Point correlator

Having determined the effect of the diffusive interfaces on the structure parameters, we now turn to the calculation of H and K in microemulsions. In the case of oil-water symmetry three-point correlation functions vanish and = 0. In order to calculate K from (77) and (83) we need the exphcit expressions for the four-point correlation functions. In the Gaussian approximation... 

The above results show that the structure of the system with the molecules self-assembled into the internal films is determined by their correlation functions. In contrast to simple fluids, the four-point correlation functions are as important as the two-point correlation functions for the description of the structure in this case. The oil or water domain size is related to the period of oscillations A of the two-point functions. The connectivity of the oil and water domains, related to the sign of K, is determined by the way four moleeules at distanees eomparable to their sizes are eorrelated. For > 0 surfactant molecules are correlated in such a way that preferred orientations... 

A. Ciach. Four-point correlation functions and average Gaussian curvature in microemuisions. Phys Rev E 55 1954-1964, 1997. 

Two point correlations We liave already mentioned several times that evolutions generally induce correlations between cell values at different sites indeed, the presence of correlations is a telltale signature of self-organization. The average density discussed above, however, is too crude a measure to capture these finer details. 

In its place we can use the simplest correlation measure, the two point correlation function,... 

Molecular vibrations, 249 Molecular weight, 33, 33 boiling point correlation, 307 calculation, 33 determination, 325 Molecules, 21, 274 energy of, 118 measuring dimensions, 245 models of, 21... 

The correlator (6) is of the utmost importance because its generating function enters into an expression which describes the angular dependence of intensity of scattering of light or neutrons [3]. It is natural to extend expression (6) for the two-point chemical correlation function by introducing the w-point correlator ya1... (kl...,kn l) which equals the joint probability of finding in a macromolecule n monomeric units Maj.Ma> divided by (n-1) arbitrary sequences... 

One more quantitative way to characterize the chemical structure of heteropolymers is based on the consideration of chemical correlation functions [1,3]. The simplest of such chemical correlators is the two-point correlator Ya,p(n) which equals the joint probability for two monomeric units divided in a polymer chain by an arbitrary directed sequence 17 containing n units to be of types a and f >... 

It is easy to derive expressions for other chemical correlators as well. For instance, the three-point correlator (Eq. 7) can be determined by the formula... 

The expressions for generating functions of two- and three-point correlators (Eq. 38), (Eq. 41) read... 

The substitution of expressions (Eqs. 46 and 47) and into formulas (Eqs. 42 and 43) permits finding the generating function of two- and three-point chemical correlators. Extension of these results to an arbitrary m-point correlator is obvious. 

Exploiting the Markovian property of random process. Rib, it is possible to derive in a standard way the expression for an arbitrary chemical correlator. In particular, for the three-point correlator the expression... 

Trace formulas like (8) are a starting point for analysing the statistical properties of quantum spectra. The statistical quantities such as the two-point correlation function can be written in terms of the density of states d(9,N), that is,... 

The two-point correlation function has been worked out explicitly by Berkolaiko et.al. (2001) and has been shown to coincide with the statistics of so-called Seba billiards, that is, rectangular billiards with a single flux line. The first few terms in a power series expansion of the form factor have been derived by Kottos and Smilansky (1999) and Berkolaiko and Keating (1999) and yield... 

The first term on the right-hand side of (2.61) is the spectral transfer function, and involves two-point correlations between three components of the velocity vector (see McComb (1990) for the exact form). The spectral transfer function is thus unclosed, and models must be formulated in order to proceed in finding solutions to (2.61). However, some useful properties of T (k, t) can be deduced from the spectral transport equation. For example, integrating (2.61) over all wavenumbers yields the transport equation for the turbulent kinetic energy ... 

However, single-point correlations are of limited value for two reasons. The first relates to the choice of the specific parameters to be correlated. Although there are some procedures in the literature that could be used for selecting the most appropriate parameter [e.g., the quadrant analysis (16,17)], these are not easy to apply in practice and the choice is usually based on a best-result basis. Another reason is that two processes having the same value of the chosen characteristic parameter can be different in terms of their overall shape. Consequently, a quantitative IVIVC is much more informative if established using all available in vitro and in vivo raw data these are termed multiple-point or point-to-point correlations. 

Level B utilizes the principles of statistical moment analysis. The mean in vitro dissolution time is compared to either the mean residence time or the mean in vivo dissolution time. Like correlation Level A, Level B utilizes all of the in vitro and in vivo data, but unlike Level A it is not a point-to-point correlation because it does not reflect the actual in vivo plasma level curve. It should also be kept in mind that there are a number of different in vivo curves that will produce similar mean residence time values, so a unique correlation is not guaranteed. 

This category relates one dissolution time point ( 50%, 90%, etc.) to one pharmacokinetic parameter such as cmax, max, or AUC. It represents a single point correlation and does not characterize the shape of the plasma level, which is... 

Model Ref. No. Data Points Correlation Coefficient RMS Error, ppm Regression Line y ax + b) ... 

I HHt" thin cucurbituril upon substrate concentrations for both HCSCCH2NH2-t 1 HH2C(CH3)3. Circles are data points correlating with concentrations given... 

Thus the Aq are independent Gaussian random variables with zero mean. Accordingly u(, t) has become a random field, i.e., a random function of the four variables, t rather than of t alone. One is interested in its stochastic properties, for instance, the two-point correlation function... 

Appendix C Four-Point Correlation Function Expression for Fluorescence Spectra Appendix D Phase-Space Doorway-Window Wavepackets for Fluorescence Appendix E Doorway-Window Phase-Space Wavepackets for Pump-Probe Signals References... 

This formula resembles Eq. (2.6) for the autocorrelation signal. We can further expand 5(l)( ,r) to second order in the pump field and express the result in terms of the four-point correlation function (2.8) (see Appendix E). 

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