309 research outputs found
Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley
A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all
roots of its characteristic polynomial have the same real part. This property
was conjectured by Postnikov and Stanley for certain families of arrangements
which are defined for any irreducible root system and was proved for the root
system . The proof is based on an explicit formula for the
characteristic polynomial, which is of independent combinatorial significance.
Here our previous derivation of this formula is simplified and extended to
similar formulae for all but the exceptional root systems. The conjecture
follows in these cases
Hyperplane arrangements in negatively curved manifolds and relative hyperbolicity
We show that certain aspherical manifolds arising from hyperplane
arrangements in negatively curved manifolds have relatively hyperbolic
fundamental group.Comment: 27 pages, minor changes, to appear in Groups, Geometry, and Dynamic
Social choice among complex objects: Mathematical tools
Here the reader can find some basic definitions and notations in order to better understand the model for social choise described by L. Marengo and S. Settepanella in their paper: Social choice among complex objects. The interested reader can refer to [Bou68], [Massey] and [OT92] to go into more depth.Arrangements, simplicial complexes, CW complexes,fundamental group, Salvetti's complex.
Social choice on complex objects: A geometric approach
Marengo and Pasquali (2008) present a model of object construction in majority voting and show that, in general, by appropriate changes of such bundles, different social outcomes may be obtained. In this paper we extend and generalize this approach by providing a geometric model of individual preferences and social aggregation based on hyperplanes and their arrangements. As an application of this model we give a necessary condition for existence of a local social optimum. Moreover we address the question if a social decision rule depends also upon the number of voting agents. More precisely: are there social decision rules that can be obtained by an odd (even) number of voting agent which cannot be obtained by only three (two) voting agent? The answer is negative. Indeed three (or two) voting agent can produce all possible social decision rules.Social choice; object construction power; agenda power; intransitive cycles; arrangements; graph theory.
Optimal strong stationary times for random walks on the chambers of a hyperplane arrangement
This paper studies Markov chains on the chambers of real hyperplane
arrangements, a model that generalizes famous examples, such as the Tsetlin
library and riffle shuffles. We discuss cutoff for the Tsetlin library for
general weights, and we give an exact formula for the separation distance for
the hyperplane arrangement walk. We introduce lower bounds, which allow for the
first time to study cutoff for hyperplane arrangement walks under certain
conditions. Using similar techniques, we also prove a uniform lower bound for
the mixing time of Glauber dynamics on a monotone system.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1605.0833
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