Bruhat intervals as rooks on skew Ferrers boards

We characterise the permutations π such that the elements in the closed lower Bruhat interval [id,π] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations π such that [id,π] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner.Our characterisation connects the Poincaré polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincaré polynomial of some particularly interesting intervals in the finite Weyl groups An and Bn. The expressions involve q-Stirling numbers of the second kind, and for the group An putting q=1 yields the poly-Bernoulli numbers defined by Kaneko.     

Mechanical properties of nanotubes of polyelectrolyte multilayers

The elastic properties of nanotubes fabricated by layer-by-layer (LbL) assembly of polyelectrolytes in the nanopores of polycarbonate track-etched membranes have been investigated by resonant contact Atomic Force Microscopy (AFM), for nanotube diameters in the range of 100 to 200 nm. The elastic modulus of the nanotubes was computed from the resonance frequencies of a cantilever resting on freely suspended LbL nanotubes. An average value of 115 MPa was found in air for Young's modulus of these nanostructures, well below the values reported for dry, flat multilayers, but in the range of values reported for water-swollen flat multilayers. These low values are most probably due to the lower degree of ionic cross-linking of LbL nanotubes and their consequently higher water content in air, resulting from the peculiar mode of growth of nanoconfined polyelectrolyte multilayers.     

Entanglement generated by dissipation and steady state entanglement of two macroscopic objects

Entanglement is a striking feature of quantum mechanics and an essential ingredient in most applications in quantum information. Typically, coupling of a system to an environment inhibits entanglement, particularly in macroscopic systems. Here we report on an experiment where dissipation continuously generates entanglement between two macroscopic objects. This is achieved by engineering the dissipation using laser and magnetic fields, and leads to robust event-ready entanglement maintained for 0.04 s at room temperature. Our system consists of two ensembles containing about 10(12) atoms and separated by 0.5 m coupled to the environment composed of the vacuum modes of the electromagnetic field. By combining the dissipative mechanism with a continuous measurement, steady state entanglement is continuously generated and observed for up to 1 h.     

A Comparison of Splittings and Integral Equation Solvers for a Nonseparable Elliptic Equation

Iterative numerical schemes for solving the electrostatic partial differential equation with variable conductivity, using fast and high-order accurate direct methods for preconditioning, are compared. Two integral method schemes of this type, based on previously suggested splittings of the equation, are proposed, analyzed, and implemented. The schemes are tested for large problems on a square. Particular emphasis is paid to convergence as a function of geometric complexity in the conductivity. Differences in performance of the schemes are predicted and observed in a striking manner. AMS subject classification (2000) 31A10, 35C15, 65R20.Received May 2004. Accepted September 2004. Communicated by Anders Szepessy.Johan Helsing: This work was supported by the Swedish Science Research Council under contract 621-2001-2799.     

Hydrogenolysis of [PhBP3]Fe[triple bond]N-p-tolyl: probing the reactivity of an iron imide with H2

This paper describes the reductive hydrogenolysis of a low-spin (S = 1/2) iron(III) imide. Pseudotetrahedral [PhBP3]FeIIIN-p-tolyl is reduced by hydrogen at ambient temperature and pressure in benzene solution. The reduction appears to proceed in a stepwise fashion. An intermediate S = 2 iron(II) anilide, [PhBP3]Fe(N(H)-p-tolyl), is observed and has been independently generated and structurally characterized. Prolonged hydrogenolysis in benzene results in the complete hydrogenolysis of the Fe-N linkage to release H2N-p-tolyl. The major iron-containing product formed from this step is the diamagnetic cyclohexadienyl complex, [PhBP3]Fe(eta5-cyclohexadienyl), which has also been independently prepared and structurally characterized. Evidence is presented to suggest that the final [PhBP3]Fe(eta5-cyclohexadienyl) product is formed via benzene insertion into a reactive [PhBP3]FeII-H intermediate.     

Dinitrogen chemistry from trigonally coordinated iron and cobalt platforms

This report establishes that trigonally coordinated "[PhBPiPr3]M" platforms (M = Fe, Co) will support both pi-acidic (N2) and pi-basic (NR) ligands at a fourth binding site. The N2 complexes of iron that are described are the first thoroughly characterized examples to exhibit a 4-coordinate geometry. Methylation of monomeric {Fe0(N2)-} and {Co0(N2)-} species successfully derivatizes the beta-N atom of the coordinated N2 ligand and affords the diazenido products {FeII(N2Me)} and {CoII(N2Me)}, respectively. One-electron oxidation of the mononuclear M0(N2)- species produces dinuclear and synthetically versatile MI(N2)MI complexes. These latter species provide clean access to the chemistry of the "[PhBPiPr3]MI" subunit. For example, addition of RN3 to MI(N2)MI results in oxidative nitrene transfer to generate [PhBPiPr3]MNR with concomitant N2 release.