Gaussian transform techniques

A working expression for the effective potential can be develojjed by using Gaussian transform techniques. We write... 

Especially in periodic systems, basis-set sizes prohibit a full DKH transformation. To a good approximation, this can be easily dealt with by using a short-ranged Coulomb operator, which naturally occurs in the Gaussian-augmented plane-wave scheme [666,667], in which the transformation can be performed in the atomic basin [649,668]. More elaborate local schemes have also been developed [669-672]. We come back to this issue in section 14.2 because it can be discussed in the general context of transformation techniques. 

Transformation techniques for non-Gaussian random fields seek to represent the non-Gaussian random field as a nonlinear transformation of a Gaussian random field ... 

The nonlinear transformation technique can be combined with any series expansion schemes described above for the underlying Gaussian... 

Nonlinear optimization techniques have been applied to determine isotherm parameters. It is well known (Ncibi, 2008) that the use of linear expressions, obtained by transformation of nonlinear one, distorts the experimental error by creating an inherent error estimation problem. In fact, the linear analysis method assumes that (i) the scatter of points follows a Gaussian distribution and (ii) the error distribution is the same at every value of the equilibrium liquid-phase concentration. Such behavior is not exhibited by equilibrium isotherm models since they have nonlinear shape for this reason the error distribution gets altered after transforming the data... 

Figure 5-28 (a) The result of Fourier transformation of a low-power rectangular pulse, (b) The result for a shaped Gaussian pulse. (Reproduced from T. D. W. Claridge, High-Resolution NMR Techniques in Organic Chemistry, Pergamon Press, Amsterdam, 1999, p. 349.)... 

In this section, we describe a statistical model of a DCT domain watermarking which is used in the implementation of the SBE scheme described in Section 5.2. We have chosen to work with the DCT, since the distribution of the DCT coefficients can be approximated by the Gaussian p.d.f.. However, the same techniques can be applied to other transforms such... 

The Fourier transforms of Eq. (51) can be performed in closed form for most commonly used basis sets. Moreover, formulas and techniques for the computation of the spherically averaged momentum density, isotropic and directional Compton profiles, and momentum moments have been worked out for both Gaussian- and Slater-type basis sets. Older work on the methods and formulas has been summarized in a review article by Kaijser and Smith [79]. A bibliography of more recent methodological work can be found in another review article [11]. Advantages and disadvantages of various types of basis sets, including many unconventional ones, have been analyzed from a momentum-space perspective [80-82]. Section 19.7 describes several illustrative computations chosen primarily from my own work for convenience. 

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