Stieltjes Transformation

The more desirable approach is to determine f(Q) from an assumed 0(P,T,Q) and the experimental adsorption isotherm. Sips (16) showed that Equation 1 could be treated by a Stieltjes transform, so that in principle an explicit function could be written for f(Q), provided the experimental isotherm function, 0, could be expressed in analytical form. Subsequently, Honig and coworkers (10, 11, 12) investigated this approach further. The difficulty is that only for certain types of assumed functions 0 and 0 is the approach practical. As a consequence the procedure has been first to restrict the choice of 0 to the Langmuir equation, and second to assume certain simple functions for 0 such as the Freundlich and Temkin isotherm equations. The system is thus forced into an arbitrary mold and again it is not certain how much reliance should be placed on the site energy distributions obtained. 

This integral, to be taken in the Stieltjes-Lebesgue sense, is known as the Stieltjes transform of the spectral density of states n( ). In general the Stieltjes transform defines two analytic fimctions, one in the upper complex plane and the other in the lower complex plane, and we refer to the former consistently with the Herglotz properties of Gqq(E). 

The preconception about the results was the most unsatisfactory aspect of this procedure, which was refined in the basic article of Gaspard and Cyrot-Lackmann the use of Stieltjes transform and continued fraction analysis poses a sound mathematical framework to the use of moments. 

A Stieltjes transform is a double Laplace tran3form. It can be calculated as follows ... 

Misra (1970) used the Langmuir equation as the local isotherm (which is also used by many because of the simplicity of the such equation), and for some specific overall isotherms they obtained the energy distribution by using the method of Stieltjes transform to solve the inverse problem. 

In this paper the noise level produced by a macroscopically steady bubbly layer on a hydrofoil is predicted analytically, based on the stochastic properties of the fluctuating quantities in the layer. This analysis uses the technique of Fourier-Stieltjes transformation as used by O.M. Phillips. In this way, the sound spectrum outside the bubble layer can be correlated to the covariance of the fluctuating quantities, as gasfraction, velocity etc., assuming for instance that the bubble layer is stochastically stationairy and almost homogeneous. 

After Fourier-StieltJes transformation the general solution for the different Fourier components d7(K,y,(<>) of is... 

In this equation the gasfraction in point (x,y,t) is correlated to the gasfraction in a point (x, y, t ). A condition for application of the Fourier-StieltJes transform was, that the wave field must be homogeneous. This means that all probability-densities are invariant under the addition of a constant vector to all space points. Strictly speaking this condition is not fulfilled for a wavy wall. But when the length scale on which the mean quantities change, is small in respect with the variation of Z in the separation variable r, then the wave field can be assumed to be almost homogeneous. A second condition on the transformation yields the stationarity of the wave field, so the second moment function will be only dependent on the separation variable x. This condition is satisfied when the mean quantities are independent in time. In this way the frequency, wave-number spectrum can be written as. 

Sips in 1948 was able to solve the integral equation for F U) using a Stieltjes transform method (see Section 2). He assumed that the internal partition function for the adsorbed phase (incorporated in K) is independent of the adsorption energy and obtained a solution of the form shown in equation (10). However the normalization integral ... 

If Bdistribution function is translated to lower adsorption energies.A difficulty arises when using the Sips procedure since the distribution function so determined is temperature dependent. Honig and Hill have repeated the analysis but with a constraint requiring the distribution to be independent of temperature thus leading to severe restrictions on the form of the total isotherm function that can be handled by the Stieltjes transform method. 

Key ST, Stieltjes Transform CA, Condensation Approximation LT, Laplace Transform... 

Toth et have noted the wide applicability of another total isotherm equation, the Toth equation,and by using the Stieltjes transform method... 

Pl being a characteristic pressure), then equation (4) can be transformed in such a way that (p) is related to a Stieltjes transform of reverse transform can be computed so yielding [Pg.62]

One of the most effective methods of evaluation of the energy distribution function %(e) relating to the overall adsorption isotherm assumed a priori was proposed by Sips [21,22], He proved that the integral equation (10) with the Langmuir local isotherm [Eq. (11)] could be rewritten as the Stieltjes transform [105] ... 

The Sips method, based the theory of Stieltjes transform and requiring the analytic extension of 9(0 [24,25]. 

To carry out the spectral analysis, it is necessary to determine the spectral properties of the involved functions. These properties may be determined through the Fourier-Stieltjes transform (Priestley 1999). 

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