D Autocorrelation

the component of the autocorrelation vector a for the distance interval between the boundaries dj (lower) and (upper) is the sum of the products of property p for atoms i and j, respectively, having a Euclidian distance d within this interval. 

In contrast to the topological autocorrelation vector, it is possible to distinguish between different conformations of a molecule using 3D autocorrelation vectors. 

The calculation of autocorrelation vectors of surface properties [25] is similar (Eq. (21), with the distance d XiXj) between two points and Xj on the molecular surface within the interval between d[ and d a certain property p, e.g., the electrostatic potential (ESP) at a point on the molecular surface and the number of distance intervals 1). 

The component of the autocorrelation vector for a certain distance interval between the boundaries 4 and du is the sum of the products of the property p x,) at a point Xi on the molecular surface with the same property p Xj) at a point Xj within a certain distance d Xj,Xj) normalized by the number of distance intervals 1. All pairs of points on the surface are considered only once. 

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MATSIe Moran autocorrelation -lag 1/weighted by atomic Sanderson electronegativities 2-D autocorrelation... 

Fig. 11.47a-d. Autocorrelation signal S oc for different pulse profiles without... 

Fig.ll.34a-d. Autocorrelation signal S oc CW(r) for different pulse profiles without background suppression upper part) and with background suppression (lower part) (a) Fourier limited Gaussian pulse, (b) rectangular pulse, (c) single noise pulse, and (d) continuous noise... 

Day P N and Truhlar D G 1991 Benchmark calculations of thermal reaction rates. II. Direct calculation of the flux autocorrelation function for a canonical ensemble J. Chem. Phys. 94 2045-56... 

In order to transform the information fi om the structural diagram into a representation with a fixed number of components, an autocorrelation function can be used [8], In Eq. (19) a(d) is the component of the autocorrelation vector for the topological distance d. The number of atoms in the molecule is given by N. 

We denote the topological distance between atoms i and j (i.e., the number of bonds for the shortest path in the structure diagram) dy, and the properties for atoms i and j are referred to as pi and pj, respectively. The value of the autocorrelation function a d) for a certain topological distance d results from summation over all products of a property p of atoms i and j having the required distance d. 

White J. A., Velasco S., Calvo Hernandez A., Luis D. On the quantum time autocorrelation function, Phys. Lett. A130, 237-9 (1988). 

Figure 20 shows a 3-D view of a generated non-Gaussian rough surface with an exponential autocorrelation and desired skewness and kurtosis of -1.75 and 5.0, respectively. The surface shows an outlook of a typical worn surface due to the negative skewness. The real values of SK and K were calculated as -1.7827 and 5.1104, a good agreement between specihed and real values. 

Figure 8.8 Typical fluorescence autocorrelation curves of R6G in ethylene glycol (a) and R123 in water (b) without the NIR laser light with calculated curves (solid line) based on Eq. (8.1) and residuals. Fluorescence autocorrelation curves of R6G in ethylene glycol (c) and R123 in water (d) under irradiation of the NIR laser at several powers up to 240 mW. The inset of Figure 8.8d shows a magnified view of a partofthe figure enclosed by a rectangle.

There must be some spatial autocorrelation in the image, that means that the local frequency have to be low or medium according to Fig. 2.6b,c (in contrast, in cases of zero- and high local frequencies, see Fig. 2.6a,d, there is no autocorrelation). 

The temporal evolution of P(r,t 0,0) is determined by the diffusion coefficient D. Owing to the movement of the particles the phase of the scattered light shifts and this leads to intensity fluctuations by interference of the scattered light on the detector, as illustrated in Figure 9. Depending on the size of the polymers and the viscosity of the solvent the polymer molecules diffuse more or less rapidly. From the intensity fluctuations the intensity autocorrelation function... 

Figure 10 (a) Free-volume persistence time extracted from the free-volume autocorrelation function (Eq. [9]) for an attractive colloidal fluid as a function of the strength of the interparticle attraction, (b) Comparison of colloidal self-diffusivity (closed symbols) with that estimated using the free-volume scaling relationship D — A(v )2 /tf discussed in the text (open symbols). Data taken from Ref. 75. 

For a rod-like probe with its absorption transition moment direction coinciding with the long molecular axis, the rotational motion in this potential well is described by the diffusion coefficient D. The decay of the autocorrelation functions is then shown to be an infinite sum of exponential terms ... 

The diffusion coefficient D is one-third of the time integral over the velocity autocorrelation function CvJJ). The second identity is the so-called Einstein relation, which relates the self-diffusion coefficient to the particle mean square displacement (i.e., the ensemble-averaged square of the distance between the particle position at time r and at time r + f). Similar relationships exist between conductivity and the current autocorrelation function, and between viscosity and the autocorrelation function of elements of the pressure tensor. 

As the salt concentration continues to decrease, however, matters change dramatically Q). The total scattering intensity decreases more abruptly, and the QLS autocorrelation function, which has been a simple single-exponential decay, becomes markedly two-exponential. The two decay rates differ by as much as two orders of magnitude. The faster continues the upward trend of D pp from higher salt, and is thus assigned the term "ordinary . The slower, which is about 1/10 of Dapp high salt, and appears to reflect a new mode of solution dynamics, is termed "extraordinary . 

Kennedy, A.D., Pendleton, B. Acceptances and autocorrelations in hybrid Monte Carlo. Nucl. Phys. B (Proc. Suppl.) 1991, 20, 118-21. 

Fittinghoff, D. N., der An, J. A., and Squier, J. 2005. Spatial and temporal characterizations of femtosecond pulses at high-numerical aperture using collinear, background-free, third-harmonic autocorrelation. Opt. Comm. 247 405-26. 

FIGURE 5.2 (a) Experimental (filled circles) wavelength tuning curve and accessible Raman freqnencies as a fnnction of the crystal temperatnre. The solid curves are a result of the calculations. (b) OPO output power versus pump power at the crystal facet (c) and (d) show the typical signal pulse spectrum and autocorrelation trace at the OPO cavity detuning of minus 36 (xm, respectively. 

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