Stereochemistry of nucleophilic additions to hexafluoro-2-butyne

Base-catalysed additions of alcohols to F-2-butyne (1) give mainly products of $\text{trans}$-addition while $\text{cis}$-addition predominates in uncatalysed additions of alcohols carried out in a diluent. The stereochemistry of addition of diethylamine is very dependent on the solvent used and $\text{cis}$- or $\text{trans}$-addition may predominate. Stepwise and concerted mechanisms are advanced to account for these observations. Nucleophilic addition of sulphur to (1) gives F-tetramethylthiophene (68%) and hydration gives CF₃CH₂COCF₃ (91%).     

Fermionic Quantization and Configuration Spaces for the Skyrme and Faddeev-Hopf Models

The fundamental group and rational cohomology of the configuration spaces of the Skyrme and Faddeev-Hopf models are computed. Physical space is taken to be a compact oriented 3-manifold, either with or without a marked point representing an end at infinity. For the Skyrme model, the codomain is any Lie group, while for the Faddeev-Hopf model it is S². It is determined when the topology of configuration space permits fermionic and isospinorial quantization of the solitons of the model within generalizations of the frameworks of Finkelstein-Rubinstein and Sorkin. Fermionic quantization of Skyrmions is possible only if the target group contains a symplectic or special unitary factor, while fermionic quantization of Hopfions is always possible. Geometric interpretations of the results are given. The first author was partially supported by NSF grant DMS-0204651 The second author was partially supported by EPSRC grant GR/R66982/01     

Solitons on Tori and Soliton Crystals

Necessary conditions for a soliton on a torus \({M = \mathbb{R}^m/\Lambda}\) to be a soliton crystal, that is, a spatially periodic array of topological solitons in stable equilibrium, are derived. The stress tensor of the soliton must be L ² orthogonal to \({\mathbb{E}}\) , the space of parallel symmetric bilinear forms on TM, and, further, a certain symmetric bilinear form on \({\mathbb{E}}\) , called the hessian, must be positive. It is shown that, for baby Skyrme models, the first condition actually implies the second. It is also shown that, for any choice of period lattice Λ, there is a baby Skyrme model which supports a soliton crystal of periodicity Λ. For the three-dimensional Skyrme model, it is shown that any soliton solution on a cubic lattice which satisfies a virial constraint and is equivariant with respect to (a subgroup of) the lattice symmetries automatically satisfies both tests. This verifies, in particular, that the celebrated Skyrme crystal of Castillejo et  al., and Kugler and Shtrikman, passes both tests.     

Porosity,Differentiability and Pansu’s Theorem

We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a \(\sigma \)-porous set. The second result states that irregular points of a Lipschitz function form a \(\sigma \)-porous set. We use these observations to give a new proof of Pansu’s theorem for Lipschitz maps from a general Carnot group to a Euclidean space.     

Asymptotic Interactions of Critically Coupled Vortices

 At critical coupling, the interactions of Ginzburg-Landau vortices are determined by the metric on the moduli space of static solutions. Here, a formula for the asymptotic metric for two well separated vortices is obtained, which depends on a modified Bessel function. A straightforward extension gives the metric for N vortices. The asymptotic metric is also shown to follow from a physical model, where each vortex is treated as a point-like particle carrying a scalar charge and a magnetic dipole moment of the same magnitude. The geodesic motion of two well separated vortices is investigated, and the asymptotic dependence of the scattering angle on the impact parameter is determined. Formulae for the asymptotic Ricci and scalar curvatures of the N-vortex moduli space are also obtained. Received: 25 June 2002 / Accepted: 7 February 2003 Published online: 17 April 2003 Communicated by A. Kupiainen     

Enantioselective epoxidation of linolenic acid catalysed by cytochrome P450(BM3) from Bacillus megaterium

Cytochrome P450(BM3), from Bacillus megaterium, catalyses the epoxidation of linolenic acid yielding 15,16-epoxyoctadeca-9,12-dienoic acid with complete regio- and moderate enantio-selectivity (60% ee). The absolute configuration of the product is tentatively assigned as 15(R),16(S)-. The Michaelis-Menten parameters kcat and Km for the reaction were determined to be 3126 +/- 226 min(-1) and 24 +/- 6 microM respectively.     

Geodesic Incompleteness in the ℂP 1 Model on a Compact Riemann Surface

It is proved that the moduli space of static solutions of the P ¹ model on spacetime ×, where is any compact Riemann surface, is geodesically incomplete with respect to the metric induced by the kinetic energy functional. The geodesic approximation predicts, therefore, that lumps can collapse and form singularities in finite time in these models.