Regularity of density for SDEs driven by degenerate L\'evy noises

By using Bismut's approach about the Malliavin calculus with jumps, we study the regularity of the distributional density for SDEs driven by degenerate additive L\'evy noises. Under full H\"ormander's conditions, we prove the existence of distributional density and the weak continuity in the first variable of the distributional density. Under the uniform first order Lie's bracket condition, we also prove the smoothness of the density.Comment: 25 page

Bifurcations of limit cycles from cubic Hamiltonian systems with a center and a homoclinic saddle-loop

It is provedin this paper that the maximum number of limit cycles of system [formula] is equal to two in the finite plane, where [formula]. This is partial answer to the seventh question in [2], posed by Arnold

On the possible ranks among matrices with a given pattern

Given are tight upper and lower bounds for the minimum rank among all matrices with a prescribed zero-nonzero pattern. The upper bound is based upon solving for a matrix with a given null space and, with optimal choices, produces the correct minimum rank. It leads to simple, but often accurate, bounds based upon overt statistics of the pattern. The lower bound is also conceptually simple. Often, the lower and an upper bound coincide, but examples are given in which they do not.Fundação para a Ciência e a Tecnologia (FCT