| 词汇 | example_english_billiard |
| 释义 | Examples of billiardThese examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. We exclude = 1 and = 0 from our consideration, as the singular set 1,0 of the corresponding billiard table coincides with its boundary. Two such billiard tables will be called geodesically equivalent if they are geodesically equivalent in the complement of the union of the corresponding singular sets. A slab of freshwater ice formed the smooth top of a billiard table made of ice blocks, with pockets and cushions crafted out of walrus-hide. In the case of geodesically equivalent billiard tables, these integrals are pairwise commuting. Whenever the billiard particle enters the hole, it gets absorbed with its orbit deleted from the phase space. Another class of integrable billiard tables appears on quadrics. It is weaker in that a uniformly distributed billiard path is always dense, but not conversely. In [6], it is shown that the topological entropy of a compact non-degenerate semi-dispersing billiard on any manifold of non-positive sectional curvature is finite. In this paper we show that the topological entropy of a compact non-degenerate semidispersing billiard on any manifold of non-positive sectional curvature is finite. That theory is already applicable for billiard systems, too. Thus, we obtained that any cylindric billiard with one-dimensional generators is necessarily tight, unless all generators are parallel. The connected configuration space of a so-called cylindric billiard system is a flat torus minus finitely many spherical cylinders. The transformation in question is a projective billiard transformation, the transversals being the normals to in the metric under consideration. The billiard transformation is defined in metric terms (equal angles), and, in particular, it is not equivariant under projective transformations of the plane. An example of this situation is given by the usual billiard in an ellipse; the billiard transformation has an invariant area form. The first observation one has to make to prove that a certain billiard is hyperbolic is that the singularities are not a big obstruction. The problem of the billiard table with a hole can be reformulated for maps of the interval as follows. Therefore, every finite piece of a trajectory of a semi-dispersing billiard makes only finitely many collisions. These sets play an important role in the study of billiards, because they generate the singularities of the billiard dynamics. This transformation has an invariant form that comes from the symplectic form in the cotangent bundle of the billiard table. This paper addresses the question of genericity of existence of elliptic islands for the billiard map associated to strictly convex closed curves. One main model they suggested was a dispersive billiard with a hole. It is proven here that under a certain geometric condition a cylindric billiard flow is completely hyperbolic. The reason why this is indeed so comes from the similar algebraic nature of the cylindric billiard dynamics. These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. |
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