6,460 research outputs found
Schmidt Games and Conditions on Resonant Sets
Winning sets of Schmidt's game enjoy a remarkable rigidity. Therefore, this
game (and modifications of it) have been applied to many examples of complete
metric spaces (X, d) to show that the set of "badly approximable points", with
respect to a given collection of resonant sets in X, is a winning set. For
these examples, strategies were deduced that are, in most cases, strongly
adapted to the specific dynamics and properties of the underlying setting. We
introduce a new modification of Schmidt's game which is a combination and
generalization of the ones of [18] and [20]. This modification allows us to
axiomatize conditions on the collection of resonant sets under which there
always exists a winning strategy. Moreover, we discuss properties of winning
sets of this modification and verify our conditions for several examples -
among them, the set of badly approximable vectors in the Euclidian space and
the p-adic integers with weights and, as a main example, the set of geodesic
rays in proper geodesic CAT(-1) spaces which avoid a suitable collection of
convex subsets.Comment: 30 pages, Comments are welcome
Uncertainty in phylogenetic tree estimates
Estimating phylogenetic trees is an important problem in evolutionary
biology, environmental policy and medicine. Although trees are estimated, their
uncertainties are discarded by mathematicians working in tree space. Here we
explicitly model the multivariate uncertainty of tree estimates. We consider
both the cases where uncertainty information arises extrinsically (through
covariate information) and intrinsically (through the tree estimates
themselves). The importance of accounting for tree uncertainty in tree space is
demonstrated in two case studies. In the first instance, differences between
gene trees are small relative to their uncertainties, while in the second, the
differences are relatively large. Our main goal is visualization of tree
uncertainty, and we demonstrate advantages of our method with respect to
reproducibility, speed and preservation of topological differences compared to
visualization based on multidimensional scaling. The proposal highlights that
phylogenetic trees are estimated in an extremely high-dimensional space,
resulting in uncertainty information that cannot be discarded. Most
importantly, it is a method that allows biologists to diagnose whether
differences between gene trees are biologically meaningful, or due to
uncertainty in estimation.Comment: Final version accepted to Journal of Computational and Graphical
Statistic
On the Decidability of Connectedness Constraints in 2D and 3D Euclidean Spaces
We investigate (quantifier-free) spatial constraint languages with equality,
contact and connectedness predicates as well as Boolean operations on regions,
interpreted over low-dimensional Euclidean spaces. We show that the complexity
of reasoning varies dramatically depending on the dimension of the space and on
the type of regions considered. For example, the logic with the
interior-connectedness predicate (and without contact) is undecidable over
polygons or regular closed sets in the Euclidean plane, NP-complete over
regular closed sets in three-dimensional Euclidean space, and ExpTime-complete
over polyhedra in three-dimensional Euclidean space.Comment: Accepted for publication in the IJCAI 2011 proceeding
Finiteness properties of cubulated groups
We give a generalized and self-contained account of Haglund-Paulin's
wallspaces and Sageev's construction of the CAT(0) cube complex dual to a
wallspace. We examine criteria on a wallspace leading to finiteness properties
of its dual cube complex. Our discussion is aimed at readers wishing to apply
these methods to produce actions of groups on cube complexes and understand
their nature. We develop the wallspace ideas in a level of generality that
facilitates their application.
Our main result describes the structure of dual cube complexes arising from
relatively hyperbolic groups. Let H_1,...,H_s be relatively quasiconvex
codimension-1 subgroups of a group G that is hyperbolic relative to
P_1,...,P_r. We prove that G acts relatively cocompactly on the associated dual
CAT(0) cube complex C. This generalizes Sageev's result that C is cocompact
when G is hyperbolic. When P_1,...,P_r are abelian, we show that the dual
CAT(0) cube complex C has a G-cocompact CAT(0) truncation.Comment: 58 pages, 12 figures. Version 3: Revisions and slightly improved
results in Sections 7 and 8. Several theorem numbers have changed from the
previous versio
Metrizable uniform spaces
Three themes of general topology: quotient spaces; absolute retracts; and
inverse limits - are reapproached here in the setting of metrizable uniform
spaces, with an eye to applications in geometric and algebraic topology. The
results include:
1) If f: A -> Y is a uniformly continuous map, where X and Y are metric
spaces and A is a closed subset of X, we show that the adjunction space X\cup_f
Y with the quotient uniformity (hence also with the topology thereof) is
metrizable, by an explicit metric. This yields natural constructions of cone,
join and mapping cylinder in the category of metrizable uniform spaces, which
we show to coincide with those based on subspace (of a normed linear space); on
product (with a cone); and on the isotropy of the l_2 metric.
2) We revisit Isbell's theory of uniform ANRs, as refined by Garg and Nhu in
the metrizable case. The iterated loop spaces \Omega^n P of a pointed compact
polyhedron P are shown to be uniform ANRs. Four characterizations of uniform
ANRs among metrizable uniform spaces X are given: (i) the completion of X is a
uniform ANR, and the remainder is uniformly a Z-set in the completion; (ii) X
is uniformly locally contractible and satisfies the Hahn approximation
property; (iii) X is uniformly \epsilon-homotopy dominated by a uniform ANR for
each \epsilon>0; (iv) X is an inverse limit of uniform ANRs with "nearly
splitting" bonding maps.Comment: 93 pages. v5: a little bit of new stuff added. Proposition 8.7,
entire section 10, Lemma 14.11, Proposition 18.4. Possibly something els
- …