منابع مشابه
On nonconvex caustics of convex billiards
Oliver Knill July 29, 1996 Abstract There are billiard tables of constant width, for which the billiard map has invariant curves in the phase space which belong to continuous but nowhere di erentiable caustics. We apply this to construct ruled surfaces which have a nowhere di erentiable lines of striction. We use it also to get Riemannian metrics on the sphere such that the caustic belonging at...
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Given a strictly convex domain Ω ⊂ R2, there is a natural way to define a billiard map in it: a rectilinear path hitting the boundary reflects so that the angle of reflection is equal to the angle of incidence. In this paper we answer a relatively old question of Guillemin. We show that if two billiard maps are C2-conjugate near the boundary, then the corresponding domains are similar, i.e. the...
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Caustics are curves with the property that a billiard trajectory, once tangent to it, stays tangent after every reflection at the boundary of the billiard table. When the billiard table is an ellipse, any nonsingular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics —the ones whose tangent trajectories are closed polygons— are d...
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In machine learning and optimization, one often wants to minimize a convex objective function F but can only evaluate a noisy approximation F̂ to it. Even though F is convex, the noise may render F̂ nonconvex, making the task of minimizing F intractable in general. As a consequence, several works in theoretical computer science, machine learning and optimization have focused on coming up with pol...
متن کاملMaximizing Orbits for Higher Dimensional Convex Billiards
The main result of this paper is, that for convex billiards in higher dimensions, in contrast with 2D case, for every point on the boundary and for every n there always exist billiard trajectories developing conjugate points at the n-th collision with the boundary. We shall explain that this is a consequence of the following variational property of the billiard orbits in higher dimension. If a ...
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ژورنال
عنوان ژورنال: Elemente der Mathematik
سال: 1998
ISSN: 0013-6018,1420-8962
DOI: 10.1007/s000170050038