Conjugate points in

An example of a geodesic in with conjugate points is given, thus providing an affirmative answer to a question of V.I. Arnold.

     

A simple and green procedure for the conjugate addition of thiols to conjugated alkenes employing polyethylene glycol (PEG) as an efficient recyclable medium

Polyethylene glycol (PEG) is found to be an inexpensive, nontoxic, environmentally friendly reaction medium for the conjugate addition of thiols to conjugated alkenes to afford the corresponding adducts in excellent yields under mild and neutral conditions. Products of undesirable side reactions resulting from polymerization are not observed. The use of PEG avoids the use of either acid or base catalysts for this conversion and moreover PEG can be recovered and reused.     

A novel synthesis, including asymmetric synthesis, of 2,4,4-trisubstituted 2-cyclopentenones based on the reaction of 1-chlorovinyl p-tolyl sulfoxides with acetonitrile and its homologues

Reaction of 1-chlorovinyl p-tolyl sulfoxides, which were synthesized from chloromethyl p-tolyl sulfoxide and ketones in high overall yields, with cyanomethyllithium (lithium α-carbanion of acetonitrile) gave adducts in high to quantitative yields. The adducts were treated with LDA followed by lithium α-carbanion of the homologues of acetonitrile to give 3,5,5-trisubstituted enaminonitriles in good to high yields. Hydrolysis of the enaminonitriles under acidic conditions gave 2,4,4-trisubstituted 2-cyclopentenones in good yields. By using the optically active chloromethyl p-tolyl sulfoxide and unsymmetrical ketones, this procedure gave the optically pure 2,4,4-trisubstituted 2-cyclopentenones. The scope and limitations of this method and the mechanism of the reactions are also discussed. These procedures offer a new and effective method for the synthesis of 2,4,4-trisubstituted 2-cyclopentenones from four components, ketones, chloromethyl p-tolyl sulfoxide, acetonitrile, and its homologues.     

Highly diastereoselective conjugate additions of monoorganocopper reagents to chiral imides

Stereoselective conjugate additions to chiral N-enoyl amides employing various monoorganocuprate reagents, Li[RCuI], are described. The presence of TMSI in the addition of Li[RCuI] in THF provided the highest stereoselectivities. Reversed major diastereomeric ratios were obtained employing Li[RCuI] in ether or conventional copper-promoted Grignard reagents. The results presented support the favored anti-s-cis conformation of the substrates using Li[RCuI]/TMSI in THF, while the copper-promoted Grignard reagents or the Li[RCuI] reagents in ether favor the opposite syn-s-cis conformation. Influence of lithium ions on the stereoselective conjugate addition of the monoorganocuprate reagent, Li[BuCuI], has been investigated and two different mechanistic pathways are presented. The results show that iodotrimethylsilane (TMSI) is crucial for the asymmetric conjugate addition of the copper reagent, but only in THF or when 12-crown-4 is used. The reaction is thought not to involve any halosilane in any critical steps in the organocopper mechanisms conducted in ether. The (CuI)₄(SMe₂)₃ complex precursor plays an instrumental role for the conjugate addition using monoorganocopper reagents.     

Conjugate base catalysed one-pot synthesis of pyrazoles from β-formyl enamides

A novel one-pot synthesis of pyrazoles has been accomplished by the reaction of β-formyl enamides with hydroxylamine hydrochloride catalysed by potassium dihydrogenphosphate in acid medium.     

Ratio-conjugation tool and its application in linear optimization models†

The ratio-conjugation-tool (RC) is constructed systematically, and its applicability for analysis of economic models is demonstrated. One of the main results establishes a general formula for RC of aggregative function. This tool is applied to the analysis of linear economic dynamics models of Neumann–Gale type which can be solved by a dynamic programming method: for each direct problem we construct the corresponding dual RC-problem. Moreover, we consider three examples of the most important economic dynamics models and look at these now from a unified general position.     

Conjugate points at infinity in infinite dimensional teichmüller spaces

We show in this article that there exists conjugate points at infinity in infinite dimensional Teichmüller spaces.     

Sobolev H1 geometry of the symplectomorphism group

For a closed symplectic manifold $(M,\omega )$ with compatible Riemannian metric g we study the Sobolev $H^1$ geometry of the group of all $H^s$ diffeomorphisms on M which preserve the symplectic structure. We show that, for sufficiently large s, the $H^1$ metric admits globally defined geodesics and the corresponding exponential map is a non-linear Fredholm map of index zero. Finally, we show that the $H^1$ metric carries conjugate points via some simple examples.     

Recent advances in asymmetric reactions catalyzed by chiral phosphoric acids

The very recent advances in chiral phosphoric acids (CPAs) catalyzed asymmetric reactions are discussed.     

On duality concepts in fractional programming

The paper (Part I) describes an approach to duality in fractional programming on the basis of another kind of conjugate functions. The connections to some duality concepts (the Lagrange-duality and duality concepts of Craven and Schaible) are investigated and some new proofs of strong duality theorems are given.