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

By M. Aizenman (Chief Editor)

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Extra resources for Communications in Mathematical Physics - Volume 217

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S to magnetic surfaces of constant curvature as well. Equipped with the mentioned above definitions, we are ready now to formulate the main results of the present paper. Let Q be a billiard table on a magnetic surface of constant curvature, and let λ(v) ≥ 0 be the Lyapunov exponent of the billiard. t. T (v). Let φ(v) = (m2 , θ2 ). We set d1 = d(v), d2 = d(φ(v)) and s = m1 m2 for the length of particle trajectory between m1 and m2 . Then T (v) is determined by the triple (d1 , d2 , s). t. t. T (d1 , d2 , s) in an auxiliary billiard table Qv , constructed from the boundary ∂Q around mi (see [GSG]).

Let Q be a billiard table on a surface of constant curvature. The billiard map φ : V → V acts on the phase space V , which consists of pairs v = (m, θ ). Here m is the position of the ball on the boundary ∂Q of Q, and θ is the angle between the outgoing velocity and the tangent to ∂Q at m. The billiard map preserves a natural probability measure µ on V . We denote the images of v after n iterations by (mn+1 , θn+1 ) = φ n (v). ) if the following conditions are satisfied: 1. The incidence angle and the curvature of the boundary κn at the bouncing points have period 2: θ2n = θ2 , θ2n+1 = θ1 , κ2n = κ2 , κ2n+1 = κ1 ; 2.

Lax equations in ten-dimensional supersymmetric classicalYang–Mills theories. Contribution to the International Seminar on Integrable Systems, in memoriam of Mikhail V. Saveliev, Max-Planck Institute for Mathematics, Bonn (1999), hep-th/9903218 10. : Solution generating in ten dimensional supersymmetric classical Yang–Mills theories. Contribution to the Dubna Memorial Volume in honor of Mikhail V. Saveliev. (1999), hep-th/9910235 11. : Progress in classically solving ten-dimensional supersymmetric reduced Yang–Mills theories.