A twistor construction of Kähler submanifolds of a quaternionic Kähler manifold

A class of minimal almost complex submanifolds of a Riemannian manifold with a parallel quaternionic structure Q, in particular of a 4-dimensional oriented Riemannian manifold, is studied. A notion of Kähler submanifold is defined. Any Kähler submanifold is pluriminimal. In the case of a quaternionic Kähler manifold of non zero scalar curvature, in particular, when is an Einstein, non Ricci-flat, anti-self-dual 4-manifold, we give a twistor construction of Kähler submanifolds M²ⁿ of maximal possible dimension 2n. More precisely, we prove that any such Kähler submanifold M²ⁿ of is the projection of a holomorphic Legendrian submanifold of the twistor space of , considered as a complex contact manifold with the natural holomorphic contact structure . Any Legendrian submanifold of the twistor space is defined by a generating holomorphic function. This is a natural generalization of Bryants construction of superminimal surfaces in S⁴=P¹. Mathematics Subject Classification (1991) Primary: 53C40; Secondary: 53C55     

Quantization: Deformation and/or Functor?

After a short presentation of the difference in motivation between the Berezin and deformation quantization approaches, we start with a reminder of Berezin’s view of quantization as a functor followed by a brief overview of deformation quantization in contrast with the latter. We end by a short survey of two main avatars of deformation quantization, quantum groups and quantum spaces (especially noncommutative geometry) presented in that perspective.     

On best cubature formulas and spline interpolation

Summary A general cubature formula with an arbitrary preassigned weight function is derived using monosplines and integration by parts. The problem of determining the best cubature is formulated in terms of monosplines of least deviation and a solution to the problem is given by Theorem 3 below. This theorem may also be viewed as an optimal property of a new kind of two-dimensional spline interpolation.This work was done while the author was working at CERN, Geneva, Switzerland     

The numerically stable reconstruction of Jacobi matrices from spectral data

Summary We present an exposé of the elementary theory of Jacobi matrices and, in particular, their reconstruction from the Gaussian weights and abscissas. Many recent works propose use of the diagonal Hermitian Lanczos process for this purpose. We show that this process is numerically unstable. We recall Rutishauser's elegant and stable algorithm of 1963, based on plane rotations, implement it efficiently, and discuss our numerical experience. We also apply Rutishauser's algorithm to reconstruct a persymmetric Jacobi matrix from its spectrum in an efficient and stable manner.Dedicated to Professor F.L. Bauer on the occasion of his 60th birthdayThis work was supported by the National Science Foundation under grant MCS-81-02344, and by the Mathematics Research Institute of the Swiss Federal Institute of Technology, Zürich     

Splineapproximationen von beliebigem Defekt zur numerischen Lösung gewöhnlicher Differentialgleichungen. Teil III

Summary In the first part [1] a general procedure is presented to obtain polynomial spline approximations for the solutions of initial value problems for ordinary differential equations; furthermore a divergence theorem is proved there. Sufficient conditions for convergence of the method are given in the second part [2]. The remaining case which has not been considered in [1] and [2] is treated in the present paper. In this special case the procedure is equivalent to an unstable two-step method with special initial values; nevertheless, convergence can be proved. Finally,A ₀-stability of the method as well as the influence of rounding errors are investigated.

     

Numerical solution of steady-state porous flow free boundary problems

Summary A new numerical method is used to solve stationary free boundary problems for fluid flow through porous media. The method also applies to inhomogeneous media, and to cases with a partial unsaturated flow.     

On the design of high order exponentially fitted formulae for the numerical integration of stiff systems

Summary Single step exponentially fitted integration formulae of orders 4 and 6 are derived. The final approximation to the required solution is obtained via a linear combination of asymptotically less accurate solutions obtained using conventional implicit integration formulae. This linear combination is performed in such a way that the final integration formula, as well as having an increased order of accuracy, integrates a certain pre-determined ordinary differential equation exactly. The algorithms developed are illustrated by means of some numerical examples.     

A general extrapolation algorithm

Summary In this paper a general formalism for linear and rational extrapolation processes is developped. This formalism includes most of the sequence transformations actually used for convergence acceleration. A general recursive algorithm for implementing the method is given. Convergence results and convergence acceleration results are proved. The vector case and some other extensions are also studied.     

Algorithmes pour suites non convergentes

Summary Two algorithms are proposed. For a given non-convergent sequence the first answers the question: How many cluster points does the sequence possess? The second one allows one to extract convergent sequences for any sequence with a finite number of cluster points.It was necessary to use the notion of strength of a cluster point. Two negative theorems show the necessity of the proposed hypotheses.

     

Une représentation intégrale de l'erreur d'interpolation

Résumé La représentation intégrale de l'erreur d'interpolation dans le cas à une ou plusieurs variables est présentée comme une généralisation de la méthode du Wronskien dans le cas à une variable. Ceci est basé sur la construction d'un système d'opérateurs différentiels associé aux fonctions de base du procédé d'interpolation et sur le noyau de Neumann.

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An integral representation of interpolation error

The integral representation of interpolation error for one or several variables is considered as a generalization of Wronskien method for one variable. This is based on the construction of a system of differential operators associated with the basis interpolated functions and on the Neumann's Kernel.