A closed locally minimal network on a convex polyhedron is an embedding of a graph provided that all edges are geodesic arcs and at each vertex exactly three adges. We study such networks on the surfaces of convex polyhedra and discuss the problem of describing the set of all convex polyhedra that have such networks. Read moreĬlosed locally minimal networks can be viewed as “branching” closed geodesics. For the property (1), we analyze the classical 24-cell: We show that the realization space has at least the dimension 48, and it has points where it is a manifold of this dimension, but it is not smoothly embedded as a manifold everywhere. However, we also identify the smallest polytopes where the dimension count (2) and thus Robertson's claim fails, among them the bipyramid over a triangular prism. From these we get easily that for various large and natural classes of polytopes the realization spaces are indeed manifolds, whose dimensions are given by NG(P). In this paper we develop Jacobian criteria for the analysis of realization spaces. While this indeed holds for many natural classes of polytopes (including simple and simplicial polytopes, as well as all polytopes of dimension at most 3),and Robertson claimed this to be true for all polytopes, Mnev's (1986/1988) Universality Theorem implies that it is not true in general: Indeed, (1) the centered realization space is not a smoothly embedded manifold in general, and (2) it does not have the dimension NG(P) in general. equations that are used in the definition of the realization space.
#DIMENSION OF CONVEX HYPERSPACES FULL#
Robertson (1988) suggested a model for the realization space of a convex d-dimensional polytope and an approach via the implicit function theorem, which - in the case of a full rank Jacobian - proves that the realization space is a manifold of dimension NG(P):=d(f_0+f_, which is the natural guess for the dimension given by the number of variables minus the number of quadratic. Networks self-similarly moving by curvature with two triple junctions. Evolution of convex lens–shaped networks under the curve shortening flow. C., Azouani, A., Georgi, M., Hell, J., Nihar, J., Koeller, A., Marxen, T., Ritthaler, S., Sáez, M., Schulze, F., Smith, B. Mean curvature flow of networks with triple junctions in the plane. Self-similar solutions of a 2-D multiple-phase curvature flow. This fact completes the classification of the self–similarly shrinking networks in the plane with at most two triple junctions, see (2007). We prove that there are no networks homeomorphic to the Greek “Theta” letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees, such that under the motion by curvature they are self–similarly shrinking. Experiments on two illustrative examples demonstrate the efficacy of our algorithm. The second step derives an over- and under-approximations of the full reachable set by including (excluding, resp.) the obtained boundary enclosure from certain convex combinations of points in that boundary enclosure. The first step computes an enclosure of the set of states reachable from the boundary of the step's initial state set.
Our reachability algorithm is iterative along the time axis and the computations in each iteration are performed in two steps. We then generalize boundary-based reachability analysis to such DDEs. Membership in this class of DDEs is determined by conducting sensitivity analysis of the solution mapping with respect to the initial states to impose a bound constraint on the time-lag term. facilitates construction of over- and under-approximations of their full reachable sets by performing reachability analysis on the boundaries of their initial sets, thereby permitting an efficient lifting of reach-set computation methods for ODEs to DDEs. This article explores reachable-set computation for a class of DDEs featuring a local homeomorphism property.
#DIMENSION OF CONVEX HYPERSPACES VERIFICATION#
This induces an interest in safety verification of delay differential equations (DDEs) used as a model of embedded control. Delays in feedback control loop, as induced by networked distributed control schemes, may have detrimental effects on control performance.
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