Back transformation procedure

The solution of the transformed equation is obtained by exponentiating this R matrix. To efficiently exponentiate this matrix we must first diagonalize it, exponentiate the eigenvalues, and back transform with the eigenvectors. This back transformation procedure is repeated for every time at which we wish to know the molecular populations. 

The formulas and remarks referring to the back-transformation of the /12 integrals are associated with step 5 in Algorithm 1. The implementation of the back-transformation procedure is not complicated and is illustrated in Algorithm 2. 

In order to restria attention to a smgle shell of scatterers, one selects a limited range of the R-space data for back-transformation to k-space, as illustrated m Figure 3B,C. In Ae ideA case, this procedure allows one to anAyze each shell separately, AAough in practice many shells cannot be adequately separated by Fourier tering (9). 

Clearly for the procedure outlined in Fig. 10.1 to work, we need to take care of several critical steps next to reasonable initial phase estimates required to formulate the initial restraints, we need a statistically valid procedure for the combination of the phases obtained by back transformation of the real space restrained map and the initial phase probability distribution. This recombination step is discussed below. 

An important discussion on rigorous and approximate procedures to solve the diffusion problem to planar, spherical, and dropping electrodes has been published by Guidelli [75]. Following this treatment, eqns. (156c) and (156e) are first divided by s1/2 and then back-transformed into the time domain eqn. (156d) is back-transformed directly. The results are... 

The earliest successful application of the idea of combining projections to reconstruct the 3D structure of a biological assembly was made by DeRosier and Klug (4). The idea-is that each 2D projection corresponds after Fourier transformation to a central section of the 3D transform of the assembly. If enough independent projections are obtained, then the 3D transform will have been fully sampled and the structure can then be obtained by back transformation of the averaged, interpolated, and smoothed 3D transform. This procedure is shown schematically for a three-dimensional object in the shape of a duck, which represents the molecule whose structure is being determined (Fig. 14.4). 

Algorithm 2 The procedure that carries out the back-transformation of the /12 integrals. 

Both the Doppler slice and the ion TOP measurement are essentially in the centre-of-mass system. Therefore the measurement directly maps out the desired 3D centre-of-mass distribution, i.e. d aj v dv dO = I 6,v) v in Cartesian velocity coordinates (d (r/diVx dvy dv ). Thus, the double differential cross-section I 6,v) is obtained by multiplying the measured density distribution in the centre-of-mass velocity space by and then transforming from the Cartesian to the polar coordinate system. This procedure has to be contrasted against the conventional neutral TOP technique (either in the universal machine or by the Rydberg-tagging method), for which the laboratory to centre-of-mass transformation must be performed, or against the 2D ionimaging technique, which involves 2D to 3D back transformation. 

The Gaussian elimination procedure can therefore be considered as two operations, a forward pass and a backward pass. The objective of the forward pass is to transform the original matrix into an upper-triangular matrix. The backward pass calculates the unknown variables using the back substitution procedure. 

By applying specific filters, it is possible to enhance or even eliminate different reflexes of the Fourier image so that the back-transformation results in a so-called corrected Image. This procedure modifies the image information somewhat, but the images gain clarity and present clearly noticeable structural details. 

The basis vectors in Table 1 are complete but not unique. Besides trivial variations in the Gram-Schmidt orthogonalization, there is a substantive difference that depends on the choice of the weighting factors q these factors determine both the result of the orthogonalization procedure, as well as the back transformation from... 

The procedure Merge transforms the internal displacement coordinates and momenta, the coordinates and velocities of centers of masses, and directional unit vectors of the molecules back to the Cartesian coordinates and momenta. Evolve with Hr = Hr(q) means only a shift of all momenta for a corresponding impulse of force (SISM requires only one force evaluation per integration step). 

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