Homogeneity index

A homogeneity index or significance coefficienf has been proposed to describe area or spatial homogeneity characteristics of solids based on data evaluation using chemometrical tools, such as analysis of variance, regression models, statistics of stochastic processes (time series analysis) and multivariate data analysis (Singer and... 

Homogeneous breakage kernels with positive homogeneity index (A>0) produce self-similar size distributions at long times... 

For the symmetric system (< )0 = 0.5) the scaling exponent for the Euler characteristic has been found in accordance with the dynamic scaling hypothesis x L(t) 3 (see Section I.G). The homogeneity index, HI, of the interface defined as [222]... 

The definition of a significance value, SV, analogous to the homogeneity index [SINGER and DANZER, 1984] enables semiquantitative assessment of the significance of inhomogeneity ... 

Equation (12-44) corrects a factor-of-two omission in Hyde s formula.] Engblom and Hyde (1995) then define the homogeneity index HI as... 

Table 4.1 Geometric properties of some periodic minimal surfaces. The "genus" of each three-periodic minimal surface (IPMS) is the genus of a unit cell of the IPMS (with symmetrically distinct sides). The "symmetry" refers to the crystallographic space group for the surface (assuming equivalent sidesl.liie surfaces are tabulated in order of deviation of the homogeneity index from the "ideal" value of 3/4. ...

So far, we have assumed the ideal value of (derived above) for the homogeneity index, H. In reality, this index varies according to the S)mimetry and topology of the IPMS (Table 4.1). If the true values of H are inserted into eq. 4.4, more accurate estimates of the local/global relations can be made for various IPMS. These estimates are plotted in Fig. 4.9 below. 

The Advanced SCLAIRTECH process uses two CSTR with CHX as a solvent and metallocene catalyst at T = 160-220 °C and P = 3-34 MPa. Process advantage is the heat recovery. The patent for the first time uses the homogeneity index, HI calculated from Tk,... 

The values of h for the P, D and G structures are close to the ideal homogeneous (and frustration-free) value of exactly 3/4 (see Table 16.1 below). The lattice parameter (o ) scales with the crystallographic genus (y) and the homogeneity index h) as follows ... 

Figure 16.21). In this case, the spatial frustration is For Type 2 bicontinuous mesophases, the chain length reflected in the homogeneity index h) for the particular is equal to the inner thickness, given by mesophase. The equations have been derived in detail elsewhere (11). For Type 1 bicontinuous mesophases, the chain length is equal to the inner thickness, as follows (16.24)...

Table 16.2. Shape parameters (equations (16.1) and (16.4)) and approximate swelling exponents s and s, cf. equation (16.8)) for known lyotropic mesophases. The (variable) constants /i and depend on the specific symmetry of the phase ( /i is the homogeneity index, ideal equal to 3/4 (cf. Table 16.1) / is the interstitial packing fraction for dense sphere and circle packings, equal to unity for ideal homogeneous packings)...

Figure 2. Lower bounds to concentration with respect to the homogeneity index in high and low...

With this information we constructed the homogeneity index, h, as the percentage of installed capacity accounted for by the most popular product within each market. Before constructing the variables measuring concentration, we had to aggregate plant level information into firm level data. With this information in hand, we were able to compute measures of concentration, and in particular the four-firm (C4) concentration ratio. ... 

The case of dyes and adhesives, both of which are high R D markets, deserve a final comment. In the latter cases, concentration is higher in Europe but the homogeneity index is lower. This is consistent with the existence of distinct technological trajectories that display economies of scope among themselves. Thus, EU firms in these markets seem to have successfully exploited these opportunities. 

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