69,796 research outputs found
On the power of the semi-separated pair decomposition
A Semi-Separated Pair Decomposition (SSPD), with parameter s > 1, of a set is a set {(A i ,B i )} of pairs of subsets of S such that for each i, there are balls and containing A i and B i respectively such that min ( radius ) , radius ), and for any two points p, q S there is a unique index i such that p A i and q B i or vice-versa. In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set of n imprecise points, modeled as pairwise disjoint balls, admits a t-spanner with edges which can be computed in time. If all balls have the same radius, the number of edges reduces to . Secondly, for a set of n points in the plane, we design a query data structure for half-plane closest-pair queries that can be built in time using space and answers a query in time, for any ε> 0. By reducing the preprocessing time to and using space, the query can be answered in time. Moreover, we improve the preprocessing time of an existing axis-parallel rectangle closest-pair query data structure from quadratic to near-linear. Finally, we revisit some previously studied problems, namely spanners for complete k-partite graphs and l
Rank rigidity for CAT(0) cube complexes
We prove that any group acting essentially without a fixed point at infinity
on an irreducible finite-dimensional CAT(0) cube complex contains a rank one
isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube
complexes. We derive a number of other consequences for CAT(0) cube complexes,
including a purely geometric proof of the Tits Alternative, an existence result
for regular elements in (possibly non-uniform) lattices acting on cube
complexes, and a characterization of products of trees in terms of bounded
cohomology.Comment: 39 pages, 4 figures. Revised version according to referee repor
Relative Riemann-Zariski spaces
In this paper we study relative Riemann-Zariski spaces attached to a morphism
of schemes and generalizing the classical Riemann-Zariski space of a field. We
prove that similarly to the classical RZ spaces, the relative ones can be
described either as projective limits of schemes in the category of locally
ringed spaces or as certain spaces of valuations. We apply these spaces to
prove the following two new results: a strong version of stable modification
theorem for relative curves; a decomposition theorem which asserts that any
separated morphism between quasi-compact and quasi-separated schemes factors as
a composition of an affine morphism and a proper morphism. (In particular, we
obtain a new proof of Nagata's compactification theorem.)Comment: 30 pages, the final version, to appear in Israel J. of Mat
Universality of Long-Distance AdS Physics from the CFT Bootstrap
We begin by explicating a recent proof of the cluster decomposition principle
in AdS_{d+1} from the CFT_d bootstrap in d > 2. The CFT argument also computes
the leading interactions between distant objects in AdS, and we confirm the
universal agreement between the CFT bootstrap and AdS gravity in the
semi-classical limit. We proceed to study the generalization to 2d CFTs, which
requires knowledge of the Virasoro conformal blocks in a lightcone OPE limit.
We compute these blocks in a semiclassical, large central charge approximation,
and use them to prove a suitably modified theorem. In particular, from the 2d
bootstrap we prove the existence of large spin operators with fixed 'anomalous
dimensions' indicative of the presence of deficit angles in AdS_3. As we
approach the threshold for the BTZ black hole, interpreted as a CFT scaling
dimension, the twist spectrum of large spin operators becomes dense. Due to the
exchange of the Virasoro identity block, primary states above the BTZ threshold
mimic a thermal background for light operators. We derive the BTZ quasi-normal
modes, and we use the bootstrap equation to prove that the twist spectrum is
dense. Corrections to thermality could be obtained from a more refined
computation of the Virasoro conformal blocks.Comment: 34+31 pages, references added, typo in higher-dimensional energy
shift corrected, discussion of coefficient density bounds expande
- …