For each positive integer g, we derive a completely integrable Hamiltonian system in g variables from the holonomic deformation of a linear differential equation with a regular singular point and an irregular singular point of Poincaré rank g + 1. For g = 1, this Hamiltonian system is equivalent to the fourth Painlevé equation.
We shall prove intrinsic ultracontractive bounds for compact manifolds with boundary, using their inner geometric properties, by the arguments of Davies and Simon 1984. In order to do so, we shall prepair two inequalities, Hardy and Lp-Sobolev (p ≥ 1).
In this paper, we use ideas introduced in an article by C. Hermite (J. Crelle 79 (1875), 324-338) to give new linear independence measures of some Abelian integrals. These approximations are obtained by a study of the theory of the Tissot-Pochhammer differential equations, the cycloelliptic curves and the computation of period matrices.