Average skewness

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 ... 

Ad average of tea samples examined hj Miller and F.skew bad the lollowing characters —... 

All the references to burn-out have thus far been concerned with uniformly heated channels, apart from some of the rod bundles where the heat flux varies from one rod to another, but which respond to analysis in terms of the average heat flux. In a nuclear-reactor situation, however, the heat flux varies along the length of a channel, and to find what effect this may have, some burn-out experiments on round tubes and annuli have been done using, for example, symmetrical or skewed-cosine axial heat-flux profiles. Tests with axial non-uniform heating in a rod bundle have not yet been reported. 

The area, volume and average depth of the ocean basins and some marginal seas are given in Table 10-1. The Pacific Ocean is the largest and contains more than one-half of the Earth s water. It also receives the least river water per area of the major oceans (Table 10-2). Paradoxically it is also the least salty (Table 10-3). The land area of the entire Earth is strongly skewed toward the northern hemisphere. 

Figure 22 shows the area ratio and average film thickness as a function of skewness. Similar trends to those shown in Fig. 21 are observed. When a smooth surface normally approaches a rough surface, a surface with positive skewness will be much more engaged in contact than the one with negative skewness. Since the average film thickness remains almost the constant, as demonstrated in Fig. 22(b), the real contact area will increase with skewness. 

In summary, the height distribution of surface roughness, characterized by the skewness and kurtosis, may present a significant influence on certain performances of mixed lubrication, such as the real contact area, the load carried by asperities, and pressure distribution, while the average film thickness and surface temperature are relatively unaffected. 

Fig. 21 —The changes of lubrication properties with the kurtosis, simulated at cr=0.1 /xm and different values of skewness, (a) Contact area ratio versus skewness, (b) Load ratio versus skewness, (c) Maximum pressure versus kurtosis. (d) Maximum temperature versus kurtosis. (e) Average film thickness versus kurtosis.

Fig. 22—The change of contact area ratio and average film thickness with skewness, simulated at o-=0.1 /u.m and for different kurtosis. (a) Area ratio versus skewness and (b) average film thickness versus skewness.

This technique assumes a Gaussian spreading function and thus does not take into account skewness or kurtosis resulting from instrumental considerations. It can, however, be modified to accommodate these corrections. The particle size averages reported here have been derived usino the technique as proposed by Husain, Vlachopoulos, and Hamielec 23). 

Table VII contains the weight-average particle diameters as calculated by this technique for all the standards employed using Column Set I. There is some difference between the values of variance used to obtain these averages and those cited in Table V. This is in fact due to the necessity to correct for skewness by...

The particle size analysis techniques outlined earlier show promise in the measurement of polydispersed particle suspensions. The asumption of Gaussian instrumental spreading function is valid except when the chromatograms of standard latices are appreciably skewed. Calc ll.ation of diameter averages indicate a fair degree of insensitivity to the value of the extinction coefficient. 

The first four terms called, respectively, the average (or expectation value), variance, skewness, and kurtosis, are equal to... 

The average defining the potential of mean force, (8.49), can be written as an average over a skewed distribution of initial momenta as described by (8.55). We can anticipate that skewed trajectories are associated with lower work, as the momenta can be biased so that important degrees of freedom tend to move in the same direction as the pulling potential. Specifically, the instantaneous contribution to the work of... 

Fig. 8.3. Histogram of work values for Jarzynski s identity applied to the double-well potential, V(x) = x2(x — a)2 + x, with harmonic guide Vpun(x, t) = k(x — vt)2/2, pulled with velocity v. Using skewed momenta, we can alter the work distribution to include more low-work trajectories. Langevin dynamics on Vtot(x(t),t) = V(x(t)) + Upuii(x(t)yt) with JcbT = 1, k = 100, was run with step size At = 0.001, and friction constant 7 = 0.2 (in arbitrary units). We choose v = 4 and a = 4, so that the barrier height is many times feT and the pulling speed far from reversible. Trajectories were run for a duration t = 1000. Work histograms for 10,000 trajectories, for both equilibrium (Maxwell) initial momenta, with zero average and unit variance, and a skewed distribution with zero average and a variance of 16.0...

A detailed numerical implementation of this method is discussed in [106]. W is the statistical weight of a trajectory, and the averages are taken over the ensemble of trajectories. In the unbiased case, W = exp -(3Wt), while in the biased case an additional factor must be included to account for the skewed momentum distribution W = exp(-/ Wt)w(p). Such simulations can be shown to increase accuracy in the reconstruction using the skewed momenta method because of the increase in the likelihood of generating low work values. For such reconstructions and other applications, e.g., to estimate free energy barriers and rate constants, we refer the reader to [117]. 

The average shape of the radical will therefore be a compromise between strain due to hydrogen-hydrogen repulsion in the completely planar state and torsional strain of the partial double bonds in the skew state. Such a compromise should result in a propeller form in which the blades are slightly feathered out of the plane. It is possible that the radical exists in two isomeric forms, one corresponding to a symmetrical propeller and the other to a propeller in which one of the blades has been tilted the wrong way. Although such isomerism has been ob-... 

As shown in Fig. 3.24, the outer contours of the hN NHO and onh NBO exhibit the expected cylindrical symmetry about the nominal hybrid direction vector marked by the dashed line. However, the inner contours are increasingly directed toward the dotted line of nuclear centers. The lack of overall cylindrical symmetry is best seen in the NBO plot of Fig. 3.24(b), where it can be seen that the H nucleus sits accurately in the pointy inner contours which are closely aligned with the dotted line of nuclear centers rather than the dashed line of nominal hybrid direction. This example illustrates that the optimized NHOs are free to adopt complex non-cylindrical shapes in which the apparent directionality varies with distance from the nucleus. Thus, the 3.9° bending does not correspond to true misalignment of the hybrid and nuclear directions at the actual bond distance, but rather reflects the fact that inner hybrid contours are slightly skewed with respect to the nominal average direction attributed to the NHO. That is, the hybrid itself (rather than the bond) is slightly bent. 

Computer software is used to improve spectral quality. The most widespread procedures deal with averaging and background subtraction. The averaging process is rather obvious. The intensities of ions peaks at each m/z, recorded along the analyte chromatographic peak profile, are summed in several spectra and divided by the number of spectra used. Averaging minimizes, for example, spectral skewing problems. 

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