Communications in Mathematical Physics - Volume 217 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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Type (A − A):  A A A A A A  [d1 , d2 ] ∪ [d1 + d2 , ∞) if d1 , d2 ≥ 0 s ∈ [0, ∞) (18) if d1A , d2A ≤ 0  [0, d A + d A ] ∪ [d A , ∞) A A if d1 ≥ 0, d2 ≤ 0, 1 2 1 Type (B − B): s∈ [d1B + d2B , ∞) if d1B + d2B ≥ 0 [0, ∞) if d1B + d2B ≤ 0, (19) Types (A − B) or (B − A): s∈ [d1A , ∞) [0, ∞) if d1A ≥ 0 if d1A ≤ 0. (20) It is worth mentioning that in Proposition 1 (resp. Eqs. (16)–(20)) the hyperbolicity of T (d1 , d2 , s) corresponds to strict inequalities (resp. inclusions in the interior). The equality case (resp.

T. t. is determined by Dv φ 2 . t. is completely determined by the triplet of parameters (d1 , d2 , s), where 2d1 (resp. 2d2 ) is the signed length of the chord generated by the intersection of the trajectory m1 m2 with the osculating circle at m1 (resp. m2 ). t. with parameters (d1 , d2 , s). 36 B. s for planar, non-magnetic billiards in some detail. Here s is the euclidean distance between consecutive bouncing points, and di = ri sin θi , i = 1, 2, where ri are the radii of curvature of the boundary ∂Q at the respective points.

Geometry 2, Encycl. of Math. Sc. Vol. 29. : Erg. Theor. Dyn. Sys. : Commun. Math. Phys. 105, 391–414 (1986) Communicated by Ya. G. Sinai Commun. Math. Phys. 217, 55 – 87 (2001) Communications in Mathematical Physics © Springer-Verlag 2001 Twisted Index Theory on Good Orbifolds, II: Fractional Quantum Numbers Matilde Marcolli1 , Varghese Mathai1,2 1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA. edu 2 Department of Mathematics, University of Adelaide, Adelaide 5005, Australia.

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