13,801 research outputs found
Combinatorial complexity in o-minimal geometry
In this paper we prove tight bounds on the combinatorial and topological
complexity of sets defined in terms of definable sets belonging to some
fixed definable family of sets in an o-minimal structure. This generalizes the
combinatorial parts of similar bounds known in the case of semi-algebraic and
semi-Pfaffian sets, and as a result vastly increases the applicability of
results on combinatorial and topological complexity of arrangements studied in
discrete and computational geometry. As a sample application, we extend a
Ramsey-type theorem due to Alon et al., originally proved for semi-algebraic
sets of fixed description complexity to this more general setting.Comment: 25 pages. Revised version. To appear in the Proc. London Math. So
Three-dimensional maps and subgroup growth
In this paper we derive a generating series for the number of cellular
complexes known as pavings or three-dimensional maps, on darts, thus
solving an analogue of Tutte's problem in dimension three.
The generating series we derive also counts free subgroups of index in
via a simple bijection
between pavings and finite index subgroups which can be deduced from the action
of on the cosets of a given subgroup. We then show that this
generating series is non-holonomic. Furthermore, we provide and study the
generating series for isomorphism classes of pavings, which correspond to
conjugacy classes of free subgroups of finite index in .
Computational experiments performed with software designed by the authors
provide some statistics about the topology and combinatorics of pavings on
darts.Comment: 17 pages, 6 figures, 1 table; computational experiments added; a new
set of author
Optimal topological simplification of discrete functions on surfaces
We solve the problem of minimizing the number of critical points among all
functions on a surface within a prescribed distance {\delta} from a given input
function. The result is achieved by establishing a connection between discrete
Morse theory and persistent homology. Our method completely removes homological
noise with persistence less than 2{\delta}, constructively proving the
tightness of a lower bound on the number of critical points given by the
stability theorem of persistent homology in dimension two for any input
function. We also show that an optimal solution can be computed in linear time
after persistence pairs have been computed.Comment: 27 pages, 8 figure
Simplicial Multivalued Maps and the Witness Complex for Dynamical Analysis of Time Series
Topology based analysis of time-series data from dynamical systems is
powerful: it potentially allows for computer-based proofs of the existence of
various classes of regular and chaotic invariant sets for high-dimensional
dynamics. Standard methods are based on a cubical discretization of the
dynamics and use the time series to construct an outer approximation of the
underlying dynamical system. The resulting multivalued map can be used to
compute the Conley index of isolated invariant sets of cubes. In this paper we
introduce a discretization that uses instead a simplicial complex constructed
from a witness-landmark relationship. The goal is to obtain a natural
discretization that is more tightly connected with the invariant density of the
time series itself. The time-ordering of the data also directly leads to a map
on this simplicial complex that we call the witness map. We obtain conditions
under which this witness map gives an outer approximation of the dynamics, and
thus can be used to compute the Conley index of isolated invariant sets. The
method is illustrated by a simple example using data from the classical H\'enon
map.Comment: laTeX, 9 figures, 32 page
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