Pseudo-projection

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

In this figure the next definitions are used A - projection operator, B - pseudo-inverse operator for the image parameters a,( ), C - empirical posterior restoration of the FDD function w(a, ), E - optimal estimator. The projection operator A is non-observable due to the Kalman criteria [10] which is the main singularity for this problem. This leads to use the two step estimation procedure. First, the pseudo-inverse operator B has to be found among the regularization techniques in the class of linear filters. In the second step the optimal estimation d (n) for the pseudo-inverse image parameters d,(n) has to be done in the presence of transformed noise j(n). 

Figure 10.2 Crystal structure of GeF2 (a) projection along the chains, and (b) environment of Ge (pseudo trigonal bipyramidal). The bond to the unshared F is appreciably shorter (179 pm) than those in the chain and there is a weaker interaction (257 pm) linking the chains into a 3D structure.

For image slicers, contiguous slices of the sky are re-arranged end-to-end to form the pseudo-slit. In that case it is obvious that the sky can be correctly sampled (according to the Nyquist sampling theorem) by the detector pixels on which the slices are projected in the same way as required for direct imaging. 

Two-dimensional NMR spectra are normally presented as contour plots (Fig. 3.11a), in which the peaks appear as contours. Although the peaks can be readily visualized by such an overhead view, the relative intensities of the signals and the structures of the multiplets are less readily perceived. Such information can be easily obtained by plotting slices (cross-sections) across rows or columns at different points along the Fi or axes. Stacked plots (Fig. 3.11b) are pleasing esthetically, since they provide a pseudo-3D representation of the spectrum. But except for providing information about noise and artifacts, they offer no advantage over contour plots. Finally, the projection spectra mentioned in the previous section may also be recorded. 

Alternatively, the right-hand side can be written in terms of a projection tensor and a convolution (Pope 2000). The form given here is used in the pseudo-spectral method. 

An interesting consequence of the existence of 5th dimension is that in the 4 dimensional observations an apparent violation of the equivalence principle must appear as one can show it by writing the geodesic equation in 5 dimensions and then projecting it into 4 dimensions. Without going into details, the sign of p5 causes the appearance of a pseudo-charge q in the 4 dimensional formalism,... 

The tI10-MoNi4 type is another superstructure based on face-centred cubic pseudo-cells. In the projection shown in Fig. 3.36, inside the true cell, the pseudo-cubic subcell (aps = 362 pm, cps = 356.4 pm) has been evidenced by dotted lines. Close-packed layers can be identified in this structure they are stacked in a 15 close-packed repeat sequence. 

From calculations of the preferred conformation of the ephedrine and pseudo ephedrine molecules the topography of the adrenergic receptor as a flat surface has been deduced (26). This view, however, does not explain the decisive effect of the additional methyl group its relative position should not influence the interaction with the proposed receptor to a large extent (Fig. 7a). However, as the differences in activities are especially high for the ephedrine and pseudo ephedrine isomers, we believe that the receptor must have an intercalated structure similar as projected for the "binding sites" of the silicones (Fig. 7b). ... 

It is obvious that the projection operators for the different species have different numbers of terms in them. The HON species have 12 terms (3 x 2 ) while the A2B-type species have four terms, and the HDT+ isotopomer has only two terms. This results in different sizes of the spin-projected basis sets, and for this reason the properties obtained in this work are not precisely comparable between the A3, A2B, and ABC systems, although a very good idea of the trends may be obtained from the data in Table XVI. While all of the above are given in terms of the original particles, it should be noted that the permutations used in the internal particle basis functions are pseudo -permutations induced by the permutations on real particles. 

Operational activities have continued under Project Prism, the international initiative designed to address diversion of the five main precursors used in the illicit manufacture of amphetamine-type stimulants, namely, ephedrine, 3,4-methylene-dioxyphenyl-2-propanone (3,4-MDP-2-P), l-phenyl-2-propanone (P-2-P), pseudo-ephedrine and safrole, as well as the equipment used in such illicit manufacture. Activities during 2004 have focused on launching operations to address weaknesses in control and monitoring mechanisms identified during 2003, such as monitoring... 

Role of Project PRISM in countering synthetic drugs and their precursors , INCB presentation to the Conference Europe-Asia Cooperation on Synthetic Drugs and their Precursors , Paris, 6-7 March 2007. These are substantial amounts. By comparison, total licit trade in ephedrine and pseudo-ephedrine is estimated at around 30 mt and 1,200 mt respectively. [Source INCB, 2005 Precursors]. The 16 mt of interdicted/suspended shipments could have been used to produce 110 mt of methamphetamine. Were this to have ended up on the illicit market, it would have increased global methamphetamine production by some 40 per cent. 

FIG. 4.2. Dependence of dimensionless pseudo-steady-state intermediate concentration and temperature excess on reactant concentration for the model with the exponential approximation (a) three-dimensional representation of universal locus (b) projection showing dependence of a5S on n (c) projection showing... 

The relative positions of the molecules in the unit cell were afterwards found by calculating some of the structure amplitudes for various positions. The arrangement in one sheet of molecules is shown in Fig. 191. (In the published account (Cox and Goodwin, 1936) use is made of the conception of a pseudo plane "of symmetry perpendicular to the plane of the molecule and perpendicular to the c axis actually the apparent halving of a in the c projection does not demand any such pseudo plane of symmetry in the molecule, but can be produced with completely asymmetric molecules, as Fig. 190 shows. However, as it turns out, the configuration and arrangement of the molecules which accounts for the intensities of the principal reflections does show a pseudo plane of symmetry in the position mentioned.)... 

A half-boat conformation of the cyclobutanone ring (in an octant projection) of 74 correctly explains the observed positive CD near 300 nm. A pseudo-axial chlorine substituent in 75 substantially enhances the positive Cotton effect. The cyclobutanone ring in 76 is nearly planar, thus the effect of exo and endo chloro substituents is effectively cancelled while the remainder of cholestane skeleton resides in a negative octant175. 

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