Harmonic average

Modehng growth where deformation is significant is more difficult. It can be assumed that a critical cutoff size exists tr> which determines which combination of granule sizes are capable of coalescence, based on their inertia. When the harmonic average of sizes of two colliding... 

The harmonic average is effective to overcome troubles due to jumps ofq from grid point (i,/) to (i +1,/). 

It can be theoretically proven that the overall sequence approaches the Poisson law when zmax increases (Feller, 1971). Moreover X() is the inverse of the harmonic average value b given by... 

The linearity property of the Fourier transform does not hold for phase delay values. Let a, be the intrinsic, single measurement, time domain standard deviation of the filtered signal (in units of amplitude). Also let o>(n) be the standard deviation of the phase of the nth harmonic averaged j times. The phase noise as a function of harmonic (frequency) is given by... 

For weakly entangled polymers or long N-chains surrounded by shorter P-chains, a tube renewal time x shorter than Xc can be expected. If the two mechanisms are assumed to be independent of each other [17], the overall relaxation time x can be put as the harmonic average of the two times ... 

The permeability constant of the composite membrane is therefore represented by the harmonic average of the permeability constants of the individual layers, the respective weights being x /Ji., the ratio of layer thickness to the total. Although composite membranes Include layers of dense films or even liquid layers in series with films, in this discussion the term is being limited to those series in which at least one of the members is a phase inversion membrane of either the integrally-skinned or skinless variety. 

We reiterate that the availability of the closed form expressions for the relaxation behavior depends on the assumption that the ratio of the Laplace transform of the first and second order spherical harmonic averages has reached its final value appreciably faster than the ratio of the first and zero order. This means that for step-on and step-off solutions, we are considering processes that take place in times of order of magnitude... 

In the three bond jump model for segmental motion there are two parameters. The time scale is set by the harmonic average correlation time, Tjj and the effective distribution of correlation times is set by the number of coupled bonds m. The sharp cut off of coupling solution of the three bond jump model is employed here. The composite spectral density for internal rotation by jumps or stochastic diffusion plus segmental motion by three bond jump is... 

Where K, is the modulus of the solid grain, is porosity, and c is a consolidation parameter that represents the sediments and fluid inside the vugs (unconsolidated material). We use c = 200. The effective compressional modulus (harmonic average of the Ki and /fj drained moduli) and the shear bulk modulus, G, of the double porosity medium can be obtained, respectively, from... 

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