Multi-Gaussian models

We will consider several of these multi-job models in Chapter 7, including Gaussian-1 and Gaussian-2 theory and their variants and several Complete Basis Set (CBS) methods. 

It is well known that in the hterature there are more than 100 isotherm equations derived based on various physical, mathematical, and experimental considerations. These variances are justified by the fact that the different types of adsorption, solid/gas (S/G), solid/liquid (S/L), and liquid/gas (L/G), have, apparently, various properties and, therefore, these different phenomena should be discussed and explained with different physical pictures and mathematical treatments. For example, the gas/solid adsorption on heterogeneous surfaces have been discussed with different surface topographies such are arbitrary, patchwise, and random ones. These models are very useful and important for the calculation of the energy distribution functions (Gaussian, multi-Gaussian, quasi-Gaussian, exponential) and so we are able to characterize the solid adsorbents. Evidently, for these calculations, one must apply different isotherm equations based on various theoretical and mathematical treatments. However, as far as we know, nobody had taken into account that aU of these different isotherm equations have a common thermodynamical base which makes possible a common mathematical treatment of physical adsorption. Thus, the main aim of the following parts of this chapter is to prove these common features of adsorption isotherms. 

While the form of this term is the same as the viscous-dissipation term in the conditional acceleration, the modeling approach is very different. Indeed, while the velocity field in a homogeneous turbulent flow is well described by a multi-variate Gaussian process, the scalar fields are very often bounded and, hence, non-Gaussian. Moreover, joint scalar... 

Unlike the velocity field, scalar fields are often perfectly correlated so that the correlation matrix p will be rank-deficient.92 When this occurs, the coefficient matrix M will not be properly defined by (6.93), and so the FP model must be modified to handle perfectly correlated scalars. As shown in Section 5.9 for the multi-variate Gaussian presumed PDF,... 

The traditional one-, two- and multi-factor equilibrium models, known as ajfine term structure models (see James and Webber, 2000 or Duffie, 1996, p. 136). These include Gaussian affine models such as Vasicek, Hull-White and Steeley, where the model describes a process with constant volatility and models that have a square-root volatility such as Cox-Ingersoll-Ross (CIR) ... 

The Bayesian time-domain approach presented in this chapter addresses this problem of parametric identification of linear dynamical models using a measured nonstationary response time history. This method has an explicit treatment on the nonstationarity of the response measurements and is based on an approximated probability density function (PDF) expansion of the response measurements. It allows for the direct calculation of the updated PDF of the model parameters. Therefore, the method provides not only the most probable values of the model parameters but also their associated uncertainty using one set of response data only. It is found that the updated PDF can be well approximated by an appropriately selected multi-variate Gaussian distribution centered at the most probable values of the parameters if the problem is... 

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