Coordinate shadows of semi-definite and Euclidean distance matrices

We consider the projected semi-definite and Euclidean distance cones onto a subset of the matrix entries. These two sets are precisely the input data defining feasible semi-definite and Euclidean distance completion problems. We classify when these sets are closed, and use the boundary structure of these two sets to elucidate the Krislock-Wolkowicz facial reduction algorithm. In particular, we show that under a chordality assumption, the "minimal cones" of these problems admit combinatorial characterizations. As a byproduct, we record a striking relationship between the complexity of the general facial reduction algorithm (singularity degree) and facial exposedness of conic images under a linear mapping.Comment: 21 page

Provably Safe Robot Navigation with Obstacle Uncertainty

As drones and autonomous cars become more widespread it is becoming increasingly important that robots can operate safely under realistic conditions. The noisy information fed into real systems means that robots must use estimates of the environment to plan navigation. Efficiently guaranteeing that the resulting motion plans are safe under these circumstances has proved difficult. We examine how to guarantee that a trajectory or policy is safe with only imperfect observations of the environment. We examine the implications of various mathematical formalisms of safety and arrive at a mathematical notion of safety of a long-term execution, even when conditioned on observational information. We present efficient algorithms that can prove that trajectories or policies are safe with much tighter bounds than in previous work. Notably, the complexity of the environment does not affect our methods ability to evaluate if a trajectory or policy is safe. We then use these safety checking methods to design a safe variant of the RRT planning algorithm.Comment: RSS 201

Automata, reduced words, and Garside shadows in Coxeter groups

In this article, we introduce and investigate a class of finite deterministic automata that all recognize the language of reduced words of a finitely generated Coxeter system (W,S). The definition of these automata is straightforward as it only requires the notion of weak order on (W,S) and the related notion of Garside shadows in (W,S), an analog of the notion of a Garside family. Then we discuss the relations between this class of automata and the canonical automaton built from Brink and Howlett's small roots. We end this article by providing partial positive answers to two conjectures: (1) the automata associated to the smallest Garside shadow is minimal; (2) the canonical automaton is minimal if and only if the support of all small roots is spherical, i.e., the corresponding root system is finite.Comment: 21 pages, 7 figures; v2: 23 pages, 8 figures, Remark 3.15 added, accepted in Journal of Algebra, computational sectio

3D simulation of complex shading affecting PV systems taking benefit from the power of graphics cards developed for the video game industry

Shading reduces the power output of a photovoltaic (PV) system. The design engineering of PV systems requires modeling and evaluating shading losses. Some PV systems are affected by complex shading scenes whose resulting PV energy losses are very difficult to evaluate with current modeling tools. Several specialized PV design and simulation software include the possibility to evaluate shading losses. They generally possess a Graphical User Interface (GUI) through which the user can draw a 3D shading scene, and then evaluate its corresponding PV energy losses. The complexity of the objects that these tools can handle is relatively limited. We have created a software solution, 3DPV, which allows evaluating the energy losses induced by complex 3D scenes on PV generators. The 3D objects can be imported from specialized 3D modeling software or from a 3D object library. The shadows cast by this 3D scene on the PV generator are then directly evaluated from the Graphics Processing Unit (GPU). Thanks to the recent development of GPUs for the video game industry, the shadows can be evaluated with a very high spatial resolution that reaches well beyond the PV cell level, in very short calculation times. A PV simulation model then translates the geometrical shading into PV energy output losses. 3DPV has been implemented using WebGL, which allows it to run directly from a Web browser, without requiring any local installation from the user. This also allows taken full benefits from the information already available from Internet, such as the 3D object libraries. This contribution describes, step by step, the method that allows 3DPV to evaluate the PV energy losses caused by complex shading. We then illustrate the results of this methodology to several application cases that are encountered in the world of PV systems design.Comment: 5 page, 9 figures, conference proceedings, 29th European Photovoltaic Solar Energy Conference and Exhibition, Amsterdam, 201

A new measure of instability and topological entropy of area-preserving twist diffeomorphisms

We introduce a new measure of instability of area-preserving twist diffeomorphisms, which generalizes the notions of angle of splitting of separatrices, and flux through a gap of a Cantori. As an example of application, we establish a sharp >0 lower bound on the topological entropy in a neighbourhood of a hyperbolic, unique action-minimizing fixed point, assuming only no topological obstruction to diffusion, i.e. no homotopically non-trivial invariant circle consisting of orbits with the rotation number 0. The proof is based on a new method of precise construction of positive entropy invariant measures, applicable to more general Lagrangian systems, also in higher degrees of freedom

Recursion Operators and Nonlocal Symmetries for Integrable rmdKP and rdDym Equations

We find direct and inverse recursion operators for integrable cases of the rmdKP and rdDym equations. Also, we study actions of these operators on the contact symmetries and find shadows of nonlocal symmetries of these equations