A new methodology for density estimation is proposed. The methodology, which builds on the one developed in [15], normalizes the data points through the composition of simple maps. The parameters of each map are determined through the maximization of a local quadratic approximation to the log-likelihood. Various candidates for the elementary maps of each step are proposed; criteria for choosing one includes robustness, computational simplicity and good behavior in high-dimensional settings. A good choice is that of localized radial expansions, which depend on a single parameter: all the complexity of arbitrary, possibly convoluted probability densities can be built through the composition of such simple maps.

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Stratified flows in hydrostatic balance are studied in both their multilayer and continuous formulations. A novel stability criterion is proposed for stratified flows, which reinterprets stability in terms not of growth of small perturbations but of the well-posedness of the time evolution. This reinterpretation allows one to extend the classic results of Miles and Howard concerning steady and planar flows to the realm of flows that are nonuniform and unsteady.

Two and three-layer models of stratified flows in hydrostatic balance are studied. For the former, nonlinear transformations are found that map [baroclinic] two-layer flows with either rigid top and bottom lids or vertical periodicity, into [barotropic] single-layer, shallow water free-surface flows. We have previously shown that two-layer flows with Richardson number greater than one are nonlinearly stable, in the following sense: when the system is well-posed at a given time, it remains well-posed through the nonlinear evolution. Here, we give a general necessary condition for the nonlinear stability of systems of mixed type. For three-layer flows with vertical periodicity, the domains of local stability are determined and the system is shown not to satisfy the necessary condition for nonlinear stability. This means that there are wave-motions that evolve into shear unstable flows.