27,809 research outputs found
Guarding art galleries by guarding witnesses
Let P be a simple polygon. We de ne a witness set W to be a set of points su
h that if any (prospective) guard set G guards W, then it is guaranteed that G guards P . We show that not all polygons admit a nite witness set. If a fi nite minimal witness set exists, then it cannot contain any witness in the interior of P ; all witnesses must lie on the boundary of P , and there
an be at most one witness in the interior of any edge. We give an algorithm to compute a minimal witness set for P in O(n2 log n) time, if such a set exists, or to report the non-existence within the same time bounds. We also outline an algorithm that uses a witness set for P to test whether a (prospective) guard set sees all points in P
Solution landscape of a reduced Landau-de Gennes model on a hexagon
We investigate the solution landscape of a reduced Landau--de Gennes model for nematic liquid crystals on a two-dimensional hexagon at a fixed temperature, as a function of λ---the edge length. This is a generic example for reduced approaches on regular polygons. We apply the high-index optimization-based shrinking dimer method to systematically construct the solution landscape consisting of multiple defect solutions and relationships between them. We report a new stable T state with index-0 that has an interior −1/2 defect; new classes of high-index saddle points with multiple interior defects referred to as H class and TD class; changes in the Morse index of saddle points with λ2 and novel pathways mediated by high-index saddle points that can control and steer dynamical pathways. The range of topological degrees, locations and multiplicity of defects offered by these saddle points can be used to navigate through complex solution landscapes of nematic liquid crystals and other related soft matter systems
A classification of the irreducible algebraic A-hypergeometric functions associated to planar point configurations
We consider A-hypergeometric functions associated to normal sets in the
plane. We give a classification of all point configurations for which there
exists a parameter vector such that the associated hypergeometric function is
algebraic. In particular, we show that there are no irreducible algebraic
functions if the number of boundary points is sufficiently large and A is not a
pyramid.Comment: 24 pages, 8 tables, 13 figure
On -Guarding Thin Orthogonal Polygons
Guarding a polygon with few guards is an old and well-studied problem in
computational geometry. Here we consider the following variant: We assume that
the polygon is orthogonal and thin in some sense, and we consider a point
to guard a point if and only if the minimum axis-aligned rectangle spanned
by and is inside the polygon. A simple proof shows that this problem is
NP-hard on orthogonal polygons with holes, even if the polygon is thin. If
there are no holes, then a thin polygon becomes a tree polygon in the sense
that the so-called dual graph of the polygon is a tree. It was known that
finding the minimum set of -guards is polynomial for tree polygons, but the
run-time was . We show here that with a different approach
the running time becomes linear, answering a question posed by Biedl et al.
(SoCG 2011). Furthermore, the approach is much more general, allowing to
specify subsets of points to guard and guards to use, and it generalizes to
polygons with holes or thickness , becoming fixed-parameter tractable in
.Comment: 18 page
Moduli of Tropical Plane Curves
We study the moduli space of metric graphs that arise from tropical plane
curves. There are far fewer such graphs than tropicalizations of classical
plane curves. For fixed genus , our moduli space is a stacky fan whose cones
are indexed by regular unimodular triangulations of Newton polygons with
interior lattice points. It has dimension unless or .
We compute these spaces explicitly for .Comment: 31 pages, 25 figure
Multiple coverings with closed polygons
A planar set is said to be cover-decomposable if there is a constant
such that every -fold covering of the plane with translates of
can be decomposed into two coverings. It is known that open convex polygons are
cover-decomposable. Here we show that closed, centrally symmetric convex
polygons are also cover-decomposable. We also show that an infinite-fold
covering of the plane with translates of can be decomposed into two
infinite-fold coverings. Both results hold for coverings of any subset of the
plane.Comment: arXiv admin note: text overlap with arXiv:1009.4641 by other author
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