Wrap-around partitioning for block bidiagonal linear systems

A stable vector algorithm for the solution of block diagonallinear systems is obtained by a permutation of the unknownscalled wrap-around partitioning combined with standard QR factorization.Wrap-around partitioning uses blocking and selects the unknownsin the blocks in turns. After a suitable orthogonal eliminationstep one ends up with a reduced system which is again blockbidiagonal and so wrap-around partitioning can be applied again.Using a simple model for vectorization overhead it is shownthat small block sizes give best performance. The minimal blocksize 2, which corresponds to cyclic reduction, is suboptimaldue to memory bank conflicts.     

Error Estimates in the Sampling From Particle Size Distributions

It is often necessary to estimate the properties of particle size distributions from limited samples taken from large populations. When the distributions are broad, and higher order moments required, as in the case of volume based particle size distributions, the inferred parameters d_3,50 (volume median diameter) and GSD (geometric standard deviation) can have high intrinsic errors not immediately obvious to the measuring scientist. We show that there is a critical number of particles, N_crit, which must be counted or else the error may blow up catastrophically. N_crit is very sensitive to the width of the distribution, and is approximately proportional to GSD¹¹ We develop formulae to estimate the random sampling error inherent in measured values of the d_3,50 and GSD for the log-normal distribution; compare the predictions to a typical experimental particle size measurement; and then generalize to the median of any arbitrary moment, d_{r, 50}.     

High-spin and low-spin iron(II) complexes with facially-coordinated borohydride ligands

Rare examples of monometallic high-spin and low-spin L3Fe(H3BH) complexes have been characterized, where the two L3 ligands are [TpPh2] and [PhBP3] ([TpPh2] = [HB(3,5-Ph2pz)3]- and [PhBP3] = [PhB(CH2PPh2)3]-). The structures are reported wherein the borohydride ligand is facially coordinated to the iron center in each complex. Density functional methods have been employed to explain the bonding in these unusual iron(II) centers. Despite the differences in spin states, short Fe-B distances are observed in both complexes and there is significant theoretical evidence to support a substantial bonding interaction between the iron and boron nuclei. In light of this interaction, we suggest that these complexes can be described as (L3)Fe(eta4-H3BH) complexes.     

Chiral N-heterocyclic carbene ligands based on a biisoquinoline template

New chiral imidazolinium salts with tunable steric features, based on a biisoquinoline template, have been developed and structurally characterised using single crystal X-ray crystallography. The trans PdI(2)(NHC)(2) complex was prepared by reaction of the parent H(4) imidazolinium salt with Pd(OAc)(2) in the presence of NaI, and the solid state structure determined by X-ray crystallography. The rigid, chiral, biisoquinoline geometry of the H(4) imidazolinium salt was found to be maintained upon ligand complexation. The sterically unencumbered parent biisoquinoline ligand has been found to give high conversion with modest enantioselectivity in the copper-catalysed asymmetric addition of diethylzinc to cyclohexenone.