Spline fitting discontinuous functions given just a few Fourier coefficients

This paper considers the use of polynomial splines to approximate periodic functions with jump discontinuities of themselves and their derivatives when the information consists only of the first few Fourier coefficients and the location of the discontinuities. Spaces of splines are introduced which include, members with discontinuities at those locations. The main results deal with the orthogonal projection of such a spline space on spaces of trigonometric polynomials corresponding to the known coefficients. An approximation is defined based on inverting this projection. It is shown that when discontinuities are sufficiently far apart, the projection is invertible, its inverse has norm close to 1, and the approximation is nearly as good as directL ₂ approximation by members of the spline space. An example is given which illustrates the results and which is extended to indicate how the approximation technique may be used to provide smoothing which which accurately represents discontinuities.     

C0 coarse geometry and scalar curvature

In this paper we introduce an alternative form of coarse geometry on proper metric spaces, which is more delicate at infinity than the standard metric coarse structure. There is an assembly map from the K-homology of a space to the K-theory of the C∗-algebra associated to the new coarse structure, which factors through the coarse K-homology of the space (with the new coarse structure). A Dirac-type operator on a complete Riemannian manifold M gives rise to a class in K-homology, and its image under assembly gives a higher index in the K-theory group. The main result of this paper is a vanishing theorem for the index of the Dirac operator on an open spin manifold for which the scalar curvature κ(x) tends to infinity as x tends to infinity. This is derived from a spectral vanishing theorem for any Dirac-type operator with discrete spectrum and finite dimensional eigenspaces.     

A bounded sequence of normal functionals has a subsequence which is nearly weakly convergent

Let (φ_n) be a norm bounded sequence in the pre-dual of a von Neumann algebra M. In general it is not true that this sequence has a weakly convergent subsequence. But given a normal state ψ, then, for any ε>0, there exists a projection e such that ψ(1−e)?ε and the restriction of (φ_n) to eMe has a subsequence which converges weakly to a normal functional on eMe.     

A new, tetragonal, helical phase of plumbocene, Cp2Pb; variations on a molecular string (Cp=C5H5)

The isolation of a tetragonal, helical phase of plumbocene, Cp₂Pb, provides a rare example of a helical metallocene and the opportunity to shed light on the relationship between the modes of aggregation found in its various polymorphs.     

An implicit edge‐based ALE method for the incompressible Navier–Stokes equations

A new finite volume method for the incompressible Navier–Stokes equations, expressed in arbitrary Lagrangian–Eulerian (ALE) form, is presented. The method uses a staggered storage arrangement for the pressure and velocity variables and adopts an edge‐based data structure and assembly procedure which is valid for arbitrary n‐sided polygonal meshes. Edge formulas are presented for assembling the ALE form of the momentum and pressure equations. An implicit multi‐stage time integrator is constructed that is geometrically conservative to the precision of the arithmetic used in the computation. The method is shown to be second‐order‐accurate in time and space for general time‐dependent polygonal meshes. The method is first evaluated using several well‐known unsteady incompressible Navier–Stokes problems before being applied to a periodically forced aeroelastic problem and a transient free surface problem. Published in 2003 by John Wiley & Sons, Ltd.     

Ionization-enhanced decomposition of 2,4,6-trinitrotoluene (TNT) molecules

The unimolecular decomposition reaction of TNT can in principle be used to design ways to either detect or remove TNT from the environment. Here, we report the results of a density functional theory study of possible ways to lower the reaction barrier for this decomposition process by ionization, so that decomposition and/or detection can occur at room temperature. We find that ionizing TNT lowers the reaction barrier for the initial step of this decomposition. We further show that a similar effect can occur if a positive moiety is bound to the TNT molecule. The positive charge produces a pronounced electron redistribution and dipole formation in TNT with minimal charge transfer from TNT to the positive moiety.     

Oxidative dehydrocoupling of N-H bonds using a redox-active main group superbase

Reaction of the redox-active base Sn(NMe(2))(2)/(n)BuLi with o-phenylene diamine leads to oxidative dehydrocoupling and rearrangement into the triazolyl anion.