Rational approximation to trigonometric operators

We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments. AMS subject classification (2000)  65F10, 65L60, 65M60, 65N22     

Exponential multistep methods of Adams-type

The paper is concerned with the construction, implementation and numerical analysis of exponential multistep methods. These methods are related to explicit Adams methods but, in contrast to the latter, make direct use of the exponential and related matrix functions of a (possibly rough) linearization of the vector field. This feature enables them to integrate stiff problems explicitly in time.     

Studies on the Fast Atom Bombardment Mass Spectrometry Fragmentation of Amino Sugars Using Synthetic Derivatives of the Trisaccharide Unit β-D-Glucopyranosyl-(1-3)-O-2-deoxy-2-amino-β-d-glucopyranosyl-(1-4)-O-d-glucopyranose

Abstract

N-Phthaloyl, N-acetyl, N-benzyl, N-acetyl-N-methyl, N,N-dimethyl, N-benzoyl, and N,N-dibenzoyl derivatives of the trisaccharide β-D-glucopyranosyl-(1-3)-O-(2-deoxy-2-amino-β-D-glucopyranosyl)-(1-4)-O-β-D-glucopyranose were synthesized and analyzed by FAB MS. The intensity ratios of the peaks resulting from cleavage of the anomeric bond of the glucosamine residue and the respective molecular ion peaks turned out to be high for the N-acyl derivatives and up to two orders of magnitude lower for the N-alkyl compounds. These results show that fragmentation at the anomeric carbon of the amino sugar may be assisted by the carbonyl group and the resulting cation is stabilized by delocalization of the positive charge.     

On the use of two QMR algorithms for solving singular systems and applications in Markov chain modeling

Recently, Freund and Nachtigal proposed the quasi-minimal residual algorithm (QMR) for solving general nonsingular non-Hermitian linear systems. The method is based on the Lanczos process, and thus it involves matrix—vector products with both the coefficient matrix of the linear system and its transpose. Freund developed a variant of QMR, the transpose-free QMR algorithm (TFQMR), that only requires products with the coefficient matrix. In this paper, the use of QMR and TFQMR for solving singular systems is explored. First, a convergence result for the general class of Krylov-subspace methods applied to singular systems is presented. Then, it is shown that QMR and TFQMR both converge for consistent singular linear systems with coefficient matrices of index 1. Singular systems of this type arise in Markov chain modeling. For this particular application, numerical experiments are reported.     

Reactions of conjugated dienes with cobalt carbonyls under hydroformylation conditions. a high pressure i.r. spectroscopic study

Summary The chemistry of cobalt carbonyls in the presence of dienes and high pressure of synthesis gas was studied by online i.r. spectroscopy. Dicobalt octacarbonyl reacts with butadiene under 95 bar CO/H₂ and 80°C to give [³-C₄H₇Co(CO)₃] (1) and [⁴-C₄H₆)₂Co₂(CO)₄] (2). Hydrogenation or hydroformylation are observed only with [HCo(CO)₄] as the starting catalyst, and only at the beginning of the reaction. The results are explained by formation of an alkenyl complex, [-C₄H₇Co(CO)₄], which either reacts with [HCo(CO)₄] to give butene and [Co₂(CO)₈], or loses CO to give (1), depending on the [HCo(CO)₄] concentration. The butene is hydroformylated. At temperatures >100°C (1) is transformed into a CO-free species, which catalyzes the oligomerisation of butadiene. Addition of tributylphosphine (L) leads to the formation of [³-C₄H₇Co(CO)₂L] (5) and [Co₂(CO)₆L₂] (6). In (5) the -allyl moiety is more labile than in (1) and a slow hydrogenation and hydroformylation of the butadiene is observed. In methanol solution the reaction of the cobalt carbonyls to give (1) is incomplete and the remaining H⁺ and [Co(CO)₄]^– catalyze the hydroformylation of butadiene. Isoprene is less reactive than butadiene but otherwise behaves similarly.     

Yttria-stabilized zirconia films deposited by plasma spraying and sputtering

Thin films of yttria-stabilized zirconia (YSZ) deposited by sputtering and plasma spraying have been analysed as solid electrolyte for oxygen gas sensors and solid oxide fuel cells. Different substrates have been considered in order to provide good adhesion, dense electrolyte films, and mechanical and thermal stability. By optimization of the sputtering parameters, a Pt/YSZ/Pt assembly of 1.5 µm thickness and with an electrical resistance of 500 M at room temperature has been obtained. The plasma-sprayed films (150 µm thickness) have shown oxygen ion conductivity with a reproducible sensitivity on oxygen partial pressure starting at 400 °C and an activation energy of ~1 eV. Lanthanum strontium manganate (350 µm thickness) was sprayed as cathode material.Presented at the OSSEP Workshop Ionic and Mixed Conductors: Methods and Processes, Aveiro, Portugal, 10–12 April 2003     

Exponential Integrators for Quantum-Classical Molecular Dynamics

We study time integration methods for equations of mixed quantum-classical molecular dynamics in which Newtonian equations of motion and Schrödinger equations are nonlinearly coupled. Such systems exhibit different time scales in the classical and the quantum evolution, and the solutions are typically highly oscillatory. The numerical methods use the exponential of the quantum Hamiltonian whose product with a state vector is approximated using Lanczos' method. This allows time steps that are much larger than the inverse of the highest frequencies.We describe various integration schemes and analyze their error behaviour, without assuming smoothness of the solution. As preparation and as a problem of independent interest, we study also integration methods for Schrödinger equations with time-dependent Hamiltonian.     

A note on conjugate-gradient type methods for indefinite and/or inconsistent linear systems

To design a conjugate-gradient type method for a given linear system one has to choose an inner product space and to compute residual polynomials which minimize the induced norm. Here we propose a unified treatment of this approach for symmetric linear systems. We mainly focus on indefinite and/or inconsistent systems.     

A Gautschi-type method for oscillatory second-order differential equations

Summary. We study a numerical method for second-order differential equations in which high-frequency oscillations are generated by a linear part. For example, semilinear wave equations are of this type. The numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly. We prove that the method admits second-order error bounds which are independent of the product of the step size with the frequencies. Our analysis also provides new insight into the m ollified impulse method of García-Archilla, Sanz-Serna, and Skeel. We include results of numerical experiments with the sine-Gordon equation. Received January 21, 1998 / Published online: June 29, 1999     

On the mechanism of the Co2(CO)8 catalyzed hydroformylation of olefins in hydrocarbon solvents. A high pressure UV and IR study

High pressure IR and UV spectroscopic experiments confirm the Heck and Breslow mechanism of the hydroformylation of 1-octene and cyclohexene with Co₂(CO)₈ as the starting catalyst. The major repeating unit is HCo(CO)₄, which is formed via the reaction of acylcobalt tetracarbonyl with H₂. The rates are 6.7 × 10^?4 mol l^?1 min^?1 and 8.8 × 10^?5 mol l^?1 min^?1 for 1-octene and cyclohexene, respectively at 80°C and 95 bar CO/H₂ = 1 in methylcyclohexane. The alternative reaction of RCOCo(CO)₄ with HCo(CO)₄ is only a minor pathway, with rates of 1.8 × 10^?5 mol l^?1 min^?1 and 1.1 × 10^?5 mol l^?1 min^?1 for 1-octene and cyclohexene, respectively. It represents an exit from the catalytic cycle. The activation of the catalyst precursor Co₂(CO)₈ is the slowest step of the reaction.