14,845 research outputs found
Decompositions of a polygon into centrally symmetric pieces
In this paper we deal with edge-to-edge, irreducible decompositions of a
centrally symmetric convex -gon into centrally symmetric convex pieces.
We prove an upper bound on the number of these decompositions for any value of
, and characterize them for octagons.Comment: 17 pages, 17 figure
Deformations of bordered Riemann surfaces and associahedral polytopes
We consider the moduli space of bordered Riemann surfaces with boundary and
marked points. Such spaces appear in open-closed string theory, particularly
with respect to holomorphic curves with Lagrangian submanifolds. We consider a
combinatorial framework to view the compactification of this space based on the
pair-of-pants decomposition of the surface, relating it to the well-known
phenomenon of bubbling. Our main result classifies all such spaces that can be
realized as convex polytopes. A new polytope is introduced based on truncations
of cubes, and its combinatorial and algebraic structures are related to
generalizations of associahedra and multiplihedra.Comment: 25 pages, 31 figure
All finitely presented groups are QSF
This is the third and last of three papers containing the complete proof that
all finitely presented groups are QSF.Comment: 57 page
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
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