A particular, yet relevant, particular case of the Penrose inequality involves null shells propagating in the Minkowski spacetime. Despite previous claims in the literature, the validity of this inequality remains open. In this paper we rewrite this inequality in terms of the geometry of the surface obtained by intersecting the past null cone of the original surface S with a constant time hyperplane and the "time height" function of S over this hyperplane. We also specialize to the case when S lies in the past null cone of a point and show the validity of the corresponding inequality in any dimension (in four dimensions this inequality was proved by Tod [1]). Exploiting properties of convex hypersurfaces in Euclidean space we write down the Penrose inequality in the Minkowski spacetime of arbitrary dimension n+2 as an inequality for two smooth functions on the sphere S n . We finally obtain a sufficient condition for the validity of the Penrose inequality in the four dimensional Minkowski spacetime and show that this condition is satisfied by a large class of surfaces.

No abstract

No abstract

Purpose The position/force control of the robot needs the parameters of the impedance model and generates the desired position from the contact force in the environment. When the environment is unknown, learning algorithms are needed to estimate both the desired force and the parameters of the impedance model. Design/methodology/approach In this paper, the authors use reinforcement learning to learn only the desired force, then they use proportional-integral-derivative admittance control to generate the desired position. The results of the experiment are presented to verify their approach. Findings The position error is minimized without knowing the environment or the impedance parameters. Another advantage of this simplified position/force control is that the transformation of the Cartesian space to the joint space by inverse kinematics is avoided by the feedback control mechanism. The stability of the closed-loop system is proven. Originality/value The position error is minimized without knowing the environment or the impedance parameters. The stability of the closed-loop system is proven.

The Penrose inequality in Minkowski is a geometric inequality relating the total outer null expansion and the area of closed, connected and spacelike codimension-two surfaces S in the Minkowski spacetime, subject to an additional convexity assumption. In a recent paper, Brendle and Wang [1] find a sufficient condition for the validity of this Penrose inequality in terms of the geometry of the orthogonal projection of S onto a constant time hyperplane. In this work, we study the geometry of hypersurfaces in n-dimensional euclidean space which are normal graphs over other surfaces and relate the intrinsic and extrinsic geometry of the graph with that of the base hypersurface. These results are used to rewrite Brendle and Wang's condition explicitly in terms of the time height function of S over a hyperplane and the geometry of the projection of S along its past null cone onto this hyperplane. We also include, in an Appendix, a self-contained summary of known and new results on the geometry of projections along the Killing direction of codimension two-spacelike surfaces in a strictly static spacetime.