241,064 research outputs found
Covering many points with a small-area box
Let be a set of points in the plane. We show how to find, for a given
integer , the smallest-area axis-parallel rectangle that covers points
of in time. We also consider the problem of,
given a value , covering as many points of as possible with an
axis-parallel rectangle of area at most . For this problem we give a
probabilistic -approximation that works in near-linear time:
In time we find an
axis-parallel rectangle of area at most that, with high probability,
covers at least points, where
is the maximum possible number of points that could be
covered
Generalized dynamical entropies in weakly chaotic systems
A large class of technically non-chaotic systems, involving scatterings of
light particles by flat surfaces with sharp boundaries, is nonetheless
characterized by complex random looking motion in phase space. For these
systems one may define a generalized, Tsallis type dynamical entropy that
increases linearly with time. It characterizes a maximal gain of information
about the system that increases as a power of time. However, this entropy
cannot be chosen independently from the choice of coarse graining lengths and
it assigns positive dynamical entropies also to fully integrable systems. By
considering these dependencies in detail one usually will be able to
distinguish weakly chaotic from fully integrable systems.Comment: Submitted to Physica D for the proceedings of the Santa Fe workshop
of November 6-9, 2002 on Anomalous Distributions, Nonlinear Dynamics and
Nonextensivity. 8 pages and two figure
Harmonic Currents of Finite Energy and Laminations
We introduce, on a complex Kahler manifold (M,\omega), a notion of energy for
harmonic currents of bidegree (1,1). This allows us to define for positive harmonic currents. We then show that for a
lamination with singularities of a compact set in P^2 there is a unique
positive harmonic current which minimizes energy. If X is a compact laminated
set in P^2 of class C^1 it carries a unique positive harmonic current T of mass
1. The current T can be obtained by an Ahlfors type construction starting with
a arbitrary leaf of X.Comment: 29 page
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