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The general implicit finite-difference approximation of second order devised by von Neumann is applied to the initial boundary value problem for a somewhat generalized wave equation. By means of eigenvalue expansion it is shown that the method is uniform convergent of order $$\log \left( {\Delta x^{ - 1} } \right)O\left( {\Delta t^2 + \Delta x^2 } \right)$$ Δt, Δx mesh widths). Moreover, the convergence on the linet=T reveals to be proportional toT.

Bounds for the pointwise or Schur product of two matrices are derived with respect to the spectral norm ‖·‖. For real symmetric and positive semidefinite matrices Q =( q ij ) one of them gives a bound of ‖| Q ‖, | Q |=(| q ij |). Two of these bounds are applied to obtain a mean value theorem for g : t → f ( A ( t )) where A ( t ) is a matrix depending on a parameter t and f is a function on the spectrum of A ( t ).

Backward differentiation methods up to orderk=5 are applied to solve linear ordinary and partial (parabolic) differential equations where in the second case the space variables are discretized by Galerkin procedures. Using a mean square norm over all considered time levels a-priori error estimates are derived. The emphasis of the results lies on the fact that the obtained error bounds do not depend on a Lipschitz constant and the dimension of the basic system of ordinary differential equations even though this system is allowed to have time-varying coefficients. It is therefore possible to use the bounds to estimate the error of systems with arbitrary varying dimension as they arise in the finite element regression of parabolic problems.

The approximation of linear systemsy?=?A(t)y+b(t) by backward differentiation methods up to order 5 is considered. It is proved that the error does not increase if the real symmetric matrixA(t) is positive definite andA?(t) is negative semi-definite.

Runge-Kutta methods have been generalized to procedures with higher derivatives of the right side ofy′=f(t,y) e.g. by Fehlberg 1964 and Kastlunger and Wanner 1972. In the present work some sufficient conditions for the order of consistence are derived for these methods using partially the degree of the corresponding numerical integration formulas. In particular, methods of Gaus, Radau, and Lobatto type are generalized to methods with higher derivatives and their maximum order property is proved. The applied technique was developed by Crouzeix 1975 for classical Runge-Kutta methods. Examples of simple explicit and semi-implicit methods are given up to order 7 and 6 respectively.

Am Beispiel des Cyclohexans wird uber die besten Reaktionsbedingungen sowie uber den Mechanismus der Photo-Sulfenchlorierung berichtet. Das Abstraktionsverhaltnis von Wasserstoff in CH3-bzw. CH2-Gruppen der n-Alkane entspricht dem bei der Photo-Chlorierung und -Cyanierung. Die Ergebnisse der Photo-Sulfenchlorierung des Cyclohexylchlorids und einige weitere Reaktionen des Cyclohexan- sulfensaurechlorids werden mitgeteilt.

Cycloaliphatische gesattigte Kohlenwasserstoffe werden mit Schwefel(II)-chlorid in Gegenwart von Licht der Wellenlangen 260–600 nm zu Cycloakan- sulfensaurechloriden umgesetzt. Die so erhaltenen Cycloalkyl- chlor-sulfane erlauben dank ihrer grosen Reaktionsfahigkeit vielfaltige Umsetzungen zu anderen Schwefelverbindungen. Die Photoreaktion ist auf gesattigte offenkettige Kohlenwasserstoffe ubertragbar.

An optical surface is generated with a very high accuracy with regard to its macro and micro geometry so that rays falling onto this surface are deflected with image-forming quality. Materials for optical elements must have high transmission or reflectivity in a given spectral range. These requirements are met by some crystals, optical glasses and plastics and some metals. There are different methods for generating optical surfaces: - precision moulding and pressing - turning or flycutting - grinding, lapping and polishing Moulding and pressing are applied to optical plastics and to glasses. Normally elements produced in this way fulfil only low requirements with regard to surface quality and homogeneity. However precision moulding of glass lenses with remarkable high quality could be achieved recently. On soft materials optical surfaces can be generated by precision diamond turning or flycutting. These techniques require high-sophisticated design of computer controlled machines. Optical glasses cannot be machined by single point diamond turning because of the hardness and brittleness. For special applications the problems are solved by turning the glass at an elevated temperature. Under certain conditions several types of optical glasses can be turned to a transparent finish. Also in the classical methods for generating optical surfaces, i.e. gringing and polishing, changes can be observed at present. This relates to rigidity and precision in machine construction, to materials and geometry of grinding and polishing tools and to polishing agents.© (1983) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

The present paper is concerned with parabolic differential equations ut-L(x,t;u)=0 where L denotes a quasilinear strongly elliptic operator of second ordr. L(x,t;u)=0 is supposed to be the Euler equation of a corresponding variational problem for which a discrete analogue is constructed. To the resulting semi-discrete problem Crank-Nicolson’s method, a backward differentiation method, and the boundary value method suggested by Greenspan are applied and their convergence is proved.