A numerical method for the Benjamin-Ono equation

This paper is concerned with the numerical solution of the Cauchy problem for the Benjamin-Ono equationu _t +uu _x −Hu _xx =0, whereH denotes the Hilbert transform. Our numerical method first approximates this Cauchy problem by an initial-value problem for a corresponding 2L-periodic problem in the spatial variable, withL large. This periodic problem is then solved using the Crank-Nicolson approximation in time and finite difference approximations in space, treating the nonlinear term in a standard conservative fashion, and the Hilbert transform by a quadrature formula which may be computed efficiently using the Fast Fourier Transform.     

Maximum-Norm Estimates for Finite-Element Methods for a Strongly Damped Wave Equation

We return to earlier work in Larsson, Thomée, and Wahlbin, IMA J. Numer. Anal. 11 (1991), concerning the numerical solution of a homogeneous linear wave equation with strong damping, arising in viscoelasticity. In that work spatial discretization by finite elements and associated fully discrete methods were analyzed in L ₂-based norms. The analysis depended on the fact that the solution may be expressed in terms of an analytic semigroup. In the present work we combine this approach with recent results on discretization of parabolic problems to derive essentially optimal order error bounds in maximum-norm for piecewise linear finite elements combined with backward Euler and Crank–Nicolson time stepping methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Filiform corrosion of aluminium alloys after a phosphate treatment

The role on filiform corrosion (FFC), after an automotive painting process with a phosphate treatment, of the near surface deformed layer (NSDL) formed during mechanical processing on AA8006 aluminium alloys is investigated. Samples subjected to specific surface pretreatments are examined before and after different steps of the phosphating process in order to identify the characteristics of the NSDL that are responsible for the susceptibility to FFC. The thickness, the structure and the composition of this NSDL are determined by SEM, RBS, XPS and transmission electron microscopy. The presence of a 200‐nm thick NSDL which is not completely eliminated during the degreasing step of the phosphating process increases dramatically the FFC susceptibility of AA8006 aluminium alloys compared with beforehand etched samples. The influence of Pb and Mg segregation on top of the surface is found to be negligible whereas the specific microstructure of the NSDL characterised by nanograins with boundaries decorated by nano‐sized oxide intermetallic particles seems to play a major role on FFC. Copyright © 2014 John Wiley & Sons, Ltd.     

An Organic–Inorganic Hybrid Exhibiting Electrical Conduction and Single‐Ion Magnetism

The first three‐dimensional (3D) conductive single‐ion magnet (SIM), (TTF)₂[Co(pdms)₂] (TTF=tetrathiafulvalene and H₂pdms=1,2‐bis(methanesulfonamido)benzene), was electrochemically synthesised and investigated structurally, physically, and theoretically. The similar oxidation potentials of neutral TTF and the molecular precursor [HNEt₃]₂[M(pdms)₂] (M=Co, Zn) allow for multiple charge transfers (CTs) between the SIM donor [M(pdms)₂]^n? and the TTF^.+ acceptor, as well as an intradonor CT from the pdms ligand to Co ion upon electrocrystallisation. Usually TTF functions as a donor, whereas in our system TTF is both a donor and an accepter because of the similar oxidation potentials. Furthermore, the [M(pdms)₂]^n? donor and TTF^.+ acceptor are not segregated but strongly interact with each other, contrary to reported layered donor–acceptor electrical conductors. The strong intermolecular and intramolecular interactions, combined with CT, allow for relatively high electrical conductivity even down to very low temperatures. Furthermore, SIM behaviour with slow magnetic relaxation and opening of hysteresis loops was observed. (TTF)₂[Co(pdms)₂] ( 2‐Co ) is an excellent building block for preparing new conductive SIMs.     

Approximate solution of u′=–A2u using a relation between exp(–tA2) and exp(itA)

Let A be a positive definite operator in a Hilbert space and consider the initial value problem for u _t =–A² u. Using a representation of the semi group exp(–A² t) in terms of the group exp(iAt) we express u in terms of the solution of the standard heat equation w _t = w _yy , with initial values v solving the initial value problem for v _y = iAv. This representation is used to construct a method for approximating u in terms of approximations of v. In the case that A is a 2^nd order elliptic operator the method is combined with finite elements in the spatial variable and then reduces the solution of the 4th order equation for u to that of the 2^nd order equation for v, followed by the solution of the heat equation in one space variable.     

Metric ultraproducts of finite simple groups

Some new results on metric ultraproducts of finite simple groups are presented. Suppose that G is such a group, defined in terms of a non-principal ultrafilter ω   on N$\mathbb{N}$ and a sequence (G_i)_i∈N${(G_i)}_{i\in \mathbb{N}}$ of finite simple groups, and that G is neither finite nor a Chevalley group over an infinite field. Then G is isomorphic to an ultraproduct of alternating groups or to an ultraproduct of finite simple classical groups. The isomorphism type of G determines which of these two cases arises, and, in the latter case, the ω  -limit of the characteristics of the groups G_i$G_i$. Moreover, G is a complete path-connected group with respect to the natural metric on G.     

Parabolic finite volume element equations in nonconvex polygonal domains

We study spatially semidiscrete and fully discrete finite volume element approximations of the heat equation with homogeneous Dirichlet boundary conditions in a plane polygonal domain with one reentrant corner. We show that, as a result of the singularity in the solution near the reentrant corner, the convergence rate is reduced from optimal second order, similarly to what was shown for the finite element method in the earlier work 2 . Optimal order convergence may be restored by mesh refinement near the corners of the domain. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Angular correlations in proton-proton collisions producing a high transverse momentum π0

In an experiment at the CERN ISR, a streamer chamber detector surrounding one of the intersection regions was triggered on large transverse momentum π⁰'s by means of an array of lead-glass counters. The directions of charged particles and γ rays converted in lead-oxide plates inside the streamer chamber were measured. Data were taken at c.m. energy of √s = 53 GeV at two production angles of the high p_Tπ⁰ (90° and 53°). They indicate an enhancement of particles mostly in the hemisphere opposite to the π⁰. In the 53° data, a shift of this enhancement towards rapidities opposite to the rapidity of the π⁰ and confined to a ±30° azimuthal region around the collision plane is observed. In addition, a short-range angular correlation is evidenced between the high p_Tπ⁰ and the other collisions products (photons or charged particles). Two-particle correlations between charged particles produced in association with the high p_Tπ⁰ are found similar to those