5,469 research outputs found
A Semidefinite Approach to the Cover Problem
We apply theta body relaxations to the -cover problem and show
polynomial time solvability for certain classes of graphs. In particular, we
give an effective relaxation where all --hole facets are valid, and
study its relation to an open question of Conforti et al. For the triangle free
problem, we show for that the theta body relaxations do not converge by
steps; we also prove for all an integrality gap of 2 for the
second theta body
Bootstrap percolation on the Hamming torus
The Hamming torus of dimension is the graph with vertices
and an edge between any two vertices that differ in a single
coordinate. Bootstrap percolation with threshold starts with a random
set of open vertices, to which every vertex belongs independently with
probability , and at each time step the open set grows by adjoining every
vertex with at least open neighbors. We assume that is large and
that scales as for some , and study the probability
that an -dimensional subgraph ever becomes open. For large , we
prove that the critical exponent is about for , and
about for . Our small
results are mostly limited to , where we identify the critical in
many cases and, when , compute exactly the critical probability that
the entire graph is eventually open.Comment: Published in at http://dx.doi.org/10.1214/13-AAP996 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Generous Pragmatism
In their introduction to the book, Nurture, about BNIM\u27s work, authors Rodolphe el-Khoury and Andrew Payne spoke about the emergence of a new pragmatism in contemporary architecture, a concern with how well buildings perform in response to the full range of social and ecological issues they are called on to organize
Black hole initial data on hyperboloidal slices
We generalize Bowen-York black hole initial data to hyperboloidal constant
mean curvature slices which extend to future null infinity. We solve this
initial value problem numerically for several cases, including unequal mass
binary black holes with spins and boosts. The singularity at null infinity in
the Hamiltonian constraint associated with a constant mean curvature
hypersurface does not pose any particular difficulties. The inner boundaries of
our slices are minimal surfaces. Trumpet configurations are explored both
analytically and numerically.Comment: version for publication in Phys. Rev.
Decision problems for word-hyperbolic semigroups
This paper studies decision problems for semigroups that are word-hyperbolic in the sense of Duncan & Gilman. A fundamental investigation reveals that the natural definition of a `word-hyperbolic structure' has to be strengthened slightly in order to define a unique semigroup up to isomorphism. The isomorphism problem is proven to be undecidable for word-hyperbolic semigroups (in contrast to the situation for word-hyperbolic groups). It is proved that it is undecidable whether a word-hyperbolic semigroup is automatic, asynchronously automatic, biautomatic, or asynchronously biautomatic. (These properties do not hold in general for word-hyperbolic semigroups.) It is proved that the uniform word problem for word-hyperbolic semigroup is solvable in polynomial time (improving on the previous exponential-time algorithm). Algorithms are presented for deciding whether a word-hyperbolic semigroup is a monoid, a group, a completely simple semigroup, a Clifford semigroup, or a free semigroup.PostprintPeer reviewe
Uniqueness and Non-uniqueness in the Einstein Constraints
The conformal thin sandwich (CTS) equations are a set of four of the Einstein
equations, which generalize the Laplace-Poisson equation of Newton's theory. We
examine numerically solutions of the CTS equations describing perturbed
Minkowski space, and find only one solution. However, we find {\em two}
distinct solutions, one even containing a black hole, when the lapse is
determined by a fifth elliptic equation through specification of the mean
curvature. While the relationship of the two systems and their solutions is a
fundamental property of general relativity, this fairly simple example of an
elliptic system with non-unique solutions is also of broader interest.Comment: 4 pages, 4 figures; abstract and introduction rewritte
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