress in order to supply quantitative information on the interface and its growth mechanism. 

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