4,201 research outputs found
Slime mould computes planar shapes
Computing a polygon defining a set of planar points is a classical problem of
modern computational geometry. In laboratory experiments we demonstrate that a
concave hull, a connected alpha-shape without holes, of a finite planar set is
approximated by slime mould Physarum polycephalum. We represent planar points
with sources of long-distance attractants and short-distance repellents and
inoculate a piece of plasmodium outside the data set. The plasmodium moves
towards the data and envelops it by pronounced protoplasmic tubes
Design of Combined Coverage Area Reporting and Geo-casting of Queries for Wireless Sensor Networks
In order to efficiently deal with queries or other location dependent information, it is key that the wireless sensor network informs gateways what geographical area is serviced by which gateway. The gateways are then able to e.g. efficiently route queries which are only valid in particular regions of the deployment. The proposed algorithms combine coverage area reporting and geographical routing of queries which are injected by gateways.\u
The convex hull of a finite set
We study -separately convex hulls of finite
sets of points in , as introduced in
\cite{KirchheimMullerSverak2003}. When is considered as a
certain subset of matrices, this notion of convexity corresponds
to rank-one convex convexity . If is identified instead
with a subset of matrices, it actually agrees with the quasiconvex
hull, due to a recent result \cite{HarrisKirchheimLin18}.
We introduce " complexes", which generalize constructions. For a
finite set , a " -complex" is a complex whose extremal points
belong to . The "-complex convex hull of ", , is the union
of all -complexes. We prove that is contained in the
convex hull .
We also consider outer approximations to convexity based in the
locality theorem \cite[4.7]{Kirchheim2003}. Starting with a crude outer
approximation we iteratively chop off "-prisms". For the examples in
\cite{KirchheimMullerSverak2003}, and many others, this procedure reaches a
" -complex" in a finite number of steps, and thus computes the
convex hull.
We show examples of finite sets for which this procedure does not reach the
convex hull in finite time, but we show that a sequence of outer
approximations built with -prisms converges to a -complex. We
conclude that is always a " -complex", which has interesting
consequences
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