A minimum time problem with a nonlinear smooth dynamics and a target satisfying an internal sphere condition is considered. Under the assumption that the minimum time function T is continuous we show that hypo(T ) satisfies a weak external sphere condition. From here, our results are divided into two parts. In the first part: if the normal cone to the hypograph of T , N hypo(T) is pointed then ...

When a symmetric, positive, isomorphism between a reeexive Banach space (that is densely and compactly embedded in a Hilbert space) and its dual varies smoothly over a Banach space, its eigenvalues vary in a Lipschitz manner. We calculate the generalized gradient of the extreme eigenvalues at an arbitrary crossing. We apply this to the generalized gradient, with respect to a coeecient in an ell...

We develop a generalized gradient expansion of the inhomogeneous dynamical mean-field theory method for determining properties of ultracold atoms in a trap. This approach goes beyond the well-known local density approximation and at higher temperatures, in the normal phase, it shows why the local density approximation works so well, since the local density and generalized gradient approximation...