39,910 research outputs found
Bounds on the maximum multiplicity of some common geometric graphs
We obtain new lower and upper bounds for the maximum multiplicity of some
weighted and, respectively, non-weighted common geometric graphs drawn on n
points in the plane in general position (with no three points collinear):
perfect matchings, spanning trees, spanning cycles (tours), and triangulations.
(i) We present a new lower bound construction for the maximum number of
triangulations a set of n points in general position can have. In particular,
we show that a generalized double chain formed by two almost convex chains
admits {\Omega}(8.65^n) different triangulations. This improves the bound
{\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by
Aichholzer et al.
(ii) We present a new lower bound of {\Omega}(12.00^n) for the number of
non-crossing spanning trees of the double chain composed of two convex chains.
The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years.
(iii) Using a recent upper bound of 30^n for the number of triangulations,
due to Sharir and Sheffer, we show that n points in the plane in general
position admit at most O(68.62^n) non-crossing spanning cycles.
(iv) We derive lower bounds for the number of maximum and minimum weighted
geometric graphs (matchings, spanning trees, and tours). We show that the
number of shortest non-crossing tours can be exponential in n. Likewise, we
show that both the number of longest non-crossing tours and the number of
longest non-crossing perfect matchings can be exponential in n. Moreover, we
show that there are sets of n points in convex position with an exponential
number of longest non-crossing spanning trees. For points in convex position we
obtain tight bounds for the number of longest and shortest tours. We give a
combinatorial characterization of the longest tours, which leads to an O(nlog
n) time algorithm for computing them
Nonlinear Dynamics of Moving Curves and Surfaces: Applications to Physical Systems
The subject of moving curves (and surfaces) in three dimensional space (3-D)
is a fascinating topic not only because it represents typical nonlinear
dynamical systems in classical mechanics, but also finds important applications
in a variety of physical problems in different disciplines. Making use of the
underlying geometry, one can very often relate the associated evolution
equations to many interesting nonlinear evolution equations, including soliton
possessing nonlinear dynamical systems. Typical examples include dynamics of
filament vortices in ordinary and superfluids, spin systems, phases in
classical optics, various systems encountered in physics of soft matter, etc.
Such interrelations between geometric evolution and physical systems have
yielded considerable insight into the underlying dynamics. We present a
succinct tutorial analysis of these developments in this article, and indicate
further directions. We also point out how evolution equations for moving
surfaces are often intimately related to soliton equations in higher
dimensions.Comment: Review article, 38 pages, 7 figs. To appear in Int. Jour. of Bif. and
Chao
Controlled coarse homology and isoperimetric inequalities
We study a coarse homology theory with prescribed growth conditions. For a
finitely generated group G with the word length metric this homology theory
turns out to be related to amenability of G. We characterize vanishing of a
certain fundamental class in our homology in terms of an isoperimetric
inequality on G and show that on any group at most linear control is needed for
this class to vanish. The latter is a homological version of the classical
Burnside problem for infinite groups, with a positive solution. As applications
we characterize existence of primitives of the volume form with prescribed
growth and show that coarse homology classes obstruct weighted Poincare
inequalities.Comment: Final version, to appear in the Journal of Topolog
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