Correlation function 324 • INDEX

Figure 9.2 displays a division of the mass range into 13 intervals. After multiplication of all the individual normalized correlation values, the composite correlation index is smaller and the method achieves more selectivity than it would if a single correlation function were calculated over the full mass range. A lower limit of 0.0001 was placed on the value for each normalized interval comparison to prevent the method from generating absurdly low values. These... 

Wun and Prins (32) have used a Gaussian form for the correlation function, and then defined a non-random index in terms of a correlation volume rather than in terms of correlation length. 

Evidently, correlation functions for different spherical harmonic functions of two different vectors in the same molecule are also orthogonal under equilibrium averaging for an isotropic fluid. Thus, if the excitation process photoselects particular Im components of the (solid) angular distribution of absorption dipoles, then only those same Im components of the (solid) angular distribution of emission dipoles will contribute to observed signal, regardless of the other Im components that may in principle be detected, and vice versa. The result in this case is likewise independent of the index n = N. Equation (4.7) is just the special case of Eq. (4.9) when the two dipoles coincide. 

The photoinduced absorbance anisotropy in a TPD experiment relaxes according to the same correlation function as in Eq. (4.16).(29) Effects of spatial variations in the excitation and probe beams, and chromophore concentration, have been treated and shown not to alter the final result.(29) NMR dipolar relaxation rates are expressed in terms of Fourier transforms of the correlation functions, 4ji< T2m[fi(0)] T2m[i2(f)]>> where fl(f) denotes the orientation of a particular internuclear vector. In view of Eq. (4.7), these correlation functions are independent of the index m, hence formally the same as in Eq. (4.16). For the analysis of NMR relaxation data, it is necessary also to evaluate Fourier transforms of the correlation functions. Methods to accomplish this in the case of deformable DNAs have been developed and applied to analyze a variety of data.(81 83)... 

Even without having the stmcture factor phases, e.g. from electron microscopy images, it is possible to get some insight into the atomic architecture of a crystal. A simple but powerful method to get this information was introduced hy A.L. Patterson about 70 years ago. Following Patterson the Fourier synthesis is carried out using the squared stmcture factor amplitudes Fha which are equal to the measured intensities for the reflections with index hkl. Moreover, all phase values must be set to zero, which leads to the following (auto-correlation) function ... 

Field-based similarities are usually evaluated by the cosine or correlation function similarity measure employed initially by Carbo and co-workers (67) to compute molecular similarities based upon quantum mechanical wavefunctions. Such a measure, which is usually called a Carbo similarity index, is given by... 

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

The shape of the vibration-rotation bands in infrared absorption and Raman scattering experiments on diatomic molecules dissolved in a host fluid have been used to determine2,15 the autocorrelation functions unit vector pointing along the molecular axis and P2(x) is the Legendre polynomial of index 2. These correlation functions measure the rate of rotational reorientation of the molecule in the host fluid. The observed temperature- and density-dependence of these functions yields a great deal of information about reorientation in solids, liquids, and gases. These correlation functions have been successfully evaluated on the basis of molecular models.15... 

The relaxation of a local mode is characterized by the time-dependent anomalous correlations the rate of the relaxation is expressed through the non-stationary displacement correlation function. The non-linear integral equations for this function has been derived and solved numerically. In the physical meaning, the equation is the self-consistency condition of the time-dependent phonon subsystem. We found that the relaxation rate exhibits a critical behavior it is sharply increased near a specific (critical) value(s) of the interaction the corresponding dependence is characterized by the critical index k — 1, where k is the number of the created phonons. In the close vicinity of the critical point(s) the rate attains a very high value comparable to the frequency of phonons. 

Becke three-parameter hybrid functional combined with Lee-Yang-Parr correlation functional Bond shortening index Bird index... 

Here B is an optical constant, or is the total polarizability of the particle, and n is the number of components in each particle. The indexes i and j refer to components of the same particle. If the assumption of independent particles was not made, then the indexes could refer to components of any two particles, and the autocorrelation expression could not be written as a simple sum of contributions from individual particles. The spatial vector r(r) refers to the center of mass of the particle. R(r). In the case of a nonspherical particle (arbitrary shape), Eq. (I0) would describe the coupled motion of the center of mass and the relative arrangement of the components of the particle. For spherical particles, translational and rotational motion arc uncoupled and we have a simplified expression for the electric field time correlation function ... 

Nieboer and McBryde (44) have developed what they term a correlating numerical index 0, which is a function of the ionic charge and electronegativity of a particular metal cation, and of the type of ligand complexing with it. Linear relations are obtained when Q is plotted against log values for complexes... 

Using the closure relationships Exj. (G.19) and (G.20) for separations below and above the hard core, we can now calculate the new direct correlation functions (r). We stress that the closure can be solved independentlj- for each index combination a/ . Furthermore, note that the HNC closure (r > a p) involves the total correlation functions (f), which can bo approximated by the current estimates (r) -... 

Variants and generalizations of the bond index were proposed by Mayer [Mayer, 1986b, 1986a Wang and Werstiuk, 2003 Ponec and Cooper, 2005]. For example, in the framework of the AIM theory, starting from the correlation function characterizing the extent of the electron sharing and defined as... 

In infrared absorption experiments, one measures the absorption coefficient k(co) as a function of a>. Since the complex refractive index is N — n + Ik, it may be shown that e" = 2me (Landau and Lifshitz, 1960). Thus in order to relate k, the measured quantity, to e" and then to the dipolar correlation function [Eq. (15.3.5)], one must knowhow the refractive index n (co) changes through the band. 

When there are more than one species present in a sample, each contributes to give a correlation function which is a sum of exponentials and a power spectrum which is a sum of Lorentzians. The intensity of each component in the composite function is proportional to the product of the molecular weight and the concentration in (w/v) units, assuming the refractive index increments of each component are identical. Two basic approaches are available to extract the particle distribution from the QLLS data. 

The ratios of TIMP-1 MMP-1 and TIMP-LMMP-3 in blood have been reported to be significantly lower in patients with rheumatoid arthritis versus patients with nonrheumatoid arthritis (CIO). In rheumatoid arthritis patients, serum C-reactive protein correlated with MMP-3 and TIMP-1 levels, but not with MMP-1 levels. The number of erosions noted on X-rays correlated with baseline levels of MMP-3, but not TIMP-1. Cunnane et al. (CIO) postulated that treatment which inhibits the production and activation of MMP-1 may preferentially limit the formation of new joint erosions and improve the clinical outcome of patients with rheumatoid arthritis. In contrast to circulating levels of MMP-1, Keyszer et al. reported that MMPrTIMP complexes in blood correlate with rheumatoid activity scores (modified Lansbury Index and Keitel Function Index) in rheumatoid arthritis nonetheless, this relationship to disease activity was weaker than that of MMP-3 or C-reactive protein (K4). 

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