65,024 research outputs found
Polychromatic Coloring for Half-Planes
We prove that for every integer , every finite set of points in the plane
can be -colored so that every half-plane that contains at least
points, also contains at least one point from every color class. We also show
that the bound is best possible. This improves the best previously known
lower and upper bounds of and respectively. We also show
that every finite set of half-planes can be colored so that if a point
belongs to a subset of at least of the half-planes then
contains a half-plane from every color class. This improves the best previously
known upper bound of . Another corollary of our first result is a new
proof of the existence of small size \eps-nets for points in the plane with
respect to half-planes.Comment: 11 pages, 5 figure
The Anti-de Sitter Gott Universe: A Rotating BTZ Wormhole
Recently it has been shown that a 2+1 dimensional black hole can be created
by a collapse of two colliding massless particles in otherwise empty anti-de
Sitter space. Here we generalize this construction to the case of a non-zero
impact parameter. The resulting spacetime, which may be regarded as a Gott
universe in anti-de Sitter background, contains closed timelike curves. By
treating these as singular we are able to interpret our solution as a rotating
black hole, hence providing a link between the Gott universe and the BTZ black
hole. When analyzing the spacetime we see how the full causal structure of the
interior can be almost completely inferred just from considerations of the
conformal boundary.Comment: 46 pages (LaTeX2e), 13 figures (eps
Coloring half-planes and bottomless rectangles
We prove lower and upper bounds for the chromatic number of certain
hypergraphs defined by geometric regions. This problem has close relations to
conflict-free colorings. One of the most interesting type of regions to
consider for this problem is that of the axis-parallel rectangles. We
completely solve the problem for a special case of them, for bottomless
rectangles. We also give an almost complete answer for half-planes and pose
several open problems. Moreover we give efficient coloring algorithms
On Cayley Identity for Self-Adjoint Operators in Hilbert Spaces
We prove an analogue to the Cayley identity for an arbitrary self-adjoint
operator in a Hilbert space. We also provide two new ways to characterize
vectors belonging to the singular spectral subspace in terms of the analytic
properties of the resolvent of the operator, computed on these vectors. The
latter are analogous to those used routinely in the scattering theory for the
absolutely continuous subspace
Computing the Walls Associated to Bridgeland Stability Conditions on Projective Surfaces
We derive constraints on the existence of walls for Bridgeland stability
conditions for general projective surfaces. We show that in suitable planes of
stability conditions the walls are bounded and derive conditions for when the
number of walls is globally finite. In examples, we show how to use the
explicit conditions to locate walls and sometimes to show that there are no
walls at all.Comment: 18 pages, 2 figures. A lot of tidying up following referee's comments
especially in section 3. To appear in Asian J. Mat
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