On the -stability of systems of differential equations in the Routh-Hurwitz and the Schur-Cohn cases.
Order and hyper-order of entire solutions of linear differential equations with entire coefficients.
Order of growth of solutions to algebraic differential equations in the unit disk.
Orders of solutions of an -th order linear differential equation with entire coefficients.
Oscillation of complex high order linear differential equations with coefficients of finite iterated order.
Oscillation of fast growing solutions of linear differential equations in the unit disc.
Oscillation of fixed points of solutions of some linear differential equations.
Oscillation of higher-order linear differential equations with entire coefficients.
Oscillation of solutions of some higher order linear differential equations.
Oscillation Results for Some Linear Differential Equations.
Oscillation results on meromorphic solutions of second order differential equations in the complex plane.
Oscillatory properties of second order linear differential equations in the complex domain
Overstability and resonance
We consider a singularity perturbed nonlinear differential equation which we suppose real analytic for near some interval and small , . We furthermore suppose that 0 is a turning point, namely that is positive if . We prove that the existence of nicely behaved (as ) local (at ) or global, real analytic or solutions is equivalent to the existence of a formal series solution with analytic at . The main tool of a proof is a new “principle of analytic continuation” for such “overstable” solutions....
Painlevé equations and complex reflections
We will explain how some new algebraic solutions of the sixth Painlevé equation arise from complex reflection groups, thereby extending some results of Hitchin and Dubrovin-- Mazzocco for real reflection groups. The problem of finding explicit formulae for these solutions will be addressed elsewhere.
Painlevé's determinateness theorem extended to proper coverings.
Painlevé’s theorem extended
Painlevé's Theorem on automorphic functions.
Periodic integrals and tautological systems
We study period integrals of CY hypersurfaces in a partial flag variety. We construct a regular holonomic system of differential equations which govern the period integrals. By means of representation theory, a set of generators of the system can be described explicitly. The results are also generalized to CY complete intersections. The construction of these new systems of differential equations has lead us to the notion of a tautological system.
Perturbation singulière en dimension trois : canards en un point pseudo-singulier nœud
On étudie les systèmes différentiels singulièrement perturbés de dimension 3 du typeoù , , sont analytiques quelconques. Les travaux antérieurs étudiaient les points réguliers où la surface lente est transverse au champ rapide vertical. C’est le domaine d’application du théorème de Tikhonov. Dans d’autres travaux antérieurs, on étudiait les singularités de certains types : plis et fronces de la surface lente, ainsi que certaines singularités plus compliquées, analogues aux points tournants...
Picard Sets for Entire Functions.