We provide further evidence to the P = W conjecture of de Cataldo, Hausel and Migliorini, by checking it in Painlevé cases. Namely, we compare perverse Leray filtration induced Hitchin map on cohomology spaces Dolbeault moduli space weight irregular character variety corresponding each I ? V systems. find that two filtrations agree. Along way, prove Geometric Katzarkov, Noll, Pandit Simpson cas...

In this paper we show that uniformly global asymptotic stability for a family of ordinary diierential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables. The same is shown for input-to-state stability and input-to-state exponential stability, and for input-to-state exponential stability and a nonlinear H 1 estimate.