Consider the Hill operator Ty = −y′′ + q′(t)y in L2(R), where q ∈ L2(0, 1) is a 1periodic real potential. The spectrum of T is is absolutely continuous and consists of bands separated by gaps γn, n > 1 with length |γn| > 0. We obtain a priori estimates of the gap lengths, effective masses, action variables for the KDV. For example, if μn are the effective masses associated with the gap γn = (λ ...

We consider Riemann mappings from bounded Lipschitz domains in the plane to a triangle. We show that in this case the Riemann mapping has a linear variational principle: it is the minimizer of the Dirichlet energy over an appropriate affine space. By discretizing the variational principle in a natural way we obtain discrete conformal maps which can be computed by solving a sparse linear system....