4,396 research outputs found
Laplacian eigenvalues functionals and metric deformations on compact manifolds
In this paper, we investigate critical points of the Laplacian's eigenvalues
considered as functionals on the space of Riemmannian metrics or a conformal
class of metrics on a compact manifold. We obtain necessary and sufficient
conditions for a metric to be a critical point of such a functional. We derive
specific consequences concerning possible locally maximizing metrics. We also
characterize critical metrics of the ratio of two consecutive eigenvalues
From singularities to graphs
In this paper I analyze the problems which led to the introduction of graphs
as tools for studying surface singularities. I explain how such graphs were
initially only described using words, but that several questions made it
necessary to draw them, leading to the elaboration of a special calculus with
graphs. This is a non-technical paper intended to be readable both by
mathematicians and philosophers or historians of mathematics.Comment: 23 pages, 27 figures. Expanded version of the talk given at the
conference "Quand la forme devient substance : puissance des gestes,
intuition diagrammatique et ph\'enom\'enologie de l'espace", which took place
at Lyc\'ee Henri IV in Paris from 25 to 27 January 201
Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle
It is shown that the problem of designing a two-reflector system transforming
a plane wave front with given intensity into an output plane front with
prescribed output intensity can be formulated and solved as the
Monge-Kantorovich mass transfer problem.Comment: 25 pages, 2 figure
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