A note on finite difference discretizations for Poisson equation on a disk

A simple second‐order finite difference treatment of polar coordinate singularity for Poisson equation on a disk is presented. By manipulating the grid point locations, we can successfully avoid finding numerical boundary condition at the origin so that the resulting matrix is simpler than traditional schemes. The treatments for Dirichlet and Neumann boundary problems are slightly different by adjusting the radial mesh width. Furthermore, the present discretization can be applied with slight modifications to Helmholtz‐type equations. The numerical evidence shows that the second‐order accuracy can also be preserved. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 199–203, 2001     

Mixed finite elements for incompressible magneto-hydrodynamics

We present a new mixed finite element discretization for three-dimensional stationary incompressible magneto-hydrodynamics. The fluid variables are discretized by standard inf–sup stable velocity–pressure pairs and the magnetic variables by a mixed approach using Nédélec's elements of the first kind. The resulting method is shown to be quasi-optimally convergent. To cite this article: A. Schneebeli, D. Schötzau, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A note on a boundary-layer collision on a rotating sphere

A recent study of the unsteady boundary layer near the equator of a spinning sphere provided strong numerical evidence that the solution develops a singularity a finite time after the motion is initiated. In this paper we reformulate and complete the asymptotic structure proposed to describe this singularity. It is then self-consistent and provides convincing evidence of the accuracy of the computations and of the existence of the singularity.

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Zusammenfassung Eine neuere Untersuchung über die nicht-stationäre Grenzschicht nahe am Äquator einer rotierenden Kugel ergab starke numerische Evidenz für das Auftreten einer Singularität in der Lösung nach einer endlichen Zeit, vom Beginn der Bewegung gerechnet. In dieser Arbeit wird die asymptotische Struktur zur Beschreibung der Singularität neu formuliert und ergänzt. Sie wird selbstkonsistent und gibt ein wichtiges Zeugnis für die Genauigkeit der Berechnungen und für die Existenz der Singularität.

     

A note on the rotating chain problem

For a non-linear boundary value problem describing the shape of a heavy rotating chain a priori estimates of the maximal deflection are obtained. Also, for the transformed chain problem an extremum Variational principle and error estimates are presented.

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Zusammenfassung Für eine nichtlineare Differentialgleichung, die die Form einer schwer rotierenden Kette beschreibt, werden a priori Abschätzungen der maximalen Ausbiegung erhalten. Dabei werden für ein transformiertes Ketten-Problem ein extremales Variationsprinzip und die Feller Abschätzung angegeben.

     

A note on quantization in rotating co-ordinates

The quantization of scalar fields in a uniformly rotating frame is reconsidered. It is pointed out that the coordinates usually used do not represent a frame describing uniform circular motion and the metric in the correct coordinates has a singularity. The result obtained earlier with the wrong coordinates, that there is no radiation, nevertheless holds.

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Zusammenfassung Die Quantisierung skalarer Felder in einem gleichförmig rotierenden Koordinatensystem wird erneut betrachtet. Dabei wird hervorgehoben, daß die gewöhnlich verwendeten Koordinaten die gleichförmige Kreisbewegung nicht beschreiben und daß die Metrik in den richtigen Koordinaten eine Singularität aufweist. Trotzdem bleibt das früher mit den falschen Koordinaten abgeleitete Resultat erhalten, daß nämlich ein strahlungsfreier Zustand existiert.

     

A note on finite abelian covers

We prove that any abelian cover over a smooth variety is defined by some cyclic equations. From the defining equations, we compute explicitly the normalization, branch locus, ramification indices, global invariants, and the resolution of singularities. As an application, we construct a new algebraic surface which is the quotient of ball.     

A note on finite element wavelets

1. IotroductionThe study of mu1ti-wavelets was initiated by Goodman, Lee and Tang[2]. Later, acharacterization of sca1ing functions and wavelets was established by Goodman and Lee[3].In [4] GoodInan constructed multi-wavelets that satisfy certain interpolating conditions.Geronimo, Hardin and Massopust[5] used fractal interpolation to construct orthogonal scal-ing functions gh1 and gh2 (with multiplicity r = 2) that are both symmetric and colltinuousbut nondifferentiable. Strang and Strela…     

A note on finite local rings

We prove that a finite local ring with one minimal left ideal is a QF-ring with one minimal right ideal, thus answering a question of Claasen and Goldbach in the negative.     

A note on surface waves created by a rotating circular cylinder

A circular cylinder, placed vertically in a liquid with a free surface, makes oscillatory rotations about its axis. We find that small amplitude, radially progressive waves are created on the surface of the liquid that move a finite distance away from the cylinder.

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Résumé Nous considérons les oscillations axiales d'un cylindre placé vérticalement dans une liquide à surface libre, et démontrons l'existence des ondes radiales à faible amplitude sur la surface s'eloignant à une distance finie du cylindre.

     

Two-layer film flow over a rotating disk

Unsteady two-layer liquid film flow on a horizontal rotating disk is analyzed using asymptotic method for small values of Reynolds number. This analysis of non-linear evolution equation elucidates how a two-layer film of uniform thickness thins when the disk is set in uniform rotation. It is observed that the final film thickness attains an asymptotic value at large time. It is also established that viscous force dominates over centrifugal force and upper layer thins faster than lower layer at large time.     

A note on TI-subgroups of finite groups

A subgroup H of a finite group G is called a TI-subgroup if H ∩ H ^x  = 1 or H for any x ∈ G. In this short note, the finite groups all of whose nonabelian subgroups are TI-subgroups are classified.     

Barodiffusion in a rotating binary mixture over an infinite rotating disk

Barodiffusion (diffusion of species brought about by pressure-gradients) in an isothermal, incompressible, Newtonian binary mixture in a steady, laminar, axisymmetric, rotating motion over an infinite, rotating, impermeable disk is considered. In view of some potential applications the simplifying features of isotopic mixtures are taken into consideration in the formulation of the problem. An exact solution for the barodiffusion problem analogous to the von Karman flow solutions is given. Results clearly identifying the small but significant separative action of the pressure-gradients in this configuration are presented for Schmidt numbers of order unity (typical of gaseous mixtures).     

A note on polynomial matrix functions over a finite field

Let F_n denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=A_Nx^N+ ?+ A₁x+A₀?F_n[x]. A function $\text{?:F}{}_{\mathrm{n}}\text{→F}{}_{\mathrm{n}}$ is called a right polynomial function iff there exists an A(x)?F_n[x] such that $\text{?(B)=A}{}_{\mathrm{N}}\text{B}{}^{\mathrm{N}}\text{+?+A}{}_1\text{B+ A}{}_0$ for every B?F_n. This paper obtains unique representations for and determines the number of right polynomial functions.     

A note on first kind convolution equations on a finite interval

This note deals with a class of convolution operators of the first kind on a finite interval. Necessary and sufficient conditions for such an operator to be Fredholm are given. The argument is based on a process of reduction of convolution-type operators on a finite interval to operators of the same type on the half line.Research supported by the Netherlands organization for scientific research (NWO).