Distribution function, scaling

Maximum wind speed Eq. 1 is the cumulative Gumbel distribution function. Scale parameter, a, and location parameter, b, are characteristic values of Gumbel distribution... 

Binder K 1981 Finite size scaling analysis of Ising-model block distribution-functions Z. Phys. B. Oondens. Matter. 43 119-40... 

Steinhauer and Gasteiger [30] developed a new 3D descriptor based on the idea of radial distribution functions (RDFs), which is well known in physics and physico-chemistry in general and in X-ray diffraction in particular [31], The radial distribution function code (RDF code) is closely related to the 3D-MoRSE code. The RDF code is calculated by Eq. (25), where/is a scaling factor, N is the number of atoms in the molecule, p/ and pj are properties of the atoms i and/ B is a smoothing parameter, and Tij is the distance between the atoms i and j g(r) is usually calculated at a number of discrete points within defined intervals [32, 33]. 

Frequency analysis is an alternative to moment-ratio analysis in selecting a representative function. Probability paper (see Figure 1-59 for an example) is available for each distribution, and the function is presented as a cumulative probability function. If the data sample has the same distribution function as the function used to scale the paper, the data will plot as a straight line. 

For any distribution, the cumulative hazard function and the cumulative distribution junction are connected by a simple relationship. The probability scale for the cumulative distribution function appears on the horizontal axis at the top of hazard paper and is determined from that relationship. Thus, the line fitted to data on hazard paper... 

The most important property of the self-organized critical state is the presence of locally connected domains of all sizes. Since a given perturbation of the state 77 can lead to anything from a trivial one-site shift to a lattice-wide avalanche, there are no characteristic length scales in the system. Bak, et al. [bak87] have, in fact, found that the distribution function D s) of domains of size s obeys the power law... 

One of the key quantities for a smectic phase is the scaled layer spacing, and this was determined from the periodicity in the longitudinal distribution function gj (r ). This distribution is unattenuated over the distances available from the simulation for the two smectic phases and the periodicity is the same although the width of the distribution is smaller in the smectic B than the... 

The distribution of a data set in the form of a histogram can always be plotted, without reference to theories and hypotheses. Once enough data have accumulated, there is the natural urge to see whether they fit an expected distribution function. To this end, both the experimental frequencies and the theoretical probabilities must be brought to a common scale a very eonvenient... 

The power of X-ray methods can be extended to investigate the local structure on a scale of a few angstroms by means of the analysis of the fine structure and the radial distribution function. 

Fig. 6.8. Scaled distribution functions of RG(N) vs. the Gaussian-normalized value of RG for chain lengths as indicated (from [45])...

Fig. 50. Small-angle neutron scattering results from different stars in a scaled form. The lines are the result of a fit with Eq. (94). Insert Related radial segment distribution functions obtained from a Fourier transformation of the theoretical scattering function. (Reprinted with permission from [150]. Copyright 1987 The American Physical Society, Maryland)...

Figure 36. The scaled distributions of mean, P(H/Y ) (a), and Gaussian, P(K/Y]2) (b), curvatures scaled with the inteface area density, computed at several time intervals of the spindal decomposition of a symmetric blend. There is no scaling at the late times because the amplitude of the thermal undulations does not depend on the average growth of the domains, and therefore the scaled curvature distributions functions broaden with rescaled time.

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