Extremal functions for Moser's inequality

Let be a bounded smooth domain in , and a function with compact support in . Moser's inequality states that there is a constant , depending only on the dimension , such that

where is the Lebesgue measure of , and the surface area of the unit ball in . We prove in this paper that there are extremal functions for this inequality. In other words, we show that the

is attained. Earlier results include Carleson-Chang (1986, is a ball in any dimension) and Flucher (1992, is any domain in 2-dimensions).

     

Absorption Spectra of 5′-carboxy-2′-hydroxychalcone and its Derivatives

Ultraviolet absorption spectra and infrared absorption spectra of 5′-carboxy-2′-hydroxychalcone and eleven derivatives substituted at side benzene nucleus are measured to elucidate the spectroscopic effects of the 5′-carboxyl group in this series of compounds.     

A chemoenzymatic approach to glycopeptide antibiotics

Many biologically active natural products are constrained by macrocyclization and modified with carbohydrates. These two types of modifications are essential for their biological activities. Here we report a chemoenzymatic approach to make carbohydrate-modified cyclic peptide antibiotics. Using a thioesterase domain from the decapeptide tyrocidine synthetase, 13 head-to-tail cyclized tyrocidine derivatives were obtained with one to three propargylglycines incorporated at positions 3-8. These cyclic peptides were then conjugated to 21 azido sugars via copper(I)-catalyzed cycloaddition. Antibacterial and hemolytic assays showed that the two best glycopeptides, Tyc4PG-14 and Tyc4PG-15, have a 6-fold better therapeutic index than the natural tyrocidine. We believe this method will also be useful for modifying other natural products to search for new therapeutics.     

Photoexcitation and photodissociation lasers. III. mechanisms of CO laser emission from the vacuum UV photodissociation of CH2COO2 and CH2COSO2 mixtures

CO laser emission was detected in the vacuum UV flash photolysis of CH₂CO. The emission is attributed to the initial photodissociation reaction

Addition of O₂ to the CH₂CO system caused a pronounced enhancement in the laser intensity. This effect is believed to be due to the removal of the CH₂ + CH₂CO reaction, which produces uninverted CO molecules. A greater laser output was obtained when SO₂ was used instead of O₂. In the O₂-added system, a total of 16 transitions ranging from Δv(8→7) to (4→3) were identified. Addition of SO₂ increased the total number of lines to 34, lasing in the range between (11→10) and (4→3). This enhancement is ascribed to the occurrence of the reaction

In addition to these chemical effects, the effects of flash energy, inert gases and total pressures have been investigated.     

Highly regioselective rearrangement of 2-substituted vinylepoxides catalyzed by gallium(III) triflate

Gallium(III) triflate catalyzed the rearrangement of 2-substituted vinylepoxides into beta,gamma-unsaturated carbonyl compounds with high regio- and chemoselectivity (>97/3) in low catalyst loading (1-5 mol %). The alkyl-substituted trimethylsilylvinyl epoxides gave beta,gamma-unsaturated ketone, but aryl-substituted vinylepoxides gave the aldehydes instead.     

Asymmetric synthesis of cis-5-tert-butylproline with metal carbenoid NH insertion

The highly stereoselective intramolecular metal carbenoid insertion reaction of sulfinimine-derived delta-amino alpha-diazoesters is used to prepare cis-5-tert-butylproline. A concerted or nearly concerted metal carbenoid N-H insertion reaction mechanism is proposed.