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تسجيل مستخدم جديدGeometric Aspects of Painleve Equations
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In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painleve equations, with a particular emphasis on the discrete Painleve equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on $mathbb{P}^1timesmathbb{P}^1$ and classified according to the degeration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeomtric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.
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