2,707 research outputs found
Algorithmic Verification of Continuous and Hybrid Systems
We provide a tutorial introduction to reachability computation, a class of
computational techniques that exports verification technology toward continuous
and hybrid systems. For open under-determined systems, this technique can
sometimes replace an infinite number of simulations.Comment: In Proceedings INFINITY 2013, arXiv:1402.661
Convex Hull Realizations of the Multiplihedra
We present a simple algorithm for determining the extremal points in
Euclidean space whose convex hull is the nth polytope in the sequence known as
the multiplihedra. This answers the open question of whether the multiplihedra
could be realized as convex polytopes. We use this realization to unite the
approach to A_n-maps of Iwase and Mimura to that of Boardman and Vogt. We
include a review of the appearance of the nth multiplihedron for various n in
the studies of higher homotopy commutativity, (weak) n-categories,
A_infinity-categories, deformation theory, and moduli spaces. We also include
suggestions for the use of our realizations in some of these areas as well as
in related studies, including enriched category theory and the graph
associahedra.Comment: typos fixed, introduction revise
A discrete methodology for controlling the sign of curvature and torsion for NURBS
This paper develops a discrete methodology for approximating the so-called convex domain of a NURBS curve, namely the domain in the ambient space, where a user-specified control point is free to move so that the curvature and torsion retains its sign along the NURBS parametric domain of definition. The methodology provides a monotonic sequence of convex polyhedra, converging from the interior to the convex domain. If the latter is non-empty, a simple algorithm is proposed, that yields a sequence of polytopes converging uniformly to the restriction of the convex domain to any user-specified bounding box. The algorithm is illustrated for a pair of planar and a spatial Bézier configuration
Shrinkage and Variable Selection by Polytopes
Constrained estimators that enforce variable selection and grouping of highly correlated data have been shown to be successful in finding sparse representations and obtaining good performance in prediction. We consider polytopes as a general class of compact and convex constraint regions. Well
established procedures like LASSO (Tibshirani, 1996) or OSCAR (Bondell and Reich, 2008) are shown to be based on specific subclasses of polytopes. The general framework of polytopes can be used to investigate the geometric structure that underlies these procedures. Moreover, we propose a specifically designed class of polytopes that enforces variable selection and grouping. Simulation studies and an application illustrate the usefulness of the proposed method
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