72 research outputs found
Minimum-weight triangulation is NP-hard
A triangulation of a planar point set S is a maximal plane straight-line
graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we
are looking for a triangulation of a given point set that minimizes the sum of
the edge lengths. We prove that the decision version of this problem is
NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the
gadgets is established with computer assistance, using dynamic programming on
polygonal faces, as well as the beta-skeleton heuristic to certify that certain
edges belong to the minimum-weight triangulation.Comment: 45 pages (including a technical appendix of 13 pages), 28 figures.
This revision contains a few improvements in the expositio
Computational Method for Phase Space Transport with Applications to Lobe Dynamics and Rate of Escape
Lobe dynamics and escape from a potential well are general frameworks
introduced to study phase space transport in chaotic dynamical systems. While
the former approach studies how regions of phase space are transported by
reducing the flow to a two-dimensional map, the latter approach studies the
phase space structures that lead to critical events by crossing periodic orbit
around saddles. Both of these frameworks require computation with curves
represented by millions of points-computing intersection points between these
curves and area bounded by the segments of these curves-for quantifying the
transport and escape rate. We present a theory for computing these intersection
points and the area bounded between the segments of these curves based on a
classification of the intersection points using equivalence class. We also
present an alternate theory for curves with nontransverse intersections and a
method to increase the density of points on the curves for locating the
intersection points accurately.The numerical implementation of the theory
presented herein is available as an open source software called Lober. We used
this package to demonstrate the application of the theory to lobe dynamics that
arises in fluid mechanics, and rate of escape from a potential well that arises
in ship dynamics.Comment: 33 pages, 17 figure
Information storing by biomagnetites
Since the discovery of the presence of biogenic magnetites in living
organisms, there have been speculations on the role that these biomagnetites
play in cellular processes. It seems that the formation of biomagnetite
crystals is a universal phenomenon and not an exception in living cells. Many
experimental facts show that features of organic and inorganic processes could
be indistinguishable at nanoscale levels. Living cells are quantum "devices"
rather than simple electronic devices utilizing only the charge of conduction
electrons. In our opinion, due to their unusual biophysical properties, special
biomagnetites must have a biological function in living cells in general and in
the brain in particular. In this paper we advance a hypothesis that while
biomagnetites are developed jointly with organic molecules and cellular
electromagnetic fields in cells, they can record information about the Earth's
magnetic vector potential of the entire flight in migratory birds.Comment: 17 pages, 3 figure
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