Spectral moment fourth

Comparison of formulae (2.51) and (2.64) allows one to understand the limits and advantages of the impact approximation in the theory of orientational relaxation. The results agree solely in second order with respect to time. Everything else is different. In the impact theory the expansion involves odd powers of time, though, strictly speaking, the latter should not appear. Furthermore the coefficient /4/Tj defined in (2.61) differs from the fourth spectral moment I4 both in value and in sign. Moreover, in the impact approximation all spectral moments higher than the second one are infinite. This is due to the non-analytical nature of Kj and Kf in the impact approximation. In reality, of course, all of them exist and the lowest two are usually utilized to find from Eq. (2.66) either the dispersion of the torque (M2) or related Rq defined in Eq. (1.82) ... 

Recall that Eq. (23) is just an expression for the fourth spectral moment and that its validity is restricted to benzenoid systems for more details on the spectral moments of benzenoids see [1]. 

However, calculation of m2 and m4 involves practical difficulties, as sample spacing and signal processing. For this work, the authors made trials to determine the second and fourth spectral moments, calculating the variation (dz/dx) and (d z/dx ) of the measured points by different methods, as described below. The profile file measured by the MAHR Concept Perthometer PGK equipment was used. The file contains 8064 points, 0.695 pm spaced. Because, in this equipment, the data contained in the file is supplied as measured, an additional filtering software has to be developed and it is described in Annex I. This user filtering process can be omitted if the equipment supplies the profile already filtered. 

FJectron correlations are intimately associated with two assumptions (1) a fourth quantum number, the electron-spin quantum number s, and ( ) the Pauli exclusion principle. In order to account for spectral data, it is necessary to postulate that electrons spin about their own axis to create a magnetic moment (G2o). Whereas the magnetic moment associated with the angular momentum may have (2Z + 1) components mi in the direction of an external magnetic field H, the spin moment may have only two components corresponding to s = ms = 1/2. Classically the magnitude of the moment fia associated with an angular momentum p is... 

Calculations of the free induetion decay (FID) up to the fourth moment are consistent with the inereased decay of the FID with increasing tip angle. The increased decay of the FID is a result of increased spectral interference amongst the various spectral transitions. 

C, within the one-phase channel. The temperature is chosen at the hydrophile-lipophile balance (HLB) temperature for a salinity of 0.49%. The SANS data taken with an oil-water pontrast are analyzed by using a random-wave model with an appropriate spectral function. The spectral function is an inverse eighth-order polynomial in wave number k, containing three length scales 1/a, 1/b, and 1/c, and has finite second and fourth moments. This three-... 

The second and fourth moments of this spectral function are given by... 

So the second requirement of the physically acceptable spectral function is that it has a finite fourth moment also. Since for a bicontinuous microemulsion the level surface defined by Eq. (21) is approximately the mid-plane passing through the surfactant monolayer in a bulk contrast experiment, the average Gaussian and average square-mean curvatures of the surfactant monolayer can he computed once a physically acceptable spectral function can be found. 

In this paper, we choose a similar spectral function for which both the second and fourth moments exist. This function is an inverse eighth-order polynomial containing... 

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