Particle-in-cell simulations of magnetosphere electrodynamics

This paper reviews the basic principles and techniques involved in formulating particle-in-cell (PIC) simulation models which can be used to address medium- and large-scale problems in magnetosphere electrodynamics. The limitations imposed by the underlying kinetic physics of a plasma are emphasized, and representative algorithms are described for full particle and hybrid (particle ions, fluid electrons) models. Issues related to the choice of initial and boundary conditions and the implementation of PIC models on massively parallel computers are discussed. Explicit examples involving the diffusion region in collisionless reconnection, plasma sheet convection, and large scale structure in magnetic reconnection are presented to illustrate the current capabilities of PIC models     

Polyhedral line transversals in space

Algorithms are developed for determining if a set of polyhedral objects inR ³ can be intersected by a common transversal (stabbing) line. It can be determined inO(n logn) time if a set ofn line segments in space has a line transversal, and such a transversal can be found in the same time bound. For a set of polyhedra with a total ofn vertices, we give anO(n ⁴ logn) algorithm for determining the existence of, and computing, a line transversal. Helly-type theorems for lines and segments are also given. In particular, it is shown that if every six of a set of lines in space are intersected by a common transversal, then the entire set has a common transversal.     

Remote actuation by optical power transmission in fibres

Some preliminary experiments on the remote actuation of a servo-valve are described in which the power transmission medium is an optical fibre. Using a semiconductor laser or a Nd:YAG laser as the power source and silicon cells as the opto-electrical transducer, switching of the servo-valve has been observed at power levels of a few milliwatts over 800 m of fibre.     

How good are convex hull algorithms?

A convex polytope P can be specified in two ways: as the convex hull of the vertex set V of P, or as the intersection of the set H of its facet-inducing halfspaces. The vertex enumeration problem is to compute V from H>. The facet enumeration problem is to compute H from V. These two problems are essentially equivalent under point/hyperplane duality. They are among the central computational problems in the theory of polytopes. It is open whether they can be solved in time polynomial in |H| + |V| and the dimension. In this paper we consider the main known classes of algorithms for solving these problems. We argue that they all have at least one of two weaknesses: inability to deal well with “degeneracies”, or, inability to control the sizes of intermediate results. We then introduce families of polytopes that exercise those weaknesses. Roughly speaking, fat-lattice or intricate polytopes cause algorithms with bad degeneracy handling to perform badly; dwarfed polytopes cause algorithms with bad intermediate size control to perform badly. We also present computational experience with trying to solve these problem on these hard polytopes, using various implementations of the main algorithms.     

On minimal 5-chromatic triangle-free graphs

It is shown that the minimum number of vertices in a triangle-free 5-chromatic graph is at least 19.