Chaotic Expansion of Elements of the Universal Enveloping Algebra of a Lie Algebra Associated with a Quantum Stochastic Calculus

The functional Ito formula, in the form df() = f( + d ) –f(),is formulated and proved in the context of a Lie algebra L associatedwith a quantum (non-commutative) stochastic calculus. Here fis an element of the universal enveloping algebra U of L, andf() + d() – f() is given a meaning using the coproductstructure of U even though the individual terms of this expressionhave no meaning. The Ito formula is equivalent to a chaoticexpansion formula for f() which is found explicitly. 1991 MathematicsSubject Classification: primary 81S25; secondary 60H05; tertiary18B25.     

Catalytic asymmetric syntheses of tyrosine surrogates

Amino acid esters 5-11 as tyrosine mimics have been synthesized in excellent enantioselectivity (up to 99.6% ee) and in good overall chemical yields. The key step in the sequence was the Burk's [Rh(COD)(2R,5R)-Et-DuPhos]BF4-catalyzed asymmetric hydrogenation of enamides with a variety of reactive functional groups.     

Syntheses of aza and fluorine-substituted 3-(piperidin-4-yl)-4,5-dihydro-1H-benzo[d][1,3]diazepin-2(3H)-ones

A practical and expedient synthesis of the title compounds is described. They were prepared by Stille reaction of nitro halopyridines 4 or nitro fluro-halobenzenes 10, followed by Michael addition of tert-butyl 4-aminopiperidine-1-carboxylate to the resulting activated vinyl compounds 5 and 11, hydrogenation (-NO₂→-NH₂), cyclic urea formation, Boc removal, and HCl salt formation. However, N3 and F1 analogs could not be made by this general strategy. Activated vinyl compounds 5a and 5d when reacted with tert-butyl 4-aminopiperidine-1-carboxylate did not stop at the desired Michael addition stage; but proceeded to produce azaindolines 8 and 9. Michael addition did not occur to compound 11d; instead, the fluorine atom was displaced.     

An efficient synthesis of N-arylated, orthogonally protected racemic aspartates

A brief and efficient synthesis of variously N-arylated racemic aspartates has been achieved by two consecutive reactions in one-pot, in which imine or equivalent, formed in situ from various anilines and ethyl glyoxylate, reacted with the Reformatsky reagent, tert-butyl 2-bromozinc acetate. Notably the two esters are orthogonally protected for the convenience of further derivatization.