Taylor function decomposition

At the beginning of this chapter, we introduced statistical models based on the general principle of the Taylor function decomposition, which can be recognized as non-parametric kinetic model. Indeed, this approximation is acceptable because the parameters of the statistical models do not generally have a direct contact with the reality of a physical process. Consequently, statistical models must be included in the general class of connectionist models (models which directly connect the dependent and independent process variables based only on their numerical values). In this section we will discuss the necessary methodologies to obtain the same type of model but using artificial neural networks (ANN). This type of connectionist model has been inspired by the structure and function of animals natural neural networks. 

Figure 33 The temperature response function r of the first-order decomposition rate constants ki and ky.. The r-function is a scalar that relates k at any given temperature to its maximal rate at a reference temperature (Tref = 30 °C, see text for details). Also shown is the modeled fit of the Lloyd and Taylor equation (Equation (12),...

The decomposition function in the Taylor s row was used in the region point (m, n) for determination of the approximation error of the scheme (2.44) ... 

Taylor s experiments180 on the decomposition of metal alkyls led him to believe that the active alkyl groups, released by decomposition of the compounds, functioned in the same way as the active metal atoms. Experiments have shown that the presence of hydrogen atoms induces oxidation of ethylene at room temperature. The theory is advanced that the free alkyl radicals act in a manner similar to hydrogen atoms or metal fogs, i.e., as active oxidation centers producing a slow homogeneous combustion of fuel. This action of free radicals may accomit for the effects of non-metallic knock suppressors as aniline, toluidine, etc. 

Within any decoupling scheme there are only a few restrictions on the choice of the transformations U. First, they have to be unitary and analytic (holomorphic) functions on a suitable domain of the one-electron Hilbert space V, since any parametrization has necessarily to be expanded in a Taylor series around W = 0 for the sake of comparability but also for later application in nested decoupling procedures (see chapter 12). Second, they have to permit a decomposition of in even terms of well-defined order in a given expansion parameter of the Hamiltonian (such as 1/c or V). It is thus possible to parametrize U without loss of generality by a power-series ansatz in terms of an antihermitean operator W, where unitarity of the resulting power series is the only constraint. In the next section this most general parametrization of U is discussed. 

To further test the weak detonation model, S. Goldstein measured the water shock velocity in the aquarium test after the detonation wave interacted with the water above the top of the X0233 cylinder. Her experimental water shock velocities, as a function of distance above the top of the explosive cylinder, are shown in Figure 2.28 along with the calculated water shock velocities. They are consistent with a flat top Taylor wave characteristic of a weak detonation and a detonation front pressure of 160 kbars. The initial water shock velocities exhibit behavior characteristic of irregular decomposition of the explosive near the shock front. The 2DL calculated aquarium pressure contours are shown in Figure 2.29. 

Equation (5.41) can therefore be considered as the ANOVA (Analysis of Variances) decomposition of T(x) and has several important properties (Sobol 2001). The expected value of all nonconstant component functions in Eq. (5.41) is zero and the terms in (5.41) are orthogonal (SoboT 2001). The notation of the zeroth, first, second order, etc. in the HDMR expansion should not be confused with the terminology of a Taylor series (see Eq. (5.3)) since the HDMR expansion is always of finite order (Rabitz and Ah 2000). The higher-order terms reflect the cooperative effects of increasing numbers of input variables acting together to influence the output 7(x). The HDMR expansion is computationally very efficient if higher-order... 

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