The many limits of mixed means

The sequences introduced by Carlson (1971) are variants of the Gauss arithmetic geometric sequences (which have been elegantly discussed by D. A. Cox (1984, 1985)). Given (complex)a ₀,b ₀ we define

What do multistep methods approximate?

Summary Standard analysis of multistep methods for ODE's assumes the application of an initialization routine that generates the starting points. Here ak-step method is considered directly as a mappingR ^kn R ⁿ . It is shown to approximate a mapping which is expressible directly in terms of the flow of the vector field. Some useful properties of that mapping are shown and for strictly stable methods these are applied to the question of invariant circles near a hyperbolic periodic solution.     

Convolution quadrature and discretized operational calculus. I

Numerical methods are derived for problems in integral equations (Volterra, Wiener-Hopf equations) and numerical integration (singular integrands, multiple time-scale convolution). The basic tool of this theory is the numerical approximation of convolution integrals

单晶硅化物及其应用

本文评述了单晶CoSi_2和NiSi_2的结构特点、各种制备方法、器件应用和发展前景。采用分子束外延(MBE)和“内延”法(Mesotaxy)制备的单晶硅化物质量,电学性能和热稳定性较好。由于首次得到理想的突变的金属——半导体接触,使对金属——半导体接触的理论分析成为可能。硅/单晶硅化物/硅结构在实际应用中非常重要,如单晶硅化物作集电极埋层,能降低集电极串联电阻,克服重掺杂埋层的横扩和自掺杂问题,提高了电路工作速度,减小了器件面积。埋层硅化物也可作为微波传输线的地线,是实现高频集成电路互连的好方法。而采用该结构制备的高速器件——金属基区晶体管(MBT)和穿透基区晶体管(PBT),具有很好的应用前景。     

A particle method for first-order symmetric systems

Summary We present and study a conservative particle method of approximation of linear hyperbolic and parabolic systems. This method is based on an extensive use of cut-off functions. We prove its convergence inL ² at the order as soon as the cut-off function belongs toW ^m+1.1.Dedicated to Professor Joachim Nitsche on the occasion of his 60th birthday     

On orthogonal linear ℓ1 approximation

Summary The problem is considered of orthogonal ₁ fitting of discrete data. Local best approximations are characterized and the question of the robustness of these solutions is considered. An algorithm for the problem is presented, along with numerical results of its application to some data sets.     

Das asymptotische Verhalten der Peanokerne einiger interpolatorischer Quadraturverfahren

Summary Interpolatory quadrature formulae consist in replacing by wherep _f denotes the interpolating polynomial off with respect to a certain knot setX. The remainder may in many cases be written as wherem=n resp. (n+1) forn even and odd, respectively. We determine the asymptotic behaviour of the Peano kernelP _X (t) forn for the quadrature formulae of Filippi, Polya and Clenshaw-Curtis.

     

Mixed finite element approximation of the vector potential

Summary We consider a mixed finite element approximation of the three dimensional vector potential, which plays an important rôle in the simulation of perfect fluids and in the calculation of rotational corrections to transonic potential flows. The central point of our approach is a saddlepoint formulation of the essential boundary conditions. In particular, this avoids the wellknown Babuka paradox when approximating smooth domains by polyhedrons. Using piecewise linear/piecewise constant elements for the vector potential/the boundary terms, we obtain optimal error estimates under minimal regularity assumptions for the solution of the continuous problem.     

The average a posteriori error of numerical methods

Summary The definition of the average error of numerical methods (by example of a quadrature formula to approximateS(f)= f d on a function classF) is difficult, because on many important setsF there is no natural probability measure in the sense of an equidistribution. We define the average a posteriori error of an approximation by an averaging process over the set of possible information, which is used by (in the example of a quadrature formula,N(F)={(f(a ₁), ...,f/fF} is the set of posible information). This approach has the practical advantage that the averaging process is related only to finite dimensional sets and uses only the usual Lebesgue measure. As an application of the theory I consider the numerical integration of functions of the classF={f:[0,1]/f(x)–f(y)||x–y|}. For arbitrary (fixed) knotsa _i we determine the optimal coefficientsc _i for the approximation and compute the resulting average error. The latter is minimal for the knots . (It is well known that the maximal error is minimal for the knotsa _i .) Then the adaptive methods for the same problem and methods for seeking the maximum of a Lipschitz function are considered. While adaptive methods are not better when considering the maximal error (this is valid for our examples as well as for many others) this is in general not the case with the average error.     

On Bateman's method for second kind integral equations

Summary Under suitable conditions, we prove the convergence of the Bateman method for integral equations defined over bounded domains inR ^d ,d1. The proof makes use of Hilbert space methods, and requires the integral operator to be non-negative definite. For one-dimensional integral equations over finite intervals, estimated rates of convergence are obtained which depend on the smoothness of the kernel, but are independent of the inhomogeneous term. In particular, for aC kernel andn reasonably spaced Bateman points, the convergence is shown to be faster than any power of 1/n. Numerical calculations support this result.