We analyze univariate oscillatory integrals for the standard Sobolev spaces Hs of periodic and non-periodic functions with an arbitrary integer s ≥ 1. We find matching lower and upper bounds on the minimal worst case error of algorithms that use n function or derivative values. We also find sharp bounds on the information complexity which is the minimal n for which the absolute or normalized er...

Hantzsch 1,4-dihydropyridines are oxidized to the corresponding pyridine derivatives using oxygen or air and a mixture of the manganese and cobalt salts of parninobenzoic acid supported on silica gel as the catalysts. Reactions are clean and the catalysts can be recovered easily and reused repeatedly.

Silica Vanadic Acid, SVA, (oxo-vanadium has been supported on silica) as a dual Lewis and Bronsted acid ability was prepared and efficiently catalyzed one-pot multi-component condensation of enolizable ketones or alkyl acetoacetates with aromatic aldehydes, acetonitrile and acetyl chloride. The described reaction was produced β-acetamido ketone or ester derivatives in high to excellent yields i...