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.     

Electrically conducting polyaromatic sulfides

Poly(p-phenylene sulfide), PPS, a melt and solution processible polymer, can be made highly conducting by doping with strong electron acceptors such as ASF5. Virgin PPS is an insulator with a relatively high ionization potential compared to polyacetylene. This, coupled with its high melting and solubilization points, restricts possible dopants to those which are particularly aggressive and capable of reacting with PPS in the solid state. These aggressive dopants induce a variety of chemical changes in the polymer backbone upon doping. While fluorination and crosslinking occur to some extent, infrared spectra and independent chemical synthesis suggest that the predominant chemical change is via intrachain bridging (cyclization) of adjacent phenyl rings. In an attempt to find processible systems with lower ionization potentials and which are less prone to chemical modification, a variety of polyaromatic sulfides were prepared. The influence of polymer structure, morphology, and dopant-induced chemical modifications upon conductivity is discussed.     

A mild, copper catalyzed route to conducting polyaniline

The aniline dimer, N-(4-aminophenyl)aniline has been polymerized cleanly under mild conditions to obtain an emeraldine base form of polyaniline using [MeB(3-(Mes)Pz)3]CuCl as the catalyst and H2O2 as the oxidant, while the subsequent acidification of the emeraldine base gives the conducting emeraldine salt form of polyaniline.     

Highly conductive meta derivatives of poly(phenylene sulfide)

Changes in the backbone structure of several meta derivatives of polyphenylene sulfide upon doping with AsF₅ have been investigated by IR spectroscopy. Poly(m-phenylene sulfide) does not form a conducting complex with AsF₅ unless reacted under conditions where carbon-carbon bonds form intramolecularly between phenyl rings. This assertion has been verified by comparison of the IR spectra of poly(m-phenylene sulfide) and a newly synthesized derivative, poly(thio-3,7-dibenzothiophendiyl), after doping with AsF₅. This new derivative forms a complex with AsF₅ which exhibits a conductivity of 18.5 S/cm. A sulfone-containing derivative has also been synthesized, poly[thio-3,7-(dibenzothiophene-5,5-dioxide)-diyl]. With this polymer, a much lower conductivity, 10^?3 S/cm, was obtained after exposure to AsF₅.     

Efficient synthesis and characterization of novel bibenzimidazole oligomers and polymers as potential conjugated chelating ligands

[Structure: see text]. A simple and mild condensation route for the synthesis of novel bibenzimidazole oligomers and polymers is reported here using methyl 2,2,2-trichloroacetimidate as a key starting material. The dimer, trimer, tetramer, and polymers of bibenzimidazole were synthesized as a new series of potential conjugated chelating ligands for metallopolymer studies. The polymers show a maximum absorption at around 400 nm. The optical band gap of the polymer was estimated to be 2.68 eV.