Double-sided complex transform

If the double-sided complex transform is performed, the number of operations increases. For each value of Vj, we now perform N multiplications for Sa k)cosInVjkh, and N multiplications are required for Sa k)sia2nVjkh. Then TV — 1 additions must be made for the cosine transform and TV — 1 for the sine transform, so a total of approximately 4N operations must be made for each wavenumber. If the full bandwidth of the spectmm is to be examined at the same resolution as the single-sided spectmm, twice as many data points must be taken so the number of operations is increased to 8TV. The number of output points is the same as for the single-sided case, so that 4TV operations are required. This number does not include the time required for the cosine, sine, square, and square root calculations required for each value of Vj. 

Suitable reactions for side-wall functionalization are mainly those that attack the TT-system of the nanotube. Primarily, these are transformations like the addition or cycloaddition known from the chemistry of double bonds. These transformations shall be discussed in the following. The direct attack to the n-system of the tube has yet another interesting aspect Unlike the functionalization of the ends it enables a control over the electronic structure of the entire tube. A suitable modification thus allows for the construction of complex electronic systems based on carbon nanotubes. At first, however, the more simple side-wall functionalizations wiU be discussed. 

Transformation of lanosterol to cholesterol (Figure 19-16) is a complex, multistep process catalyzed by enzymes of the endoplasmic reticulum (microsomes). A cytosolic sterol carrier protein is also required and presumably functions as a carrier of steroid intermediates from one catalytic site to the next but may also affect activity of the enzymes. The reactions consist of removal of the three methyl groups attached to C4 and C14, migration of the double bond from the 8,9- to the 5,6-position, and saturation of the double bond in the side chain. Conversion of lanosterol to cholesterol occurs principally via 7-dehydrocholesterol and to a minor extent via desmosterol. 

Although for the catalytic transformations of organosilicon compounds only hydrosilylation is well known as industrially important process, in the last 20 years other reactions of silicon compounds catalyzed by transition metal complexes have been discovered and developed. They include double (bis)silylation of alkenes and alkynes, silylative coupling of alkenes and alkynes with vinylsi-lanes, dehydrocoupling of hydrosilanes, silylformylation and silylcarbonylation of unsaturated compounds, and dehydrogenative silylation of alkenes and alkynes with hydrosilanes. Only the latter, as related to hydrosilylation (and very often its side reaction), has been discussed here (13). 

During formation of MMCs various thermodynamic side efiects driven by a thermodynamically favoured terms can occur. This includes conformational changes, modification of functional groups and also macrochain breakage. Examples of conformational changes are chain transformation in poly(oxyethylene)-transition metal complexes [61,62], double helix model of poly(oxyethylene)-alkali metal ion complexes [63], conformational modifications of poly(2-vinylpyridine) or poly(amidoamines) during complex formation [64,65], and others. Important to mention here is that chain destruction can occur in type I polymers during their formation [3,66,67]. 

The iabiie beta acids, present in hops, are sensitive to oxidation reactions, initiated by air (auto-oxidation). Thus, the beta acids are converted partiaiiy in the hop plant and to a greater extent during storage of the hops. The oxidation is almost quantitative in the brewing process. A very complex reaction mixture is formed. The experimental data show that auto-oxidation occurs via radical mechanisms, which transform the polyfunctional beta acids to a large number of oxidized compounds. In particular the three 3-methyl-2-butenyl side chains are very sensitive to oxidation, either at the double bonds or in the allylic positions. The native compounds are usually oxidized further or transformed by hydration and/or elimination. These reactions are responsible for the complexity. Reaction between two side chains leads to bicyclic and tricyclic derivatives. It is remarkable that in all known oxidized compounds derived from the beta acids, at least one 3-methyl-2-butenyl group remains unchanged. 

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