The formation and molecular structure of acetylcyclopentadienylsodium-tetrahydrofuranate

Acetylcyclopentadienylsodium has been isolated in crystalline form as a THF adduct from a reaction between cyclopentadienylsodium and methyl acetate in THF solution. The product has been characterized by means of a single-crystal X-ray diffraction study. {[C₅H₄CMeO]Na·THF}_n crystallizes in the monoclinic space group P2₁/c with unit cell parameters a 6.698(3), b 16.095(4), c 10.661(3) Å, β 92.93(3)° and D_c 1.17 g cm^?3 for Z = 4. Least-squares refinement led to a final R value of 0.080 based on 661 independent observed reflections. The coordination sphere around each sodium atom consists of the oxygen atoms from two C₅H₄CMeO ligands, the oxygen atom of the THF molecule, and an ion contact pair between the sodium and the five ring carbon atoms of the C₅H₄CMeO ligand.     

The charge-charge flux-overlap model for the interpretation of atomic polar tensors and infrared intensities

The interpretation of infrared intensities using the concepts of atomic polar tensors divided according to contributions from charge-charge flux-overlap (CCFO) terms appears in general to be quite useful. Here we shall illustrate this analysis for the F and H atoms in chemically related molecules to show how their properties vary for CF, CH, OF and OH bonds as R varies in RCF, RCH and ROH molecules.     

Sharp Morawetz estimates

We prove sharp Morawetz estimates — global in time with a singular weight in the spatial variables — for the linear wave, Klein-Gordon, and Schrödinger equations, for which we can characterise the maximisers. We also prove refined inequalities with respect to the angular integrability.     

Irregularities of distribution with respect to strips

Summary A lower bound is established for the strip discrepancy of a broad class of point distributions. The bound implies unbounded strip discrepancy for equally weighted point distributions under favorable conditions. The methods of proof use notions from integral geometry.     

Optimal and robust contracts for a risk-constrained principal

The theory of risk measurement has been extensively developed over the past ten years or so, but there has been comparatively little effort devoted to using this theory to inform portfolio choice. One theme of this paper is to study how an investor in a conventional log-Brownian market would invest to optimize expected utility of terminal wealth, when subjected to a bound on his risk, as measured by a coherent law-invariant risk measure. Results of Kusuoka lead to remarkably complete expressions for the solution to this problem. The second theme of the paper is to discuss how one would actually manage (not just measure) risk. We study a principal/agent problem, where the principal is required to satisfy some risk constraint. The principal proposes a compensation package to the agent, who then optimises selfishly ignoring the risk constraint. The principal can pick a compensation package that induces the agent to select the principal’s optimal choice.     

Indecomposable continua and the Julia sets of polynomials, II

We find necessary and sufficient conditions for the connected Julia set of a polynomial of degree d?2 to be an indecomposable continuum. One necessary and sufficient condition is that the impression of some prime end (external ray) of the unbounded complementary domain of the Julia set J has nonempty interior in J. Another is that every prime end has as its impression the entire Julia set. The latter answers a question posed in 1993 by the second two authors.We show by example that, contrary to the case for a polynomial Julia set, the image of an indecomposable subcontinuum of the Julia set of a rational function need not be indecomposable.     

Fastest Coupling of Random Walks

A new coupling of one-dimensional random walks is describedwhich tries to control the coupling by keeping the separationof the two random walks of constant sign. It turns out thatamong such monotone couplings there is an optimal one-step couplingwhich maximises the second moment of the difference (assumingthis is finite), and this coupling is ‘fast’ inthe sense that for a random walk with a unimodal step distributionthe coupling time achieved by using the new coupling at eachstep is stochastically no larger than any other coupling. Thisis applied to the case of symmetric unimodal distributions.     

Unified semi‐analytical wall boundary conditions for inviscid,laminar or turbulent flows in the meshless SPH method

Wall boundary conditions in smoothed particle hydrodynamics (SPH) is a key issue to perform accurate simulations. We propose here a new approach based on a renormalising factor for writing all boundary terms. This factor depends on the local shape of a wall and on the position of a particle relative to the wall, which is described by segments (in two‐dimensions), instead of the cumbersome fictitious or ghost particles used in most existing SPH models. By solving a dynamic equation for the renormalising factor, we significantly improve traditional wall treatment in SPH, for pressure forces, wall friction and turbulent conditions. The new model is demonstrated for cases including hydrostatic conditions for still water in a tank of complex geometry and a dam break over triangular bed profile with sharp angle where significant improved behaviour is obtained in comparison with the conventional boundary techniques. The latter case is also compared with a finite volume and volume‐of‐fluid scheme. The performance of the model for a two‐dimensional laminar flow in a channel is demonstrated where the profiles of velocity are in agreement with the theoretical ones, demonstrating that the derived wall shear stress balances the pressure gradient. Finally, the performance of the model is demonstrated for flow in a schematic fish pass where both the velocity field and turbulent viscosity fields are satisfactorily reproduced compared with mesh‐based codes. Copyright © 2012 John Wiley & Sons, Ltd.     

Identities for the Ramanujan zeta function

We prove formulas for special values of the Ramanujan tau zeta function. Our formulas show that L(Δ,k)$L(\mathrm{\Delta },k)$ is a period in the sense of Kontsevich and Zagier when k?12$k?12$. As an illustration, we reduce L(Δ,k)$L(\mathrm{\Delta },k)$ to explicit integrals of hypergeometric and algebraic functions when k∈{12,13,14,15}$k\in \{12,13,14,15\}$.