Harmonic mean values

By definition at an expansion ratio variable by height, the total foam resistance in Eq. (4.32), appears to be its harmonic mean value. That is why the experimental expansion ratio (Eq. 4.32) at its equilibrium distribution along the height of the cells is compared to the calculated harmonic mean ng value A2Al... 

In both foams, from Triton-X-100 and NaDoS solutions, the bubble sizes during 5 min of centrifugation did not exceed avL = 2 to 2.5-1 O 2 cm. The dispersity of a NaDoS foam at the 15 to 20th minute of centrifugation was avL = 310"2 cm. For small angular velocity of rotation co = 52.3 s 1 and A1= 1.5 cm, the highest expansion ratio exceeded the lowest by a factor of 5 times and by a factor of 1.5 its harmonic mean value (Table 6.2). The destruction of a foam layer with A/ = 1.5 cm from NaDoS begins at capillary pressure 8.95 kPa which corresponds as well to a harmonic mean value of 10.9 kPa. 

While many data are suggestive of chain length dependence, the data are not usually suitable for or have not been tested with respect to model discrimination. Values of ,u have been determined for a variety of small monomeric radicals to be ca I09 M s 1.4 Taking kt0 as Jk,lj and a as 1.0 in the geometric expression yields values of ,iJ as shown in Figure 5.4a.49 Use of the Smoluchowski mean or the harmonic mean approximation prediets a shallower dependence of k 1 on the chain length (Figure 5.4b). All expressions yield the same dependence for j=i. 

The value of y at the center of the pattern is the arithmetic mean of the values of y at the remaining four nodes of the pattern. This formula gives a difference analog of the formula for the mean value of a harmonic function. 

The ° mn coefficients are the mean values of the generalized spherical harmonics calculated over the distribution of orientation and are called order parameters. These are the quantities that are measurable experimentally and their determination allows the evaluation of the degree of molecular orientation. Since the different characterization techniques are sensitive to specific energy transitions and/or involve different physical processes, each technique allows the determination of certain D mn parameters as described in the following sections. These techniques often provide information about the orientation of a certain physical quantity (a vector or a tensor) linked to the molecules and not directly to that of the structural unit itself. To convert the distribution of orientation of the measured physical quantity into that of the structural unit, the Legendre addition theorem should be used [1,2]. An example of its application is given for IR spectroscopy in Section 4. 

Fig. 3.8 Plot of 8 0 values vs Mg numbers for oceanic basalts filled circles) and continental basalts open circles). The shaded field denotes the 20 range of a MORE mean value of +S.l%o, the clear vertical field denotes the range for primary basaltic partial melts in equilibrium with a peridotitic source (Harmon and Hoefs, 1995)...

Besides the above-mentioned measures for the location there are several other possible parameters. The geometric mean is the arithmetic mean of the logarithms of the data (or the n root of the product of all data). The harmonic mean is the reciprocal of the mean of the reciprocals of all single values. 

Harmonic Mean - the reciprocal of the average of the reciprocals of the observed values. 

Take a mean value of 80 (i.e., 0.83 eV). Numerical calculations show that T] < 0.2 V is the condition up to which 9.38 yields the experimental version of Tafel s law (of course, the value depends on the Fs chosen and the allowed T, for the applicability of 9.38 will be roughly halved at the lower limit and doubled at the higher one. In any case, this harmonic approximation, which is involved in the Weiss—Marcus theory, cannot be applied to the experimental current-potential data, which in reality extend over 0.2 V and even 1.0 V (for hydrogen and oxygen evolution). 

Microwave Absorption of radiation due to dipole change during rotation (A = 0.1—30 cm 300-1 GHz in frequency) Mean value off—2 terms potential function IO- >s 10-2 pa (10- torr) Mean value of/-2 does not occur at rB even For harmonic motion. Dipole moment necesanry. Only one component may be detected. Analysis difficult for large molecules of low symmetry... 

Results of this type have proved of value in experimental investigations involving surface-volume relations. Of particular interest is the fact that specific surface is inversely proportional to the first moment of the surface-weighted size distribution, and this moment, in turn, is equal to the harmonic mean of the volume-weighted size distribution. 

Table I lists some of the basic mathematical expressions of importance in droplet statistics. The expressions are given in terms of an arbitrary ptb-weighted size distribution. The specific forms are obtained for various integral values of p. For example, the substitution of p = 2 into the equations of Table I yields the cumulative distribution, arithmetic mean, variance, geometric mean, and harmonic mean of the surface-weighted size distribution. Analogous expressions valid for frequencies or mass distributions are obtained by setting p equal to 0 or 3, respectively.

Solving the matrix set (4.293) at various frequencies, one finds that the harmonic suppression effect is most pronounced under the adiabatic condition cox —> 0. Henceforth, we focus on it. In this limit, suppression of a kth harmonic means its complete vanishing the value of p) turns into zero. In... 

Assume initially that the excess pressure 8 is determinable by some function 8 = f(x) for t = 0 at any depth to the impervious stratum. Let the depth of this stratum be L/2. If the excess of pressure at the surface is simply harmonic, 8 = 0O sin cot for x = 0 and any value of t, we have the upper surface boundary condition. The value of 60 is of course the maximum oscillation from the mean value 760 mm Hg. At L/2 no flow can take place. This boundary can be accomplished by the following mathematical artifice. Instead of assuming the impervious layer to exist at L/2, assume that it exists at a depth L. At the depth L, let there be a forced oscillation of the same magnitude as that at the surface, so that the region then encompassed will be symmetrical about L/2, and no flow will take place across a plane parallel to the surface at this point. The boundary conditions to be satisfied by Eq (14-20) will then be ... 

The value of Sc is difficult to determine unless we assume that the surface of a given volume of particles is proportional to the harmonic mean. If this assumption is valid, then such contacts as we have described are also proportional to this mean (see Chapter 3). Let us now write... 

From a mathematical perspective (see Equation 4.1), CA simply represents the weighted harmonic mean of the individual ECx values, with the weights just being the fractions / , of the components in the mixture. This has important consequences for the statistical uncertainty of the CA-predicted joint toxicity. As the statistical uncertainty of the CA-predicted ECx is a result of averaging the uncertainties of the single substance ECx values, the stochastic uncertainty of the CA prediction is always smaller than the highest uncertainty found in all individual ECx values. Perhaps contrary to intuition, the consideration of mixtures actually reduces the overall stochastic uncertainty, which is a result of the increased number of input data. 

Mean value of r-2 does not occur at even for harmonic motion. 

A basic means of modelling approximate reaction paths is the adiabatic mapping or coordinate driving approach [123,149]. The energy of the system is calculated by minimizing the energy at a series of fixed (or restrained, e.g. by harmonic forces) values of a reaction coordinate, which may be the distance between two atoms, for example. More extensive and complex combinations of geometrical variables can be chosen. This approach is only valid if one... 

Using one of the suitable discretization schemes discussed above, it is possible to relate values of variables and their gradient at CV faces to the node values. It is also necessary to use suitable interpolation schemes to estimate other relevant quantities like effective diffusion coefficients, (F) at required locations. Either algebraic mean or harmonic mean can be used to estimate the value of effective diffusion coefficients at cell faces. For example, the effective diffusion coefficient at face e can be written (for a uniform grid) ... 

The harmonic mean to estimate effective diffusion coefficients (second expression on RHS of Eq. (6.30)) can handle abrupt changes in values of F without requiring an excessively fine grid in the vicinity of the change (see Patankar, 1980 for more details). 

This temperature oscillation is harmonic with a periodic time of t0 and an amplitude of Ail around the mean value i)rn. At the surface x = 0 the heat transfer condition is... 

In which Atfe 1/2 is the thermal conductivity at temperature f 1//2- This requires a suitable mean value to be chosen, the arithmetic, geometric or harmonic mean of the thermal conductivities at the known temperatures and tA+1 or nd The type of mean value formation does not play a decisive role if A is only weakly dependent on d or if the step size Ax is chosen to be very small. D. Marsal [2.53] recommends the use of the harmonic mean, so... 

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