An isoperimetric inequality for the Heisenberg groups

We show that the Heisenberg groups of dimension five and higher, considered as Riemannian manifolds, satisfy a quadratic isoperimetric inequality. (This means that each loop of length L bounds a disk of area ~ L ².) This implies several important results about isoperimetric inequalities for discrete groups that act either on or on complex hyperbolic space, and provides interesting examples in geometric group theory. The proof consists of explicit construction of a disk spanning each loop in . Submitted: April 1997, Final version: November 1997     

Control of the conjugation length and solubility in electroluminescent polymers

A new type of cyclolinear polymer, poly(phenylene vinylene‐alt‐cyclotriphosphazene), was synthesized through Heck‐type coupling reactions to produce π‐conjugated macromolecules with excellent solubility and precise electronic control of the band‐gap energy. This synthesis method is capable of producing well‐defined alternating polymers. The method is highly adaptable and can be readily used for other chromophore systems. The resulting polymers were also capable of accommodating a wide variety of substituents on the cyclophosphazene rings with minimal effect on the electronic properties. The band gap and electron affinities of the polymer were varied through the manipulation of the π‐conjugated unit located between the insulating phosphazene rings. Each chromophore matched the intended conjugation length consistently throughout the macromolecules. The polymers were good film formers because of the chosen substituents on the phosphazene rings. The absorbance of the polymers indicated minimal spectral shift from the monomer absorbance. This suggested an effective insulation of each chromophore unit from its neighbors by the phosphazene rings. Solution photoluminescence efficiencies were found to be up to 44.1%. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 69–76, 2006     

Dissociation patterns in N2O following electron impact

A number of dissociation channels in N₂0 have been observed by time-of-flight spectroscopy following electron impact excitation. The metastable atoms and molecules produced were directly detected. Excited N₂ molecules were produced in the A³Σ_u⁺, B³Π_g and B′³Σ_u^? states in conjunction with ground state oxygen atoms. A number of additional dissociation channels were monitored by observing the production of oxygen and nitrogen atoms in Rydberg states. The results indicate the existence of potential minima in some of the repulsive surfaces involved.     

THE CRUCIAL ROLE OF INORGANIC RING CHEMISTRY IN THE DEVELOPMENT OF NEW POLYMERS

The field of phosphazene high polymers has developed into a large area of more than 700 different types of macromolecules with novel combinations of properties and diverse applications. Small-molecule phosphazene rings have played a major role in these developments, first as starting materials for polymer synthesis, second as synthetic and structural models for the high polymers, and third as components of hybrid inorganic-organic macromolecules. These three aspects are reviewed, with examples taken from our recent work, together with some thoughts on the development of this and related fields in the future.     

Solvent effects on oxygen atom transfer reaction between manganese(V)-oxo corrole and alkene

Pseudo-first order reaction rate constants of 5,10,15-tris（pentafluorophenyl）corrole Mn（V）-oxo (F₁₅CMn（V）-oxo),5,15-bis（pentafluorophenyl）-10-（phenyl）corrole Mn（V）-oxo(F₁₀CMn（V）-oxo),5,15- bis（phenyl）-10-（pentafluorophenyl）corrole Mn（V）-oxo（F₅CMn（V）-oxo） and 5,10,15-tris（phenyl）corrole Mn（V）-oxo（F₀CMn（V）-oxo） with a series of alkene substrates in different solvents were determined by UV-vis spectroscopy.The results indicated that the oxygen atom transfer pathway between Mn（V）-oxo corrole and alkene is solvent-dependent.     

Identifying Models of the Octave Projective Plane

We provide a convenient identification between two models of the projective plane over the alternative field of octaves: Aslaksen's coordinate approach and the classic approach via Jordan algebras. We do this by modifying a 1949 lemma of P. Jordan.     

β-Octafluorinated tetrakis(ethoxycarbonyl)porphyrin

Tetrakis(alloxycarbonyl)porphyrin and its β-octafluoro-substituted derivatives were synthesized via Lindsey method and transformed to their zinc complexes.Single crystal X-ray structures of corresponding Zn(Ⅱ) porphyrins revealed that β-octafluorination will give more compactness of porphyrin moieties in the crystal structure owing to the hydrogen bonding interactions involving β-fluorine atoms.An unusual six-coordinated Zn(Ⅱ) was found via intramolecular coordination of oxygen atom of meso-substituents with central Zn(Ⅱ).     

Nonarithmetic uniformization of some real moduli spaces

Some real moduli spaces can be presented as real hyperbolic space modulo a non-arithmetic group. The whole moduli space is made from some incommensurable arithmetic pieces, in the spirit of the construction of Gromov and Piatetski-Shapiro.     

Hyperbolic surfaces with prescribed infinite symmetry groups

For any countable group whatsoever, there is a complete hyperbolic surface whose isometry group is .