Statistical theory, component

Certainly two-dimensional techniques have far greater peak capacity than onedimensional techniques. However, the two-dimensional techniques don t utilize the separation space as efficiently as one-dimensional techniques do. These theories and simulations utilized circles as the basis function for a two-dimensional zone. This was later relaxed to an elliptical zone shape for a more realistic zone shape (Davis, 2005) with better understanding of the surrounding boundary effects. In addition, Oros and Davis (1992) showed how to use the two-dimensional statistical theory of spot overlap to estimate the number of component zones in a complex two-dimensional chromatogram. 

Davis, J.M., Giddings, J.C. (1983). Statistical theory of component overlap in multicomponent chromatograms. Anal. Chem. 55, 418-424. 

The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (JJ, by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chain-junction entanglements. 

RR is similar to PCR in that the independent variables are transformed to their principal components (PCs). However, while PCR utilizes only a subset of the PCs, RR retains them all but downweighs them based on their eigenvalues. With PLS, a subset of the PCs is also used, but the PCs are selected by considering both the independent and dependent variables. Statistical theory suggests that RR is the best of the three methods, and this has been generally borne out in multiple comparative studies [30,36-38]. Thus, some of our published studies report RR results only. 

In other statistical theories of rubber elasticity (see e.g. reviews 29,34)) the Gaussian statistics is not valid even at small deformations and the intramolecular energy component is dependent on deformation. 

Hydrate experimental conditions have been defined in large part by the needs of the natural gas transportation industry, which in turn determined that experiments be done above the ice point. Below 273.15 K there is the danger of ice as a second solid phase (in addition to hydrate) to cause fouling of transmission or processing equipment. However, since the development of the statistical theory, there has been a need to fit the hydrate formation conditions of pure components below the ice point with the objective of predicting mixtures, as suggested in Chapter 5. 

Although factorial designs are very useful for studying multiple variables at various levels, typically they will not be applicable to cosolvent solubility studies because of the constraint that all of the components must add to 100%. Forthis reason, mixtures of experimental designs are typically used. The statistical theory behind mixture designs has been extensively published [81-85], There... 

Maximum Number of Components Resolvable by Gel Filtration and Other Elution Chromatographic Methods, J. C. Giddings, Anal. Chem., 39, 1927 (1967). Statistical Theory for the Equilibrium Distribution of Rigid Molecules in Inert Porous Networks. Exclusion Chromatography, J. C. Giddings, E. Kucera, C. P. Russell, and M. N. Myers, J. Phys. Chem., 72, 4397 (1968). 

Statistical Theory of Component Overlap in Multicomponent Chromatograms, J. M. Davis and J. C. Giddings, Anal. Chem., 55, 418 (1983). 

Percolation theory is a statistical theory that studies disordered or chaotic systems where the components are randomly distributed in a lattice. A cluster is defined as a group of neighboring occupied sites in the lattice, being considered an infinite or percolating cluster when it extends from one side to the rest of the sides of the lattice, that is, percolates the whole system [38],... 

To Buckingham et al. is due the theory of the Kerr effect in dilute solutions, whereas Kielich developed a statistical theory of the effect for multi-component systems of an arbitrary degree of concentration and showed the Kerr constant of real solutions to be a non-additive quantity. 

GC Runger and FB Alt. Choosing principal components for multivariate statistical process control. Commun. Statist. - Theory Methods, 25(5) 909-922, 1996. 

The statistical theory shows that the mean number p of observed peaks Is related to the expected number of components m by... 

The statistical theory also shows that the number of one-component or singlet peaks (an observed peak consisting of only one underlying component peak) Is given by... 

We use the variable m to represent the total number of components In the synthetic chromatogram Instead of in. The fonner value Is known In our computer-generated chromatograms but not In a complex mixture subjected to chromatography. In either case, only the mean component number m may be estimated by the statistical theory. 

Several aspects of this formulation require emphasis. First, defining the SCR yields a particular statistical theory hence, a wide variety of theories are possible. Second, one should anticipate that a typical scattering event will have both direct and statistical components. Third, an analysis of the degree of statistical behavior in an exact dynamical calculation requires three sequential steps (1) the SCR must first be defined (2) the component of the scattering that does not pass through the SCR is then eliminated from consideration and (3) the product distribution associated with the component passing through the SCR is compared with the results predicted from Eq. (2.17) the latter often requires additional numerical computations. 

These studies indicate a direct connection between requirements of exponential divergence and adherence to statistical theories. Note, however, that the particular statistical theory obeyed by the dynamics need not be a simple analytic theory such as RRKM or phase space theory. It may appear, therefore, that the an essential simplicity of statistical theory, that is, the ability to bypass long-lived trajectory calculations in favor of an easily computed result, has been lost. This is indeed not the case, that is, it is easy to see that this approach affords a method for obtaining contributions from both the unstable (statistical) trajectory component as well the direct component with a minimum of computation. This technique, the minimally dynamic 33,34 approach, will now be sketched. 

The statistical theory is concerned with the actual velocities of individual particles in stationary, homogeneous turbulence. Under this assumption the statistics of the motion of one typical particle provides a statistical estimate of the behavior of all particles, and that of two particles an estimate of the behavior of a cluster of particles. In the atmosphere one may expect the cross-wind component (v) of turbulence to be nearly homogeneous since... 

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