On estimating a density using Hellinger distance and some other strange facts

Résumé On s'intéresse ici aux possibles vitesses d'estimation d'une densité à support compact dans ^m sous des hypothèses de régularité, lorsque la perte est mesurée par le carré de la distance de Hellinger (on regardera aussi le cas connu des normes pour 1q2) et le risque est le risque minimax sur la famille. On donne une méthode générale permettant de traiter les problèmes dans le cadre de la théorie de l'approximation sous des conditions concernant l'entropie métrique et l' -capacité des familles à estimer. Les rapports entre régularité et entropie métrique étant bien connus, nous pourrons aussi traiter les cas classiques et d'autres qui le sont moins. Sous des conditions de bornes inférieures les vitesses sont celles observées pour la norme mais elles diffèrent dans le cas général. On montre aussi que les restrictions sur la compacité du support ou la régularité sont indispensables et que leur absence mène à l'impossibilité d'obtenir une estimation raisonnable en ce sens que n'importe quelle suite d'estimateurs sera arbitrairement mauvaise en un point au moins. Un résultat analogue est vrai sous des conditions de régularité.

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This work was carried out during a visit of the author at the Mathematical Sciences Research Institute at Berkeley     

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.     

Consistency of a linear system of inequalities

We show that a consistency check of a linear system of inequalitiesAxb reduces to check whetherQb0 for a certain matrixQ. It is a direct consequence of the Farkas-Minkowski theorem. Thus, when one has to check consistency for different values ofb, one need not run a numerical algorithm for eachb.On leave at the Electronics Research Laboratory of the University of California at Berkeley in a CNRS/NSF Exchange Program.     

Observables on hypergraphs

Observables on hypergraphs are described by event-valued measures. We first distinguish between finitely additive observables and countably additive ones. We then study the spectrum, compatibility, and functions of observables. Next a relationship between observables and certain functionals on the set of measures M(H) of a hypergraph H is established. We characterize hypergraphs for which every linear functional on M(H) is determined by an observable. We define the concept of an effect and show that observables are related to effect-valued measures. Finally, we define operational transformations from M(H) to itself and show that they can be described as a certain combination of effects.On leave from University of Berne, Institute of Mathematical Statistics, Sidlerstrasse 5, CH-3012 Berne, Switzerland.     

Coloring of universal graphs

LetG be a graph andr a cardinal number. Extending the theorem of J. Folkman we show that if eitherr or clG are finite then there existsH with clH = clG andH (G) _r ¹ . Answering a question of A. Hajnal we show that countably universal graphU ₃ satisfiesU ₃ (U₃) _r ¹ for every finiter.     

The lamellar inclusion problem in plane elasticity

Within the framework of linear isotropic plane elasticity, a general method giving the elastic field in an internally stressed three-phase material is determined. The internal stress source may be located either in or near the lamellar component. General closed-form formulas solving the limiting case of a bimetallic interface are also obtained. As an example of application, the interaction between a lamellar inclusion and a dilatation centre is analyzed.

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Riassunto Nel presente lavoro viene elaborato un metodo per determinare il campo elastico in un materiale a tre fasi deformato internamente. Ciascuna fase e' supposta omogenea ed isotropa e si assume uno stato di deformazione piana, o di sforzo piano, trattabile mediante la teoria lineare dell'elasticita'. La sorgente di deformazione puo' essere localizzata all'interno oppure all'esterno del componente lamellare. Formule risolutive esatte si ricavano nel caso limite di mezzo bifasico. Come esempio di applicazione, viene analizzata l'interazione tra un'inclusione lamellare ed un centro di dilatazione.

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Work supported by the Ministero della Publica Istruzione, Italy     

Bianchi type-V perfect fluid models

A study of the nontilted diagonal Bianchi type-V perfect fluid models is undertaken in the nonlocally rotationally symmetric case. It is shown that in such a model, the singularity is necessarily velocity dominated and the upper bound for the ratio of the shear to the expansion is 1/3. A class of such exact models is also presented; the structure of the singularity and the effect of the anisotropy on the scale are discussed. Lastly, an attempt is made to solve exactly the equations of motion for the coordinates of the null geodesics.     

Rigorous justification of the Friedlander-Keller formulas

In the paper an asymptotic expansion (as , the wave number) is proved for the Green function of the problem of diffraction by a smooth convex body in two cases: when one of the points of the source or observer lies on the boundary while the other is an arbitrary distance from the boundary and also when both points lie off the boundary but not far from it. The two-dimensional Dirichlet problem is considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 49–60, 1984.     

The problem of radial excitations of light mesons

Strong decays of the ground and radial-excited (L=0) states of light mesons are calculated within a relativistic (Lorentz-contraction-modified) version of the model of spontaneous production of aq \(\bar q\) -pair from vacuum.