Reachability analysis of linear systems using support functions

This work is concerned with the algorithmic reachability analysis of continuous-time linear systems with constrained initial states and inputs. We propose an approach for computing an over-approximation of the set of states reachable on a bounded time interval. The main contribution over previous works is that it allows us to consider systems whose sets of initial states and inputs are given by arbitrary compact convex sets represented by their support functions. We actually compute two over-approximations of the reachable set. The first one is given by the union of convex sets with computable support functions. As the representation of convex sets by their support function is not suitable for some tasks, we derive from this first over-approximation a second one given by the union of polyhedrons. The overall computational complexity of our approach is comparable to the complexity of the most competitive available specialized algorithms for reachability analysis of linear systems using zonotopes or ellipsoids. The effectiveness of our approach is demonstrated on several examples.     

Toute forme modérément ramifiée d’un polydisque ouvert est triviale

Let k be a complete, non-Archimedean field and let X be a k-analytic space. Assume that there exists a finite, tamely ramified extension L of k such that X _L is isomorphic to an open polydisc over L ; we prove that X is itself isomorphic to an open polydisc over k. The proof consists in using the graded reduction (a notion which is due to Temkin) of the algebra of functions on X, together with some graded counterparts of classical commutative algebra results : Nakayama’s lemma, going-up theorem, basic notions about étale algebras, etc.     

The small quantum cohomology of a weighted projective space, a mirror D-module and their classical limits

We first describe a mirror partner (B-model) of the small quantum orbifold cohomology of weighted projective spaces (A-model) in the framework of differential equations: we attach to the A-model (resp. B-model) a quantum differential system (that is a trivial bundle equipped with a suitable flat meromorphic connection and a flat bilinear form) and we give an explicit isomorphism between these two quantum differential systems. On the A-side (resp. on the B-side), the quantum differential system alluded to is naturally produced by the small quantum cohomology (resp. a solution of the Birkhoff problem for the Brieskorn lattice of a Landau–Ginzburg model). Then we study the degenerations of these quantum differential systems and we apply our results to the construction of (classical, limit, logarithmic) Frobenius manifolds.     

Semigroups That Are Almost N 3

A construction is given for semigroups S such that S ³ has just two elements.     

Pressure-enforced plasticity in MAX phases: from single grain to polycrystal investigation

Ti₄AlN₃, Ti₃AlC₂ and Ti₃Al_0.8Sn_0.2C₂ MAX phases were plastically deformed at room temperature (RT) under gaseous confining pressure. Microstructures of as-grown and deformed samples are carefully analysed using scanning electron microscopy (SEM), atomic force microscopy (AFM) and transmission electron microscopy (TEM). It is demonstrated that high level of plastic deformation can be reached under confining gas pressure; the later suppresses the brittle failure at RT to the profit of plasticity. Multiscale characterization techniques are shown to provide a unique insight into all the scales of the plastic deformation; in particular, the effect of the mesoscale. Indeed, grain shape and orientation relative to the compression axis are shown to play a key role in the deformation process, intergranular stresses leading to a complex stress field in the polycrystalline samples. The TEM results show that dislocation activity highly depends on the grain orientation. The observation of dislocation entanglements unambiguously demonstrates that dislocations may be organized in such a configuration so that their glide in the basal plane can be hindered when deep plastic regime is reached.