Richard courant and the finite element method: A further look

The finite element method has in recent years become one of the most popular and effective numerical procedures for boundary value and eigenvalue problems. In the early 1960s it was found that the method was an independent rediscovery of a simpler idea proposed in 1943 by Richard Courant. The importance of Courant's 1943 paper has been emphasized by G. Strang in 1973 and by others. This note seeks to describe briefly Courant's finite element work which led to his publication of 1943. It is shown that Courant used the finite element ideas as early as 1922 in a proof employing Dirichlet's principle.     

Prolongement d'un courant positif plurisousharmonique

The purpose of this Note is to prove an extension result for a positive plurisubharmonic (Psh) current T defined in the complement of a closed complete pluripolar set A, under the hypothesis that $H_{2p}(\overline{\mathrm{supp}T}\cap A)=0$ and that $d{d}^{c}T$ has a locally finite mass. We prove in the first part an Oka type inequality for a positive Psh current. We then prove an extension result for such a current across an irreductible analytic set, thereby generalizing a result of Siu concerning positive closed current and one of our previous results for a negative Psh current. To cite this article: K. Dabbek, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Richard Rado

Richard Rado was born in Berlin; he was the second son of LeopoldRado, from Budapest. At one stage of his education he had todecide whether to become a concert pianist or a mathematician.He chose the latter in the belief that he could continue withmusic as a hobby, but that he could never treat mathematicsin that way. He studied at the University of Berlin, but alsospent some time in Göttingen. He took a DPh at Berlin withhis thesis ‘Studien zur Kombinatorik’ [3] underIssai Schur in 1933. During this period he was also influencedby Erhard Schmidt. On 16 March 1933, he married Luise Zadek, the elder daughterof Hermann Zadek, whom he had earlier come to know when he neededa partner to play piano duets. It was indeed a remarkable partnership. As Hitler came to power in 1933, the Rados, being Jewish, madetheir way to England, Richard having obtained a scholarshipof £300 p.a. from Sir Robert Mond through the recommendationof Professor Lindemann (later Lord Cherwell), who had interviewedhim in Berlin, to enable him to study at Cambridge.     

Monoid generalizations of the Richard Thompson groups

The groups G_k,1 of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call M_k,1, and to inverse monoids, called ; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids M_k,1 have connections with circuit complexity (studied in other papers). Here we prove that M_k,1 and are congruence-simple for all k. Their Green relations J and D are characterized: M_k,1 and are J-0-simple, and they have k−1 non-zero D-classes. They are submonoids of the multiplicative part of the Cuntz algebra O_k. They are finitely generated, and their word problem over any finite generating set is in P. Their word problem is coNP-complete over certain infinite generating sets.     

Determination of the thermal stresses in anisotropic shells of revolution

Using the relations of the improved model of layered anisotropic shells based on the straight line assumption taking account of the thermal compression over the thickness we obtain a resolvent system of equations for shells whose properties depend on temperature. We carry out a study of the stresses in a two-layer cylindrical shell formed by winding as a function of the winding angle. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 82–85.     

Lattice points in bodies of revolution II

In [3] it was shown that when a three-dimensional smooth convex body has rotational symmetry around a coordinate axis one can find better bounds for the lattice point discrepancy than what is known for more general convex bodies. To accomplish this, however, it was necessary to assume a non-vanishing condition on the third derivative of the generatrix. In this article we drop this condition, showing that the aforementioned bound holds for a wider family of revolution bodies, which includes those with analytic boundary. A novelty in our approach is that, besides the usual analytic methods, it requires studying some Diophantine properties of the Taylor coefficients of the phase on the Fourier transform side.     

Trois formulations d'un modèle de plasma quasi-neutre avec courant non nul

In this Note, we propose three formulations of a model describing a quasi-neutral plasma with non-vanishing current. In order to study and compare the numerical efficiency of each formulation, two test-problems are implemented in one dimension. The first is a periodic perturbation of a uniform stationary plasma. The second is a case of plasma expansion in vacuum between two electrodes. To cite this article: P. Crispel et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Optimally reinforced shells of revolution

The design of filament-wound multilayer shells of revolution of arbitrary shape subjected to axisymmetric surface and body forces is examined. Three cases of existence of a solution of the optimal problem are established and certain particular types of solutions are presented.     

Axisymmetric deformation of bimetallic shells of revolution in the supercritical region

Geometrically nonlinear relationships of the theory of thin layered shells are applied to analyze axisymmetric strain of bimetallic shells of revolution in a temperature field. One-dimensional nonlinear boundary-value problems are solved by a combination of the linearization method and the discrete orthogonalization method. A numerical approach is proposed to solve the boundary-value problems in the supercritical strain region.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 52–56, 1990.