The catalysis of alkene isomerisation,oligomerisation, and polymerisation by cationic nitrosylrhodium complexes

The complex [Rh(NO)(NCMe)₄][BF₄]₂ catalyses the isomerisation of terminal to internal alkenes, the oligomerisation of branched alkenes such as 2-methylpropene, and the stereospecific polymerisation of buta-1,3-diene to trans-1,4-polybutadiene.     

Field theories that have no manifestly Lorentz-invariant formulation

A class of theories for which the Lorentz algebra closes only on the mass-shell, and which therefore cannot be written in a manifestly Lorentz-invariant form, is presented. Examples include certain supergravity theories in six and ten dimensions. These results make the corresponding phenomenon in the case of supersymmetry algebras less surprising. The also shed light on the problem of finding off-mass-shell dual string model amplitudes.     

Fixed point subalgebras of rational higher-dimensional non-commutative tori

We consider the fixed point subalgebras of higher-dimensional rational non-commutative tori induced by finite order automorphisms derived from a diagonal extension of the canonical action on rotation algebras. We give an alternative characterization and determine their -theory.

     

Algebraic Cycles and Even Unimodular Lattices

The number (up to isomorphism) of positive-definite, even, unimodularlattices of rank 8r grows rapidly with r. However, Bannai [1]has shown that, when counted according to weight, those withnon-trivial automorphisms make up a fraction of the whole, whichgoes rapidly to zero as r. Therefore it is of some interestto produce families of positive-definite, even, unimodular latticeswith large automorphism groups and unbounded ranks. Suppose that G is a finite group and V is an irreducible Q[G]-modulesuch that VR is still irreducible. Then, as observed by Gross[8], the space of G-invariant symmetric bilinear forms on Vis one-dimensional and is necessarily generated by a positive-definiteform, unique up to scaling by non-zero positive rationals. Thompson[23] showed that, if V is also irreducible modp for all primesp, then it contains an invariant lattice (unique up to scaling)which is even and unimodular with appropriate scaling of thequadratic form. Examples arising in this manner are the E₈-latticeof rank 8, the Leech lattice of rank 24 and the Thompson–Smithlattice of rank 248. Gow [6] has also constructed some examplesassociated with the basic spin representations of 2A_n and 2S_n.     

Dynamic wave-front generation for the characterization and testing of optical systems

We describe a simple method for generating known optical aberrations dynamically, using a ferroelectric liquid-crystal spatial light modulator. Aberrations inherent in the optical system are measured and corrected, and as an example Kolmogorov turbulence is simulated for aperture sizes D/r(0) from 0 to 30, varying at frame rates up to 2.5 kHz. A measure of wave-front generation efficiency is introduced and is shown to be better than 86% for Kolmogorov phase screens with D/r(0) in the range from 0 to 30.     

The importance of inter-molecular forces in crystal structure analysis: Solid state study of the diag-to-lat isomerization of (η5-C5H4Me)Re(CO)2Br2

Single crystal X-ray structures of diag and lat-(⁵-C₅H₄Me)Re(CO)₂Br₂ have been determined. The diag form crystallizes in the triclinic space group , a = 6.751(2), b = 8.537(1), c = 9.758(1) Å = 96.70(1), = 93.15(2), = 104.96(2)°, V = 534.8(2) ų, Z = 2. The lat form is monoclinic, P2₁/c, a = 11.820(1), b = 7.133(1), c = 12.924(1) Å = 98.278(8)°, V = 1078.2(2) ų, Z = 4. The unit cell volumes (per Z) of the two isomers differ by ca 1%. Molecular modelling reveals an energy difference between the two isomers of 1 kJ mol^–1 or less. The XRD powder diffraction patterns for the lat isomer produced by crystallization from solution or prepared by thermal isomerization are identical. The evidence thus suggests that the solid state isomerization reaction is a novel example of a reaction which yields a molecular structure determined by crystal packing forces.