Coherent adequate forcing and preserving CH

We develop a general framework for forcing with coherent adequate sets on [Formula: see text] as side conditions, where [Formula: see text] is a cardinal of uncountable cofinality. We describe a class of forcing posets which we call coherent adequate type forcings. The main theorem of the paper is that any coherent adequate type forcing preserves CH. We show that there exists a forcing poset for adding a club subset of [Formula: see text] with finite conditions while preserving CH, solving a problem of Friedman [Forcing with finite conditions, in Set Theory: Centre de Recerca Matemática, Barcelona, 2003–2004, Trends in Mathematics (Birkhäuser-Verlag, 2006), pp. 285–295.].

Multi-level gradient-based methods and parametrisation in aerodynamic shape design

The present study focuses on multi-level approaches in the context of discrete gradient-based methods for aerodynamic shape design. More precisely, the minimisation is done alternatively on different control subspaces according to multigrid-like cycles, providing at each sub-level a particular gradient preconditioning. Starting from an existing multi-level gradient-based formulation associated to shape grid-points coordinates, a possible generalisation to more compact shape representations is proposed through the construction of adequate sets of embedded shape sub-parametrisations. The behaviour of the new formulation is illustrated on different 2D inverse problems for inviscid flows.