Geometric means

WASH-1400 introduced this method for unknown common mode effects. The procedure was not presented in the best light and was severely criticized by the Lewis Commiiiee or lack of a physical basis, although the approach is not unreasonable. Basically, the procedure considered the failure rate of a system, with a common mode, to be i ctween two bounds. The lower bound is the one with no 

The overall system failure rate including common modes was estimated to be the geometric mean of these extremes (equation 

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The parameter should be unity if molecular diameters also obey a geometric mean law [193] and is often omitted. Equation X-44, if applied to the Young equation with omission of leads to the relationship [192]... 

Equation XVI-21 provides for the general case of a molecule having n independent ways of rotation and a moment of inertia 7 that, for an asymmetric molecule, is the (geometric) mean of the principal moments. The quantity a is the symmetry number, or the number of indistinguishable positions into which the molecule can be turned by rotations. The rotational energy and entropy are [66,67]... 

Van der Waals and especially van Laar simplified these expressions by assuming a geometric mean for and an aritlmietic mean for... 

With these simplifications, and with various values of the as and bs, van Laar (1906-1910) calculated a wide variety of phase diagrams, detennining critical lines, some of which passed continuously from liquid-liquid critical points to liquid-gas critical points. Unfortunately, he could only solve the difficult coupled equations by hand and he restricted his calculations to the geometric mean assumption for a to equation (A2.5.10)). For a variety of reasons, partly due to the eclipse of the van der Waals equation, this extensive work was largely ignored for decades. 

Flalf a century later Van Konynenburg and Scott (1970, 1980) [3] used the van der Waals equation to derive detailed phase diagrams for two-component systems with various parameters. Unlike van Laar they did not restrict their treatment to the geometric mean for a g, and for the special case of b = hgg = h g (equalsized molecules), they defined two reduced variables. 

For examples of different types of similarity measures, see Table 6-2. The Tanimoto similarity measure is monotonic with that of Dice (alias Sorensen, Czekanowski), which uses an arithmetic-mean normaJizer, and gives double weight to the present matches. Russell/Rao (Table 6-2) add the matching absences to the nor-malizer in Tanimoto the cosine similarity measure [19] (alias Ochiai) uses a geometric mean normalizer. 

The resulting similarity measures are overlap-like Sa b = J Pxi ) / B(r) Coulomblike, etc. The Carbo similarity coefficient is obtained after geometric-mean normalization Sa,b/ /Sa,a Sb,b (cosine), while the Hodgkin-Richards similarity coefficient uses arithmetic-mean normalization Sa,b/0-5 (Saa+ b b) (Dice). The Cioslowski [18] similarity measure NOEL - Number of Overlapping Electrons (Eq. (10)) - uses reduced first-order density matrices (one-matrices) rather than density functions to characterize A and B. No normalization is necessary, since NOEL has a direct interpretation, at the Hartree-Fodt level of theory. 

As Eq. (29) shows, corresponds to the arithmetic mean and to the geometric mean of the homodimeric parameters AA and BB. 

Raw data are collected observations that have not been organized numerically. An average is a value that is typical or representative of a set of data. Several averages can be defined, the most common being the arithmetic mean (or briefly, the mean), the median, the mode, and the geometric mean. 

The geometric mean of a set of N numbers is the A th root of the product of the numbers The root mean square (RMS) or quadratic mean of a set of numbers is defined by ... 

The application of these ideas to the mixing of low molecular weight liquids has been the object of extensive research. As a result of these investigations, the appropriate kind of hybridization of individual molecular properties in W12 is found to be the geometrical mean ... 

Dispersive Interactions. For pairs of nonpolar polymers, the intermolecular forces are primarily of the dispersive type, and in such cases the energy of interaction between unlike segments is expected to be closely approximated by the geometric mean of the energies of interaction between the two like pairs (98). In this case, the Flory-Huggins interaction energy between this polymer pair can be expressed in terms of the solubiUty parameters 5 of the pure components. 

Distribution Averages. The most commonly used quantities for describing the average diameter of a particle population are the mean, mode, median, and geometric mean. The mean diameter, d, is statistically calculated and in one form or another represents the size of a particle population. It is usefiil for comparing various populations of particles. 

Another frequentiy used average is the geometric mean, which is particularly usehil for log-normal or wider (spanning over a decade) distributions. The geometric mean diameter, d is calculated usiag the logarithm values of the measured diameters ... 

Carleman s Inequality The arithmetic and geometric means just defined satisfy the inequality... 

The optimum intermediate pressure for the two-stage refrigeration cycles is determined as the geometric mean between evaporation pressure (pi) and condensing pressure (p/, Fig. 11-79) ... 

When the equihbrium line is not straight, Treybal Liquid Extraction, 2d ed., McGraw-Hill, New York, 1963) recommends that the geometric mean value of m be used. The geometric mean of the slope of the equilibrium hue at the concentration leaving the feed state and at the raffinate concentration leaving the raffinate stage is... 

For a symmetrical separation of component h from c, Brian Staged Cascades in Chemical Processing, Prentice-Hall, Englewood Cliffs, N.J., 1972) reported that the ratio of wash solvent to extraction solvent W /S should be set equal to the geometric mean of the two slopes of the equilibrium lines [Eq. (15-35)]. 

NOTE National standards, other than those based on annual arithmetic means or annual geometric means, are not to he exceeded more than once a year. 

The reactance thus obtained can be doubled for singlephase systems. For a three-phase system the configuration of the three phases with respect to each other will play a significant role and the linear centre spacing S has to be modified to an effective or geometric mean spacing S, where... 

For determining in solid or hollow round sections it is essential to first determine the self geometric mean distance, of the conductors which varies with the thickness / (annulus) of the conductor, approaches its outer radius, ri. in an infinitely thin conductor and to O.TTSri in a solid bar. This variation, in the form of D lr is drtiwn in Figure 28.21, as a function of r,// . 

If air quality data at a receptor for any one averaging time are lognormally distributed, these data will plot as a straight line on log probability graph paper (Fig. 4-9) which bears a note Sg = 2.35. Sg is the standard geometric deviation about the geometric mean (the geometric mean is the Nth root of the product of the n values of the individual measurements). 

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