44 research outputs found
Splitting definably compact groups in o-minimal structures
An argument of A.Borel shows that every compact connected Lie group is
homeomorphic to the Cartesian product of its derived subgroup and a torus. We
prove a parallel result for definably compact definably connected groups
definable in an o-minimal expansion of a real closed field. As opposed to the
Lie case, however, we provide an example showing that the derived subgroup may
not have a definable semidirect complement.Comment: final version 13 page
On the computational complexity of a game of cops and robbers
We study the computational complexity of a perfect-information two-player game proposed by Aigner and Fromme (1984) [1]. The game takes place on an undirected graph where n simultaneously moving cops attempt to capture a single robber, all moving at the same speed. The players are allowed to pick their starting positions at the first move. The question of the computational complexity of deciding this game was raised by Goldstein and Reingold (1995) [9]. We prove that the game is hard for PSPACE.©2012 Elsevier B.V. All rights reserved
On definably proper maps
In this paper we work in o-minimal structures with definable Skolem functions
and show that a continuous definable map between Hausdorff locally definably
compact definable spaces is definably proper if and only if it is proper
morphism in the category of definable spaces. We give several other
characterizations of definably proper including one involving the existence of
limits of definable types. We also prove the basic properties of definably
proper maps and the invariance of definably proper in elementary extensions and
o-minimal expansions.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1401.084
Discrete subgroups of locally definable groups
We work in the category of locally definable groups in an o-minimal expansion
of a field. Eleftheriou and Peterzil conjectured that every definably generated
abelian connected group G in this category is a cover of a definable group. We
prove that this is the case under a natural convexity assumption inspired by
the same authors, which in fact gives a necessary and sufficient condition. The
proof is based on the study of the zero-dimensional compatible subgroups of G.
Given a locally definable connected group G (not necessarily definably
generated), we prove that the n-torsion subgroup of G is finite and that every
zero-dimensional compatible subgroup of G has finite rank. Under a convexity
hypothesis we show that every zero-dimensional compatible subgroup of G is
finitely generated.Comment: Final version. 17 pages. To appear in Selecta Mathematic
Condensation and topological phase transitions in a dynamical network model with rewiring of the links
Growing network models with both heterogeneity of the nodes and topological
constraints can give rise to a rich phase structure. We present a simple model
based on preferential attachment with rewiring of the links. Rewiring
probabilities are modulated by the negative fitness of the nodes and by the
constraint for the network to be a simple graph. At low temperatures and high
rewiring rates, this constraint induces a Bose-Einstein condensation of paths
of length 2, i.e. a new phase transition with an extended condensate of links.
The phase space of the model includes further transitions in the scaling of the
connected component and the degeneracy of the network.Comment: 12 pages, 14 figure
Higher homotopy of groups definable in o-minimal structures
It is known that a definably compact group G is an extension of a compact Lie
group L by a divisible torsion-free normal subgroup. We show that the o-minimal
higher homotopy groups of G are isomorphic to the corresponding higher homotopy
groups of L. As a consequence, we obtain that all abelian definably compact
groups of a given dimension are definably homotopy equivalent, and that their
universal cover are contractible.Comment: 13 pages, to be published in the Israel Journal of Mathematic