397,607 research outputs found
Degree-regular triangulations of the double-torus
A connected combinatorial 2-manifold is called degree-regular if each of its
vertices have the same degree. A connected combinatorial 2-manifold is called
weakly regular if it has a vertex-transitive automorphism group. Clearly, a
weakly regular combinatorial 2-manifold is degree-regular and a degree-regular
combinatorial 2-manifold of Euler characteristic - 2 must contain 12 vertices.
In 1982, McMullen et al. constructed a 12-vertex geometrically realized
triangulation of the double-torus in \RR^3. As an abstract simplicial
complex, this triangulation is a weakly regular combinatorial 2-manifold. In
1999, Lutz showed that there are exactly three weakly regular orientable
combinatorial 2-manifolds of Euler characteristic - 2. In this article, we
classify all the orientable degree-regular combinatorial 2-manifolds of Euler
characteristic - 2. There are exactly six such combinatorial 2-manifolds. This
classifies all the orientable equivelar polyhedral maps of Euler characteristic
- 2.Comment: 13 pages. To appear in `Forum Mathematicum
Coloring Complexes and Combinatorial Hopf Monoids
We generalize the notion of coloring complex of a graph to linearized
combinatorial Hopf monoids. These are a generalization of the notion of
coloring complex of a graph. We determine when a combinatorial Hopf monoid has
such a construction, and discover some inequalities that are satisfied by the
quasisymmetric function invariants associated to the combinatorial Hopf monoid.
We show that the collection of all such coloring complexes forms a
combinatorial Hopf monoid, which is the terminal object in the category of
combinatorial Hopf monoids with convex characters. We also study several
examples of combinatorial Hopf monoids.Comment: 37 pages, 5 figure
Combinatorial Calabi flows on surfaces
For triangulated surfaces, we introduce the combinatorial Calabi flow which
is an analogue of smooth Calabi flow. We prove that the solution of
combinatorial Calabi flow exists for all time. Moreover, the solution converges
if and only if Thurston's circle packing exists. As a consequence,
combinatorial Calabi flow provides a new algorithm to find circle packings with
prescribed curvatures. The proofs rely on careful analysis of combinatorial
Calabi energy, combinatorial Ricci potential and discrete dual-Laplacians.Comment: 17 pages, 5 figure
Combinatorial Voting
We study elections that simultaneously decide multiple issues, where voters have independent private values over bundles of issues. The innovation is in considering nonseparable preferences, where issues may be complements or substitutes. Voters face a political exposure problem: the optimal vote for a particular issue will depend on the resolution of the other issues. Moreover, the probabilities that the other issues will pass should be conditioned on being pivotal. We prove that equilibrium exists when distributions over values have full support or when issues are complements. We then study large elections with two issues. There exists a nonempty open set of distributions where the probability of either issue passing fails to converge to either 1 or 0 for all limit equilibria. Thus, the outcomes of large elections are not generically predictable with independent private values, despite the fact that there is no aggregate uncertainty regarding fundamentals. While the Condorcet winner is not necessarily the outcome of a multi-issue election, we provide sufficient conditions that guarantee the implementation of the Condorcet winner. © 2012 The Econometric Society
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