Hlder Norm Estimate for a Hilbert Transform in Hermitean Clifford Analysis

A Hilbert transform for Hlder continuous circulant (2 × 2) matrix functions, on the d-summable (or fractal) boundary Γ of a Jordan domain Ω in R2n , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the Hlder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the Hlder exponents, the diameter of Γ and a specific d-sum (d d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary.     

Synthesis and Polymerization of the Organometallic Monomer Series Based on Cymantrenylethyl Acrylate

{η⁵ -C₅H₄[CH(CH₃)OC(O)CH = CH₂])Mn(CO)3, {η⁵—C₅[CH-(CH₃)OC(O)C(CH₃)=CH₂]]Mn(CO)₃, and {η⁵—C₅H₄[CH(CH₃)-OC(O)CH=C(CH₃)2])Mn(CO)₃ were synthesized (63, 57, and 51%, respectively) from {η⁵—C₅H₄[CH(CH₃)OH])Mn(CO)₃, toluene-sulfonic acid, and the acrylic, methacrylic, and dimethylacrylic acids, and from (η⁵-C₅H₄[CH(CH₃)OH]}Mn(CO)₃, pyridine, and the acrylic, methacrylic, and dimethylacrylic acyl chlorides [26, 48, and 25% (impure), respectively]. No product was obtained when NaH was used as the base in the latter method. The acrylate and methacrylate monomers were bulk homopolymerized at 65°C with AIBN (75% yield, M_n = 88,550 g/mol; 78% yield, M_n = 349,350 g/mol, respectively). The dimethylacrylate did not polymerize under these conditions. The polymers lost vinylcymantrene upon heating to 257 and 279°C, respectively. The polymers did not exhibit a clear T_g but were observed to soften at 85 and 160°C, respectively, and they could be pulled into fibers.     

Efficacious Entry Into Substituted 5,6-Dihydro-4-Hydroxy-2H-Pyran-2-Ones and 2,3-Dihydro-4H-Pyran-4-Ones Utilizing Ketonic Dianions

Methyl acetoacetate and 2,4-pentanedione dianions were condensed with aldehydes and ketones to afford a 1,3,5-trioxygenated carboskeleton. Intramolecular cyclization of the aldol adducts delivered the title compounds in good yield.     

Two Convenient Syntheses of 2-Aryl-1-Cyano-1-Propenes From Quaternary Ammonium Salts

Two readily available and different quaternary ammonium salts have been prepared by the alkylation of vinamidinium salt reduction products. These salts have been successfully converted to 2-aryl-1-cyano-1-propenes and this methodology represents a clean and efficient procedure for the construction of such systems.     

Protection of A 2,6-Dialkylaniline: Synthesis of 4-Dimethylmethoxysilyl-2,6-Diethylaniline

The synthesis of 4-dimethylmethoxysilyl-2,6-diethylani-line (1) from 2,6-diethylaniline (2) is described. An intermediate in the process, disilazane 5 appears to be the first 2,6-dialkylaniline to be protected with 1,1,5,5-tetramethyl-1,4-dichlorodisilethylene (4).     

A Convenient Synthesis Of 2-Methyl-5-(Methylthio)Benzothiazole

A convenient and efficient four step synthesis of 2-methyl-5-(methylthio)benzothiazole is presented.     

Approaches to the Synthesis of Gibberellic Acid. Construction of the B,C and D Ring System

One of the most challenging aspects of the synthesis of gibberellic acid² and structurally related compounds such as steviol³ is the construction of the bicyclo[3.2.1]octane system with the unusual features of an exocyclic methylene group adjacent to a bridgehead tertiary hydroxyl group. In gibberellic acid this structural feature comprises the C, D ring system, which is fused to the five-membered B ring.