415 research outputs found
On the semiclassical Laplacian with magnetic field having self-intersecting zero set
This paper is devoted to the spectral analysis of the Neumann realization of
the 2D magnetic Laplacian with semiclassical parameter h > 0 in the case when
the magnetic field vanishes along a smooth curve which crosses itself inside a
bounded domain. We investigate the behavior of its eigenpairs in the limit h
0. We show that each crossing point acts as a potential well,
generating a new decay scale of h 3/2 for the lowest eigenvalues, as well as
exponential concentration for eigenvectors around the set of crossing points.
These properties are consequences of the nature of associated model problems in
R 2 for which the zero set of the magnetic field is the union of two straight
lines. In this paper we also analyze the spectrum of model problems when the
angle between the two straight lines tends to 0
Plane waveguides with corners in the small angle limit
The plane waveguides with corners considered here are infinite V-shaped
strips with constant thickness. They are parametrized by their sole opening
angle. We study the eigenpairs of the Dirichlet Laplacian in such domains when
this angle tends to 0. We provide multi-scale asymptotics for eigenpairs
associated with the lowest eigenvalues. For this, we investigate the eigenpairs
of a one-dimensional model which can be viewed as their Born-Oppenheimer
approximation. We also investigate the Dirichlet Laplacian on triangles with
sharp angles. The eigenvalue asymptotics involve powers of the cube root of the
angle, while the eigenvector asymptotics include simultaneously two scales in
the triangular part, and one scale in the straight part of the guides
Using rewriting techniques to produce code generators and proving them correct
AbstractA major problem in deriving a compiler from a formal definition is the production of correct and efficient object code. In this context, we propose a solution to the problem of code-generator generation.Our approach is based on a target machine description where the basic concepts used (storage classes, access modes, access classes and instructions) are hierarchically described by tree patterns. These tree patterns are terms of an abstract data type. The program intermediate representation (input to the code generator) is a term of the same abstract data type.The code generation process is based on access modes and instruction template-driven rewritings. The result is that each program instruction is reduced to a sequence of elementary machine instructions, each of them representing an instance of an instruction template.The axioms of the abstract data type are used to prove that the rewritings preserve the semantics of the intermediate representation
Spectral asymptotics of the Dirichlet Laplacian in a conical layer
The spectrum of the Dirichlet Laplacian on conical layers is analysed through
two aspects: the infiniteness of the discrete eigenvalues and their expansions
in the small aperture limit. On the one hand, we prove that, for any aperture,
the eigenvalues accumulate below the thresh-old of the essential spectrum: For
a small distance from the essential spectrum, the number of eigenvalues farther
from the threshold than this distance behaves like the logarithm of the
distance. On the other hand, in the small aperture regime, we provide a
two-term asymptotics of the first eigenvalues thanks to a priori localization
estimates for the associated eigenfunctions. We prove that these eigenfunctions
are localized in the conical cap at a scale of order the cubic root of the
aperture angle and that they get into the other part of the layer at a scale
involving the logarithm of the aperture angle
Eigenpairs of a Model Schrödinger Operator with Neumann Boundary Conditions
International audienceThe considered Schrödinger operator has a quadratic potential which is degenerate in the sense that it reaches its minimum all along a line which makes the angle \theta with the boundary of the half-plane where the problem is set. We exhibit localization properties for the eigenfunctions associated with its lowest eigenvalues below its essential spectrum. We investigate the densification and the asymptotics of the eigenvalues below the essential spectrum in the limit \theta\to 0
Discrete spectrum of a model Schrödinger operator on the half-plane with Neumann conditions
International audienceWe study the eigenpairs of a model Schrödinger operator with a quadratic potential and Neumann boundary conditions on a half-plane. The potential is degenerate in the sense that it reaches its minimum all along a line which makes the angle \theta with the boundary of the half-plane. We show that the first eigenfunctions satisfy localization properties related to the distance to the minimum line of the potential. We investigate the densification of the eigenvalues below the essential spectrum in the limit \theta \to 0 and we prove full asymptotic expansion for these eigenvalues and their associated eigenvectors. We conclude the paper by numerical experiments obtained by a finite element method. The numerical results confirm and enlighten the theoretical approach
New early devonian charophyta from Gondwana
Early Devonian charophytes are reported from Australia (Buchan, Victoria) and Europe (Landeyran, southern France): Moellerina australica n. sp. Feist and Pinnoputamen occitanicum n. sp. Feist. Sedimentological data and associated faunas from these localities accord with both species having inhabited lacustrine or estuarine environments. A critical review of Devonian biotopes confirms that, as with present day species, Mid Palaeozoic charophytes could not have lived in open marine habitats. Originating in Baltica during the Silurian, charophytes appeared in Gondwana in the earliest Devonian.<br /
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