Truncation function

The natural expansion has here also another important optimum convergency property. If this expansion is interrupted after r terms, the renormalized truncated function Wr has the smallest total deviation from the exact solution ... 

The construction of the phase diagram of a heteropolymer liquid in the framework of the WSL theory is based on the procedure of minimization of the Landau free energy T presented as a truncated functional series in powers of the order parameter with components i a(r) proportional to Apa(r). The coefficients of this series, known as vertex functions, are governed by the chemical structure of heteropolymer molecules. More precisely, the values of these coefficients are entirely specified by the generating functions of the chemical correlators. Hence, before constructing the phase diagram of the specimen of a heteropolymer liquid, one is supposed to preliminarily find these statistical characteristics of the chemical structure of this specimen. Here a pronounced interplay of the statistical physics and statistical chemistry of polymers is explicitly manifested. 

Figure 1. The exact and the truncated functional for the thermodynamic potential. For the latter, a propagator near to the stationary point (which may be unphysical) can give a physically reasonable approximation.

Equation (3) implies an integration between - oo and -I- oo with an infinitely high resolution. But we have a limit for the maximal retardation of the interferogram and this means that we are multiplying the interferogram by a truncation function T 8), which fulfills the conditions ... 

The rectangular window function is a simple truncating function which can be applied to transformed data. This function has zero values above some pre-selected cut-off frequency, /c, and unit values at lower frequencies. Using various cut-off frequencies for the truncating function and applying the inverse transform results in the smoothed spectra shown in Figure 13. 

Equation (5.9) shows that in order to measure the complete spectrum, we would have to scan the moving mirror of the interferometer an infinitely long distance, with (5 varying between -oo and +cx) centimeters. In practice, the optical path length difference is finite. By restricting the maximum retardation to /, we are effectively multiplying the complete interferogram by the boxcar truncation function (see Fig. 5.3a left)... 

By sampling a finite path difference A another instrumental effect is introduced to the interferogram. Effectively, the complete interferogram (from —oo to oo) is multiplied by a boxcar truncation function, D x), which is... 

Apodization is the modification of the interferogram by multiplication with an apodization function (Griffiths and de Haseth 2007). If the interferogram is unweighted, the shape of a spectral line is the convolution of the spectrum with a sine function, which is the Fourier transform of the boxcar truncation function. 

The spectrum B( ) is affected by this change, consequently it must be renamed B (i ). This is equivalent to multiplying E(S)oet over the original limits by a truncation function U(o) so that... 

Figure 3. A schematic representation of a rectangular truncation function and its Fourier transform, the sine function.

When a rectangular truncation function is used on the inter-ferogram, the inherent resolution of the spectrum can be shown to be... 

Note that it represents the xenon concentration after shutdown for the special case when the concentration at shutdown (X,) is precisely zero and normalized so that the maximum value of the function (9.292) is unity. Thus we find that the displacement transformation (9.287) along with the specification (9.289) transforms the truncated function (9.285)... 

A flexible, three-center model labeled F3C has been compared with other water models within the context of biomolecular simulations. This model contains a smooth truncation function acting on all interactions, so dielectric properties may not be well represented. This model stabilizes the hydrogen bond network, as evident from the oxygen-oxygen radial distribution functions, suggesting that the correct local ordering of liquid water is reproduced. A version of this model with a different truncation scheme has also been evaluated. " ... 

By restricting the maximum retardation of the interferogram to A centimeters, we are effectively multiplying the complete interferogram (between 8 = oo and 8 = +oo) by a truncation function, D(5), which is unity between 8 = —A and +A, and zero at aU other points, that is. 

In view of the shape of this function, D(5) is often called a boxcar truncation function. By analogy to Eq. 2.13, the spectmm in this case is given by the equation... 

The comparator in the figure determines the section where the train op-erates at the time. The comparator is represented by a nonlinear function (Device 56) and a truncation function TRUNC() installed in the TAGS. 

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