Union Of Convex Polyhedra Research Articles

Structural Correspondences between the Low-Energy Nanoclusters of Silica and Water

Nanoclusters of two well-known tetrahedrally coordinated systems, (SiO2)N silica and (H2O)N water, are compared in the size range N = 8−26 with respect to energetic stability and T-atom topology (where T = Si and O, respectively). Because of the similar cage-forming tendencies of both systems in a number of bulk phases (e.g., clathrates), we focus on clusters with an even number of T-atoms for which cluster structural decomposition into a union of convex polyhedra (i.e., face-sharing cages with T-atom-vertices) is always possible. For (H2O)N, in the range N = 8−24, the even-N ground-state clusters are almost exclusively formed from face-sharing cubes and pentaprisms of T-atoms. We show that (SiO2)N clusters having the same T-atom topology as these prismatic water clusters are also low in energy (per SiO2 unit with respect to the corresponding lowest energy (SiO2)N cluster), and have the same groundstate structure for N = 12. Along with similarities, an important difference is discussed, namely that, rather than a converging tendency for the two systems to have similar prismatic topologies with increasing size, for (H2O)N at N = 24, self-solvating amorphous clusters compete strongly with the prismatic ground states, becoming ground states themselves for N > 24. Contrary to this shift in water cluster structural preference, (SiO2)24 clusters with the same T-atom topology as the amorphous (H2O)24 clusters are found to be high in energy with respect to both the (SiO2)24 ground-state and (SiO2)24 prismatic isomers. We discuss possible reasons for the existence and eventual breakdown of the topological analogy between (H2O)N and (SiO2)N nanoclusters with relation to the different bonding characteristics in each system.

Inaccurate linear equation system with a restricted-rank error matrix

It is well-known that the solution set of an interval linear equation system is a union of convex polyhedra the number of which increases, in general, exponentially with the problem size. As a consequence, the problem of finding the interval hull of the solution set is NP-hard as J. Rohn and V. Kreinovich proved in [13]. The purpose of this paper is to show that the solution set analysis can be simplified substantially provided the rank of the error matrix is restricted even if the assumption of interval character of data errors is replaced by a more general one. Especially, in the case of a rank-one error matrix we have to look into at most two convex subsets. Besides, a dual approach to describing the solution set is discussed. The original version of this approach was suggested in [7].

The Union of Convex Polyhedra in Three Dimensions

We show that the number of vertices, edges, and faces of the union of k convex polyhedra in 3-space, having a total of n faces, is O(k3 + kn log k). This bound is almost tight in the worst case, as there exist collections of polyhedra with $\Omega(k^3+kn\alpha(k))$ union complexity. We also describe a rather simple randomized incremental algorithm for computing the boundary of the union in O(k3 + kn log k log n) expected time.

The algorithmic analysis of hybrid systems

We present a general framework for the formal specification and algorithmic analysis of hybrid systems. A hybrid system consists of a discrete program with an analog environment. We model hybrid systems as finite automata equipped with variables that evolve continuously with time according to dynamical laws. For verification purposes, we restrict ourselves to linear hybrid systems, where all variables follow piecewise-linear trajectories. We provide decidability and undecidability results for classes of linear hybrid systems, and we show that standard program-analysis techniques can be adapted to linear hybrid systems. In particular, we consider symbolic model-checking and minimization procedures that are based on the reachability analysis of an infinite state space. The procedures iteratively compute state sets that are definable as unions of convex polyhedra in multidimensional real space. We also present approximation techniques for dealing with systems for which the iterative procedures do not converge.

Collision detection for spacecraft proximity operations

A new collision detection algorithm has been developed for use when two spacecraft are operating in the same vicinity. The two spacecraft are modeled as unions of convex polyhedra, where the resulting polyhedron may be either convex or nonconvex. The relative motion of the two spacecraft is assumed to be such .that one vehicle is moving with constant linear and angular velocity with respect to the other. Contacts between the vertices, faces, and edges of the polyhedra representing the two spacecraft are shown to occur when the value of one or more of a set of functions is zero. The collision detection algorithm is then formulated as a search for the zeros (roots) of these functions. Special properties of the functions for the assumed relative trajectory are exploited to expedite the zero search. The new algorithm is the first algorithm that can solve the collision detection problem exactly for relative motion with constant angular velocity. This is a significant improvement over models of rotational motion used in previous collision detection algorithms.