Statistical overlap

The use of the Poisson distribution for this purpose predates the statistical overlap theory of Davis and Giddings (1983), which also utilized this approach, by 9 years. Connors work seems to be largely forgotten because it is based on 2DTLC that doesn t have the resolving power (i.e., efficiency or the number of theoretical plates) needed for complex bioseparations. However, Martin et al. (1986) offered a more modem and rigorous theoretical approach to this problem that was further clarified recently (Davis and Blumberg, 2005) with computer simulation techniques. Clearly, the concept and mathematical approach used by Connors were established ahead of its time. 

Davis (1993) extended the statistical overlap theory to generalized -dimensional separations with the consistent result that the separations get much better, but as dimensionality increases the efficiency of using that separation space decreases. For -dimensional separations, Davis says... 

These are most important realizations that will guide the evolution of multiple dimension chromatographic systems and detectors for years to come. The exact quantitative nature of specific predictions is difficult because the implementation details of dimensions higher than 2DLC are largely unknown and may introduce chemical and physical constraints. Liu and Davis (2006) have recently extended the statistical overlap theory in two dimensions to highly saturated separations where more severe overlap is found. This paper also lists most of the papers that have been written on the statistical theory of multidimensional separations. 

Davis, J.M. (2005). Statistical-overlap theory for elliptical zones of high aspect ratio in comprehensive two-dimensional separations. J. Sep. Sci. 28, 347-359. 

Davis, J.M., Pompe, M., Samuel, C. (2000). Justification of statistical overlap theory in programmed temperature gas chromatography thermodynamic origin of random distribution of retention times. Anal. Chem. 72, 5700-5713. 

Liu, S., Davis, J.M. (2006). Dependence on saturation of average minimum resolution in two-dimensional statistical-overlap theory peak overlap in saturated two-dimensional separations. J. Chromatogr. A 1126, 244—256. 

The practical peak capacity of these spaces can be characterized by 2D statistical-overlap theory. Consider a relatively simple problem of probability. If, on average, m circles of diameter d() are distributed randomly in a large area A, then the average number p of clusters of isolated and overlapping circles approaches (Roach, 1968)... 

It is important to realize that statistical-overlap theory is not constrained by the contour of area A, which does not have to be rectangular as in earlier studies (in addition to previous references, see Davis, 1991 Martin, 1991,1992). In other words, Equations 3.2 and 3.3 should apply to the spaces WEG, FAN, and PAR. In this chapter, the number of clusters of randomly distributed circles in such areas is compared to the predictions of Equations 3.2 and 3.3a to assess the relationship between nP and practical peak capacity. Similarly, the number of peak maxima formed by randomly distributed bi-Gaussians in such areas is compared to the predictions of Equations 3.2 and 3.3b, and to Fig. 3.2, to make another assessment. 

In general, statistical-overlap theory has been developed and applied to several systems, mostly one-dimensional ones. Two reviews summarize many developments (Pietrogrande et al., 2000 Felinger and Pietrogrande, 2001). With regards to other multidimensional separations, theories have been reported for -dimensional... 

The agreement provides the answer to the question posed earlier in the chapter a measure of practical peak capacity, as assessed by statistical-overlap theory, is nv... 

Davis, J.M. (1997b). Extension of statistical overlap theory to poorly resolved separations. 

Samuel, C., Davis, J.M. (2002). Statistical-overlap theory of column switching in gas chromatography application to flavor and fragrance compounds. Anal. Chem. 74, 2293. 

As mentioned previously, the gas-side hydrophobic layer does not eliminate completely the liquid drain from the gas side, a drain that may be - if only marginally -favoured by the additional liquid-side hydrophilic layer. The situation is clearly tied to the presence of cracks, which, as stated above, develop during the sintering and cooling cycles (see Fig. 9.10). The number of cracks is limited thanks to a proper temperature profile of the thermal treatment. Further, if the layer is itself comprised of a number of sub-layers, each one thermally treated before an additional sub-layer is applied, the cracks in each sub-layer do not statistically overlap, so that the popu-... 

As all live plant and animal studies incorporate the distributive process, they are currently assailable only through statistical analysis of experimental data. These restrictions focus the area of modeling and statistical overlap clearly at the vitro stage of reversible and irreversible inhibition, areas where all techniques enter the arena. 

Furthermore, the problem of comparing excited states sometimes arises, and quantities familiar from probability theory can be exploited for this purpose. When needed, we apply the so-called classical fidelity measure, or statistical overlap, computed in our case from the CT probability distribution (14.37) as follows ... 

In the case of amorphous materials, the reciprocal space does not contain localized scattered beams. It is fuU of intensity with some faint maxima (see Section 1.1.2.4 and Figure 1.3). The objective aperture is thus illuminated in any of its positions. It produces bright dots due to the aperture Airy disc combined with the beams scattered by statistical overlapping of atom pairs. The dots are thus not localized in the real space. 

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