27,564 research outputs found
Simplex and Polygon Equations
It is shown that higher Bruhat orders admit a decomposition into a higher
Tamari order, the corresponding dual Tamari order, and a "mixed order." We
describe simplex equations (including the Yang-Baxter equation) as realizations
of higher Bruhat orders. Correspondingly, a family of "polygon equations"
realizes higher Tamari orders. They generalize the well-known pentagon
equation. The structure of simplex and polygon equations is visualized in terms
of deformations of maximal chains in posets forming 1-skeletons of polyhedra.
The decomposition of higher Bruhat orders induces a reduction of the
-simplex equation to the -gon equation, its dual, and a compatibility
equation
On contractible edges in convex decompositions
Let be a convex decomposition of a set of points in
general position in the plane. If consists of more than one polygon, then
either contains a deletable edge or contains a contractible edge
Weighted skeletons and fixed-share decomposition
AbstractWe introduce the concept of weighted skeleton of a polygon and present various decomposition and optimality results for this skeletal structure when the underlying polygon is convex
Tessellations of hyperbolic surfaces
A finite subset S of a closed hyperbolic surface F canonically determines a
"centered dual decomposition" of F: a cell structure with vertex set S,
geodesic edges, and 2-cells that are unions of the corresponding Delaunay
polygons. Unlike a Delaunay polygon, a centered dual 2-cell Q is not determined
by its collection of edge lengths; but together with its combinatorics, these
determine an "admissible space" parametrizing geometric possibilities for the
Delaunay cells comprising Q. We illustrate its application by using the
centered dual decomposition to extract combinatorial information about the
Delaunay tessellation among certain genus-2 surfaces, and with this relate
injectivity radius to covering radius here.Comment: 56 pages, 8 figure
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