Cutoff functions

Integrating paa (s) with a cutoff function gc(s) introduced by Ando et al. [5]... 

The definition of 9ftey( ) is then extended to < 0 by imposing that it must be an even function of . In Eq. (24), the function/c( / c) is a cutoff function tending toward zero when —> oo. One often chooses a Lorenzian cutoff function ... 

In the Ohmic dissipation model with a Lorentzian cutoff function, y(oo) is given by Eq. (28), and Eq. (102) reads... 

The values of the critical exponents r and a and the cutoff functions /+ (N/N ) and/ (N/N ) depend only on the dimension of space in which gelation takes place. The percolation model has been solved analytically in one dimension (d=, see Sections 1.6.2 and 6.1.2) and critical exponents have been derived for two dimensions (d = 2). The mean-field model of gelation corresponds to percolation in spaces with dimension above the upper critical dimension (d>6). The cutoff function in the mean-field model [see Eq. (6.77)] is approximately a simple exponential function [Eq. (6.79)]. The exponents characterizing mean-field gelation are o — 1/2 and... 

Since the Fisher exponent r for percolation in any dimension is limited to the interval 2 < r < 5/2, the first two moments of the distribution mo and mi do not diverge ahthe gel point. These two moments are dominated by the smaller polymers with a small contribution from the larger ones. Below the gel point, the sol fraction is unity [Psoiip) = mi = 1, see Eq. (6.42)] and the gel fraction is zero [Fgei(/ ) = 1 - P oi(p) = Oj- Above the gel point, we can approximate the cutoff function as a step function that goes to zero... 

The cutoff function f x) assures that the integral in the above equation converges to a number. This approach therefore determines the way in... 

The cutoff functions/+(/V/A ) and/-(A /A ) are expected to only depend on space dimension. In dimensions 1 < 6, the cutoff functions... 

The scaling variable z can be used to construct universal plots of molar mass distributions. There are two approaches to making these universal plots. To study the cutoff function itself, multiply both sides of Eqs (6.93) and (6.94) by -----------------------------------------------------------------... 

The analytical form of the cutoff function in three-dimensional critical percolation is not known. However, experimental and numerical universal plots have been constructed using the methods described above and constitute a convincing proof of the validity of the scaling ansatz (see Fig. 6.26). 

Comparing this expression with Eq. (6.93) identifies r = 2 and the onedimensional percolation cutoff function... 

Using the proportionality between the characteristic degree of polymerization N and a universal distribution function for hyperbranched polymers can be constructed by plotting n p, M jNl, against N/N. The facr that the cutoff function contains the power law in z makes the apparent power law exponent 3/2 in the molar mass distribution different from r = 2. The cutoff function for gelation has no power law so the apparent exponent is equal to t. 

The cutoff functions are plotted in Fig. 6.28. For critical gelation in three dimensions (and in the more general case of 1 [Pg.234]

A very important caveat with the universal molar mass distributions and cutoff functions is that calculated molar mass distributions of linear condensation, hyperbranched condensation and mean-field percolation all assume no intramolecular reactions occur. Intramolecular reactions are... 

Show that T = 5/2, find N and evaluate the cutoff function f NjN ) for a general functionality /. 

Mayer /-function, [dimensionless], p. 99 cutoff function above the gel point, [dimensionless], p. 227 cutoff function below the gel point, [dimensionless], p. 227... 

PAGE 251 change values to p=0.4, 0.45, 0.48. Add "Hint Plot the cutoff function... 

The function fc(R) is a cutoff function and passes quickly but smoothly from unity to zero as R —> Rc, where Rc is a cutoff distance. The function giOijk) is an angular term of the form... 

Equation 14.29 defines the density correlation function C(r), where p(f) is the density of material at position r, and the brackets represent an ensemble average. In Equation 14.30, A is a normalization constant, D is the fractal dimension of the object, and d is the spatial dimension. Also in Equation 14.30 are the limits of scale invariance, a at the smaller scale defined by the primary or monomeric particle size, and at the larger end of the scale h(rl ) is the cutoff function that governs how the density autocorrelation function (not the density itself) is terminated at the perimeter of the aggregate near the length scale As the structure factor of scattered radiation is the Fourier transform of the density autocorrelation function. Equation 14.30 is important in the development below. 

To gain more than a qualitative notion on how the structure factor is related to the structure of the object, an exact form for the correlation function must be given. A fractal object (cluster) will have C(r) as in Equation 14.30. Beyond that, it is crucial to know the form of the correlation function cutoff function This function must decrease faster than any power law to... 

In another study, soot from the same flame was thermophoretically captured and examined via transmission electron microscopy (TEM). The correlation function of soot clnsters was determined and the cutoff function extracted from that. Once again, the P = 1 exponential failed completely. The best cutoff was the overlapping spheres, which is edifying given its physical origin. However, the P = 2 Gaussian worked very well failing only when h(rl ) had decreased by a factor of 100. 

Here we use the standard Weyl constructions, described for the spectral theory of molecules, for example in [59]. For more modern constructions see [5,55]. For example, one may use, for the free eigenfunction i/rj of the state with the energy Ej, the modified function ffj with the cutoff function /(r), that is equal to 1 for some large region 2 (A) and vanishes outside a neighborhood of this region, 2(A + 8). In this case... 

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