Computing CDFs

The main steps from the recorded pattern toward the CDFs are schematically depicted in Fig. 3.5. It shows a representative recorded SAXS pattern, the corrected flber diagram I s 2, S3), absolute values of the CDF z(ri2, / 3) and a slice of the CDF along the meridian, z(0, r ). Plotting the z(0, r3) curve is the easiest way of analyzing the structure along the principal axis of the material (injection-molding direction). 

The multidimensional CDF z(r) shows peaks wherever there are domain surface contacts between domains in /o(r) and in its displaced ghost. Such peaks hi (ru, r ) are called [6] distance distributions. Distance is the ghost displacement. It is sometimes useful to replace the index i by a sequence of indexes that indicate the sequence of domains that have been passed along the displacement path until the considered domain surface contact occurs. For instance hcairu, r ) indicates the passing of an amorphous and a crystalline domain. Thus this peak is a long period peak, hca will 

Like in the correlation method, a set of measured data points is fed to a regression algorithm [2, 9]. Instead of the 7 highest points of a ID peak, now all those points in a cap are used, whose intensity is above a user-defined level. The ID quadratic polynomial from the correlation method is replaced by a 2D (bivariate) polynomial 

The long period L measures the distance in straining direction between neighboring crystallites. Let L(t) the long period at time t, and Lq = L(0) the long period at the beginning of a deformation experiment, then a nanoscopic strain 

Similarly a lateral nanoscopic strain s j can be estimated based on the variations 

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CDFs are computed from scattering data which are anisotropic and complete in reciprocal space. Thus the minimum requirement is a 2D SAXS pattern of a material with fiber symmetry taken in normal transmission geometry (cf. p. 37, Fig. 4.1). Required pre-evaluation of the image is described in Chap. 7. 

Figure 8.27. Steps preceding the computation of a CDF with fiber symmetry from recorded raw data The image is projected on the fiber plane, the equivalent of the Laplacian in real space is applied, the background is determined by low-pass filtering. After background subtraction the interference function is received... 

The quantitative analysis of a multiphase topology comprises the formulation of structure models and the fitting of measured data. Fitting is discussed in Chap. 11. In this section the setup of topological models is discussed. The problem arises from the fact that most structural models of particle correlation are anisotropic and the visualization of structure in anisotropic materials by means of the CDF shows that suitable models must be rather complex. Thus a direct fit of anisotropic data would require fitting of a measured 3D or 2D function by a complex model. Both the effort to setup such models, and the computational effort to fit the data are very high. 

Figure 5 Numerical computation of the difference between two profiles (left PDF, right CDF), from four actual data points, observed at corresponding times.

When computed from a corresponding PDF, the PDF clearly represents independent observations any analysis of these is also valid for the corresponding CDF. 

Computational methods combined with a novel approach in the application of scattering physics were recently employed by Barbi et al. in a synchrotron SAXS study of the nanostructure of Nafion as a function of mechanical load. A new method of multidimensional chord-distribution function (CDF) analysis was used to visualize the multiphase nano-... 

Figure 10.11. S-N data for SUJ2 steel along with cdfs computed from a fatigue crack growth model [15].

Hence, by assuming nearly Gaussian distributed random variables we are able to compute the exercise probabilities Tip [.ST] directly by performing tbe lEE approach instead of miming a generalized Edgeworth series expansion. Overall, the lEE approach (4.2) can be seen as an equivalent to tbe generalized EE tecbnique, especially adapted to compute tbe cdf s used in finance theory. 

Complete information about the specimen would be available only by tomographic methods with a stepwise rotation of the sample (see e.g. Schroer, 2006) or using inherent symmetry properties of the sample. Under the assumption of fibre symmetry of the stretched specimen around the tensile axis, from the slices through the squared FT-structure the three-dimensional squared FT-structure in reciprocal space can be reconstructed and hence also the projection of the squared FT-structure in reciprocal space. The Fourier back-transformation of the latter delivers slices through the autocorrelation function of the initial structure. Stribeck pointed out that the chord distribution function (CDF) as Laplace transform of the autocorrelation function can be computed from the scattering intensity l(s) simply by multiplying I(s) by the factor L(s) = prior to the Fourier back-... 

The final periodic-review system we consider is actually the simplest, the base-stock system. In this system, we once again place an order at each review interval, but in this case, the review interval is set equal to the smallest discrete time unit covered by the system. Thus, for this inventory system, the period and the time unit are one-in-the-same. For example, if we manage the system using days as our time units—i.e., if lead time is specified in days—then in a base-stock system, we would review the inventory position X at the start of each day and place an order of size Q = S - x, to be received L days later. Thus, for this system, the review interval is T = 1 and the base-stock level is set to Sb = fr (CSL), where F is the cdf of the DLTR distribution, which in this case is the distribution of demand over L + 1 time units. If the demand rate per time unit, D, is normally distributed, then we can compute... 

Computer manufacturer XYZ in the US wants to evaluate the risks from its supplier ABC in Southern China. Inaccurate delivery time is one of the major problems affecting XYZ s supply chain. This represents an N-type risk for XYZ. After analyzing the historical data, XYZ found that ABC s delivery time fits a generalized hyperbolic distribution with parameters X = -0.5, a = 3 = 0, 8 = 1, and p = 0.1. This actually is a special case of generalized hyperbolic distribution called the Cauchy distribution. The PDF and CDF functions of a Cauchy distribution are as follows ... 

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