A study of the electric field in an open magnetospheric model

The qualitative properties of an open magnetosphere and its electric field are examined and compared to a simple model of a dipole in a constant field and to actual observations. Many of these properties are found to depend on the separatrix, a curve connecting neutral points and separating different field-line regimes. In the simple model, the electric field in the central polar cap tends to point from dawn to dusk for a wide choice of external fields. Near the boundary of the polar cap electric equipotentials curve and become crescent-shaped, which may explain the correlation of polar magnetic variations with the azimuthal component of the interplanetary magnetic field, reported by Svalgaard. Modifications expected to occur in the actual magnetosphere are also investigated: in particular, it appears that bending of equipotentials may be reduced by cross-field flow during the merging of field lines and that open field lines connected to the polar caps emerge from a long and narrow slot extending along the tail

On the Parameterized Complexity of Red-Blue Points Separation

We study the following geometric separation problem: Given a set R of red points and a set B of blue points in the plane, find a minimum-size set of lines that separate R from B. We show that, in its full generality, parameterized by the number of lines k in the solution, the problem is unlikely to be solvable significantly faster than the bruteforce nO(k) -time algorithm, where n is the total number of points. Indeed, we show that an algorithm running in time f(k)nᵒ(k/log k) , for any computable function f, would disprove ETH. Our reduction crucially relies on selecting lines from a set with a large number of different slopes (i.e., this number is not a function of k). Conjecturing that the problem variant where the lines are required to be axis-parallel is FPT in the number of lines, we show the following preliminary result. Separating R from B with a minimum-size set of axis-parallel lines is FPT in the size of either set, and can be solved in time O∗(9|B|) (assuming that B is the smaller set)

Towards the computation of the convex hull of a configuration from its corresponding separating matrix

In this paper, we cope with the following problem: compute the size of the convex hull of a configuration C, where the given data is the number of separating lines between any two points of the configuration (where the lines are generated by pairs of other points of the configuration). We give an algorithm for the case that the convex hull is of size 3, and a partial algorithm and some directions for the case that the convex hull is of size bigger than 3.Comment: 10 pages, 3 figures; To appear in the Australasian Journal of Combinatoric