Gaussian distribution weight function

Fig. 2.2. Average electrostatic potential mc at the position of the methane-like Lennard-Jones particle Me as a function of its charge q. mc contains corrections for the finite system size. Results are shown from Monte Carlo simulations using Ewald summation with N = 256 (plus) and N = 128 (cross) as well as GRF calculations with N = 256 water molecules (square). Statistical errors are smaller than the size of the symbols. Also included are linear tits to the data with q < 0 and q > 0 (solid lines). The fit to the tanh-weighted model of two Gaussian distributions is shown with a dashed line. Reproduced with permission of the American Chemical Society...

It is noted that the phonon wavefunction is a superposition of plane waves with q vectors centered at In the literature, several weighting functions such as Gaussian functions, sine, and exponential functions have been extensively used to describe the confinement functions. The choice of type of weighting function depends upon the material property of nanoparticles. Here, we present a brief review about calculated Raman spectra of spherical nanoparticle of diameter D based on these three confinement functions. In an effort to describe the realistic Raman spectrum more properly, particle size distribution is taken into account. Then the Raman intensity 7(co) can be calculated by ... 

In the literature (Chalons et al, 2010), only a bivariate EQMOM with four abscissas represented by weighted Gaussian distributions with a diagonal covariance matrix has been considered. However, it is likely that brute-force QMOM algorithms can be developed for other distribution functions. Using the multi-Gaussian representation as an example, the approximate NDF can be written as... 

This heterogeneity may, however, also be expressed in terms of polymer lifetimes instead of propagation rates. By assuming a Gaussian distribution of chain lifetimes Mussa (35) has been able to derive molecular weight distributions of the type observed experimentally. This treatment appears to imply a chain termination which has the form of an error function round a mean value. On the whole it seems that a variable propagation rate is the more likely. 

V Vu w(a,p,y) w(N, r) W t) wq volume of a polymer segment. 6.1.1.3 scattering volume. 1.2.2 unit cell volume. 3.3.1 crystallite orientation distribution function. 3.6.3 end-to-end distribution of a Gaussian chain. 5.2.1 [5.12] slit-length weighting function. 5.6.1 constant value of W(t) with infinite slit approximation. 5.6.3... 

The first term originates from the configurations that require the two directed polymers to meet at a vertex. In contrast, the second term counts the no encounter cases. There are no energy costs at the two end points. The Boltzmann weight is random and for a Gaussian distribution of energy, y" = y . The moments of the partition function can be written as... 

A nuclear magnetic resonance line is usually found to have one of two ideal lineshapes - Gaussian, or more often, Lorentzian. A Gaussian line is found when there is a random distribution of static fields within the sample. A Lorentzian line by contrast arises because the spin lifetime follows a first-order decay law. Weighting functions can be applied to a free-induction decay to generate... 

Log normal distribution (logarithmic normal distribution). A statistical probabiUty-density function, characterized by two parameters, that can sometimes provide a faithful representation of a polymer s molecular-weight distribution or the distribution of particle sizes in ground, brittle materials. It is a variant of the familiar normal or Gaussian distribution in which the logarithm of the measured quantity replaces the quantity itself. Its mathematical for is... 

Therefore, in an attempt to obtain simple analytic expressions for the distribution functions of stiff polymer chains, condition (5.2a) is relaxed. The relaxation of this condition is in the original spirit of the use of Wiener integrals. If this condition were imposed for flexible polymer chains, the Wiener measure would be 2[t s)] exp (—3L/2/) and would give equal weight (measure) to all continuous configurations of the polymer. Thus the use of (5.2a) would not yield the correct gaussian distribution for flexible chains. 

Width of Moiecuiar Weight Distributions. The width of the Gaussian distribution function may be expressed in terms of the standard deviation of the mole fraction MWD function cr or the mass fraction MWD function... 

Of course, there is a distribution in end-to-end distances for random coils, even of the same molecular weight. The distribution of end-to-end distances can be treated by Gaussian distribution functions (see Chapter 9). The most important result is that, for relaxed random coils, there is a well-defined maximum in the frequency of the end-to-end distances, this distance is designated as To. 

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