Comparison of Distribution Functions

The Normal (or Gaussian) and Weibull distribution functions are most commonly used in engineering design. The normal distribution is usually expressed in one of the following two functional forms  

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Figure 35. Comparison of distribution functions extracted from SAXS data versus from TEM observations. There are some systematic errors in the TEM analysis. However some larger scale particles appear to be lost in the SAXS analysis. As TEM particle counting favors larger particles this could also be a systematic effect. After Rieker et al. (1999).

In the paper, we describe the neural network to optimize the manufacturing data output. For the detail study, we watch the amormt of time necessary to obtain the result. In the conclusion, the neural network method result is compared with the presently used one. The comparison of both methods and their quality assessment is based on the comparison of distribution functions that describe the time to display the data output. The best solution is going to be applied in the Production module of the K2 ERP system. 

T = (/i + l)ln[iici(t)/i ci(t2)], aiid normalized to the unity distribution function of clusters over dimensions G (u) (/ " G (u)du = 1). Figure 18. illustrates the comparison of distribution function Gi(u) (with d = 3) obtained from approximate solution of (13.1.3), with the experimental one (Hermann uid Rhodin 1966) (for the systems Au-Si02, Au-NaCl, and Ni-KCl). The comparison of experimental data on Sn-C (Blackman and Guzzon 1959) and Au-NaCl (Hermann and Rhodin 1966) with theoretical curve (d = 3, /i = 2) is presented in Fig. 19 and shows a good agreement. 

Figure 9. Comparison of damping function for (a) simulated copolymer with a hole in the distribution and (b) a natural copolymer...

This can be clearly seen in the comparison of viscosity functions of polystyrene 1 and polystyrene 2. If we add a very high-molecular weight component to polystyrene 2, we obtain polystyrene 3. It is therefore evident that the bimodal molar mass distribution causes the shear thinning to increase further, although not as far as the polydispersity change of the molar mass distribution between polystyrene 1 and polystyrene 2. In the case of higher shear rates, all flow curves proceed to similar viscosity functions. 

Kowalczyk, P. et al., Estimation of the pore size distribution function from the nitrogen adsorption isotherm. Comparison of density functional theory and the method of Do and co-workers. Carbon. 2003,41(6), 1113-1125. 

We shall restrict the comparison to the flux and birth-rate density formulations of integral transport theory and to two sets of distribution functions one consists of the flux and source importance function, and the other set consists of the solutions of the transport equations in the formulation under consideration. 

Fig. 2.16 Comparison of distribution ratios for the extraction of U(VI) by [bmim][PF6], [bmim] [NTf2] and 1.1 M TBP/diluent as a function of the initial nitric acid concentration. The diluents are n-dodecane (DD), [bmim][NTf2] and [bmimilPFg] (Vasudeva Rao et al. 2008)...

The comparison of Figs. 3.23 and 3.24 shows that the films thickness decrease reveals the features similar to those at the increase of distribution function width. This means that the decrease of film thickness may be considered as equivalent to the disordering of the system. The reason for that is that the fluctuations due to film thickness decrease have the same nature as those at the system disordering. In other words, in thinner films of relaxor ferroelectrics, the part of long range order decreases so that dipole glass state may appear in free-standing films. The complement state in such situation may be the electret-like one with remnant polarization induced by built-in field. Latter state is more profitable in the thinnest possible films with thickness less than some critical value. 

Thus D(r) is given by the slope of the V versus P plot. The same distribution function can be calculated from an analysis of vapor adsorption data showing hysteresis due to capillary condensation (see Section XVII-16). Joyner and co-woikers [38] found that the two methods gave very similar results in the case of charcoal, as illustrated in Fig. XVI-2. See Refs. 36 and 39 for more recent such comparisons. There can be some question as to what the local contact angle is [31,40] an error here would shift the distribution curve. 

Figure 8-6. Comparison of the radial distribution function of the ctiair, boat, and twist conformations of cyclohexane (hydrogen atoms are not considered).

The properties of fillers which induence a given end use are many. The overall value of a filler is a complex function of intrinsic material characteristics, eg, tme density, melting point, crystal habit, and chemical composition and of process-dependent factors, eg, particle-si2e distribution, surface chemistry, purity, and bulk density. Fillers impart performance or economic value to the compositions of which they are part. These values, often called functional properties, vary according to the nature of the appHcation. A quantification of the functional properties per unit cost in many cases provides a vaUd criterion for filler comparison and selection. The following are summaries of key filler properties and values. 

Figure 4.1-13 Comparison of the experimental without (—) and with (—) triphenylphosphine at (solid line) and fitted (dashed line) (a) EXAFS 80 °C and in the presence of triphenylphosphine and (b) pseudo-radial distribution functions and reagents at 50 °C for 20 min (—). Repro-...

Comparison of Eq. (184) with Eq. (183) shows the effect of size distribution for the case of fast chemical reaction with simultaneous diffusion. This serves to emphasize the error that may arise when one applies uniform-drop-size assumptions to drop populations. Quantitatively the error is small, because 1 — is small in comparison with the second term in the brackets [i.e., kL (kD)112). Consequently, Eq. (184) and Eq. (183) actually give about the same result. In general, the total average mass-transfer rate in the disperser has been evaluated in this model as a function of the following parameters ... 

Preliminary measurements with space-resolved PMC techniques have shown that PMC images can be obtained from nanostructured dye sensitization cells. They showed a chaotic distribution of PMC intensities that indicate that local inhomogeneities in the preparation of the nanostructured layer affect photoinduced electron injection. A comparison of photocurrent maps taken at different electrode potentials with corresponding PMC maps promises new insight into the function of this unconventional solar cell type. 

The vertebrates show many morpho-functional variants on a basic theme (Chap. 2). Some of these, such as the pattern of distribution of the genetically distinct chemosensory neurones within die VN epithelium, will be related to the level of complexity of the animal. In some groups, the VNO can be equally complex, whilst the accessory areas of the brain will differ in complexity, as in the advanced reptiles and mammals. Eventually, detailed comparisons of the genomic repertoire of the various accessory systems should reveal the extent of the operational distinctions amongst them. Of particular interest would be the events which account for the suppression of AOS morphogenesis, and those which compensate for its absence. 

Figure 2. Comparison of the simulated velocity distribution (histogram) with the Maxwell— Boltzmann distribution function (solid line) for kgT —. The system had volume V — 1003 cells of unit length and N = 107 particles with mass m = 1. Rotations (b were selected from the set Q — tt/2, — ti/2 about axes whose directions were chosen uniformly on the surface of a sphere.

Figure 3. Comparison of the measured momentum distributions of the outermost valence orbital for wafer [6-8] with spherically averaged orbital densities from Hartree-Fock limit and correlated wave functions [6].

A comparison of the measured mean relative size with the model in Eq. (101) is shown in Fig. 28. The material was monosize glass beads. The data fit the model quite well, with the exception of fine 0.038-mm powder. It is evident that the steady-state size distribution is a function of the liquid content, and consequently, as shown by Sherrington (S9), there is an optimal granulating liquid for maximum granulation efficiency, that is, percentage of the product-grade material. 

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