The generalized EE scheme

we approximate the logarithm of the characteristic function by a Taylor series expansion as follows 

There always exists a one-to-one mapping between the moments and the cumulants of a probability distribution function with the cumulants cumin) given by 

Petrov [64] and Blinnikov and Moessner [8] find a general fundamental expression for the n-th derivative of a function h g x)) given by 

we obtain the standardized cumulants by computing the derivative 

The summation extending over the set km satisfies the following noimega-tive integer equation  

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The EE (lEE) scheme we derive in section (3.1) (section (4.1)) is exact in the sense that the Taylor expansion is valid for infinity terms. Hence, the generalized EE is an equivalent expression for the pdf (cdf). Now, perform-... 

The general reaction scheme for fi ee-radical polymerization can be expressed as follows ... 

Kobayashi and co-workers reported similar enantioselectivity switch in the bi-nol-yterrbium(III) triflate complex-catalyzed cycloaddition reactions [69] between N-benzylidenebenzylamine N-oxide and 3-crotonoyl-2-oxazolidinone [70]. The reaction in the presence of MS 4 A showed an exclusively high enantioselectivity of 96% ee, while that in the absence of MS 4 A (-50% ee) or in the presence of pyridine N-oxide (-83% ee) had the opposite enantioselectivity (Scheme 7.24). This chirality switch happens generally for the combination of a wide variety of nitrones and dipolarophiles. 

In 2000, Gennaii et al. discovered a new family of chiral Schiff-base ligands, with the general structure, Af-alkyl-p-(A -salicylideneamino)alkanesulfonamide, depicted in Scheme 2.28. These ligands were successfully implicated in the copper-catalysed conjugate addition of ZnEt2 to cyclic enones (Scheme 2.28) and, less efficiently, to acyclic enones such as benzalacetone (50% ee) or chalcone... 

In the same study, these authors have prepared another series of amino-sulf(ox)ide ligands based on the (Nor)ephedrine and 2-aminodiphenylethanol skeletons, bearing two chiral centres in the carbon backbone.Their application to the iridium-catalysed hydrogen-transfer reduction of acetophenone generally gave better yields, but the enantioselectivity never exceeded 65% ee (Scheme 9.4). 

A direct comparison of the stereochemical efficiency of the fragmentation reaction versus the tandem reaction (Scheme 53) was studied by Porter et al. as a function of the steric effect based on the Taft parameters for different substituents [146]. In general, the tandem reactions perform better and provide higher levels of ee s than the fragmentation reactions. This effect could be due to the tinbromide by-product catalyzing a non-stereoselective process as has been uncovered by the same authors (vide supra) and by Sibi and Ji in their diastereoselective studies [147]. 

The use of chiral auxiliaries has been developed into elegant three-step sequences to achieve high ee s (Figure 2). In the general scheme a ketone is derivatized with a chiral amine. Low temperature lithiation and alkylation followed by hydrolysis produces the alkylated ketone in moderate to excellent ee s. The auxiliaries most often used are (S)-valine tert-butyl ester (Koga), l-amino-2-methoxymethylpyrrolidine (Enders) and (S)-2-amino-1-... 

In principle, the approach outlined above for the a-oxoamides can be applied to any reaction, ground or excited state, which converts an achiral reactant into a chiral product, and Toda, Tanaka, and coworkers have investigated a wide variety of such processes [ 15,16]. A complete discussion of their work is beyond the scope of this review, and we illustrate the general approach taken with one final example. As shown in Scheme 4, irradiation of crystalline complexes of ene-diones 20a-f with chiral host (R,R)-(-)-9b led to cyclized products 21a-f in the variable yields and ee values indicated in Table 1 [22]. Remarkably, for reasons that were not clear (there was no accompanying X-ray crystallography), the R=n-propyl derivative 20g was found to give a completely different photoproduct, spiro compound 22 (69% yield, 97% ee, stereochemistry unknown), a result that once again illustrates the rather capricious nature of the use of chiral hosts for asymmetric induction. 

Early investigations have demonstrated that aldehydes and ketones can be enantioselectiveiy a-alkyl-ated via Michael reactions of the corresponding enamines, prepared from proline-derived secondary amines.149-156 However, optical purities of the products were generally low and never exceeded 59% ee.iS1 This kind of asymmetric a-alkylation could later be improved, allowing for example the preparation of compound (141) with high ee (Scheme 51).156-160... 

In the first term, the rate constant is multiplied by cHe / K% + cH ), the fraction of total enol present in neutral form E, and in the second term, k A is multiplied by K /(K + cH ), the fraction of total enol present in basic form Ee [Equation (4)]. The observed coefficient for general base catalysis is now seen to arise from the pre-equilibrium reaction shown in the second line of Scheme 3. Replacing (A /ch )cha by (K /K A)cAe we find k e=kf A / A. Thus, pre-equilibrium deprotonation of the enol by the general base followed by carbon protonation of the ensuing enol anion is operationally equivalent to general base catalysis. 

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