Nonuniform averaging

Color Difference Evaluation. Shade evaluation is comparable in importance to relative strength evaluation for dyes. This is of interest to both dye manufacturer and dye user for purposes of quaUty control. Objective evaluation of color differences is desirable because of the well-known variabihty of observers. A considerable number of color difference formulas that intend to transform the visually nonuniform International Commission on Illumination (CIE) tristimulus color space into a visually uniform space have been proposed over the years. Although many of them have proven to be of considerable practical value (Hunter Lab formula, Friele-MacAdam-Chickering (FMC) formula, Adams-Nickerson formula, etc), none has been found to be satisfactorily accurate for small color difference evaluation. Correlation coefficients for the correlation between average visually determined color difference values and those based on measurement and calculation with a formula are typically of a magnitude of approximately 0.7 or below. In the interest of uniformity of international usage, the CIE has proposed two color difference formulas (CIELAB and CIELUV) one of which (CIELAB) is particularly suitable for appHcation on textiles (see Color). 

Heterogeneity, nonuniformity and anisotropy are terms which are defined in the volume-average sense. They may be defined at the level of Darcy s law in terms of permeability. Permeability, however, is more sensitive to conductance, mixing and capillary pressure than to porosity. 

Heterogeneity, nonuniformity and anisotropy are defined as follows. On a macroscopic basis, they imply averaging over elemental volumes of radius e about a point in the media, where e is sufficiently large that Darcy s law can be applied for appropriate Reynolds numbers. In other words, volumes are large relative to that of a single pore. Further, e is the minimum radius that satisfies such a condition. If e is too large, certain nonidealities may be obscured by burying their effects far within the elemental volume. 

To determine the average porosity of a homogeneous but nonuniform medium, the correct mean of the distribution of porosity must be evaluated. The porosities of natural and artificial media usually are normally distributed. The average porosity of a heterogeneous nonuniform medium is the volume-weighted average of the number average ... 

The average nonuniform permeability is spatially dependent. For a homogeneous but nonuniform medium, the average permeability is the correct mean (first moment) of the permeability distribution function. Permeability for a nonuniform medium is usually skewed. Most data for nonuniform permeability show permeability to be distributed log-normally. The correct average for a homogeneous, nonuniform permeability, assuming it is distributed log-normally, is the geometric mean, defined as ... 

Recognizing that the information transmission is often both nonuniform (since the left and right fronts do not always propagate outward from the perturbation with a single well-defined velocity), and dependent on choice of initial states, one can instead define maximal and minimal average propagation speeds, and A " " ... 

If the catalyst active centers are nonuniform, a time variation of the average value of Kp may be caused by the change of the proportion between the centers with various reactivity during polymerization. However, in the case of chromium oxide catalysts the experimental data show that the... 

The density of the atmosphere varies greatly from place to place, as does its composition and temperature. The average composition of dry air (air from which water vapor has been removed) is shown in Table 4.4. One reason for the nonuniformity of air is the effect of solar radiation, which causes different chemical reactions at different altitudes. The density of air also varies with altitude. For example, the air outside an airplane cruising at 10 km is only 25% as dense as air at sea level. 

In a test section operated in a quality region with the same H1B, L, D, G, and p, the critical powers will be approximately the same for both the uniform and nonuniform flux distributions, provided the peak-to-average flux ratio is not greater than 1.6 (Janssen and Kervinen, 1963). 

In practice, this simple formula will hardly ever work, especially if the free energy changes appreciably with . Consider, for example, two states of the systems, , and j such that A,1 f <) - AA( j) = 5/, BT. Then, on average, the former state is sampled only seven times for every 1,000 configurations sampled from the latter state. Such nonuniform sampling is undesirable, as it leads to a considerable loss of statistical accuracy. For the free energy profile shown in Fig. 3.1, transitions between... 

Figure 16 shows two cases for lateral solids distribution, a fairly uniform distribution illustrated on the left-hand side, and a relatively concentrated distribution near the center source, shown on the right-hand side. It is clear that for both cases, areaa, represents the excess of solids over the average for w near the central feed, at the expense of deficit for areaa2 near the peripheral region. Graphically the nonuniformity index is thus... 

In the application of XAS to the study of fuel cell catalysts, the limitations of the technique must also be acknowledged the greatest of which is that XAS provides a bulk average characterization of the sample, on a per-atom basis, and catalyst materials used in low temperature fuel cells are intrinsically nonuniform in nature, characterized by a distribution of particle sizes, compositions, and morphologies. In addition, the electrochemical reactions of interest in fuel cells take place at the surface of catalyst par-... 

The evaporation bed samples were generally a complex mixture of many compounds and good records of Inputs to the beds were not always kept. Also some of the beds were nonuniform In their distribution of the pesticides as well as soil depth. All beds had been packed with sandy loam and when sandy loam was fortified with 11 of the most frequently found pesticides, the average... 

In addition to a tight distribution of the thickness variation within a wafer, the average of a group of individual thicknesses must also be targeted within a certain range. Statistically, the control of the WIWNU is the control of standard deviation of individual thicknesses, and the control of final thickness post CMP is the control of the mean. The variation of the mean from wafer to wafer is called wafer-to-wafer nonuniformity (WTWNU). All the thicknesses mentioned in this section are actually the means of many individual thickness measurements in each wafer. Control is not easy, for reasons discussed in the following. 

---