728 research outputs found
The definability criterions for convex projective polyhedral reflection groups
Following Vinberg, we find the criterions for a subgroup generated by
reflections \Gamma \subset \SL^{\pm}(n+1,\mathbb{R}) and its finite-index
subgroups to be definable over where is an integrally
closed Noetherian ring in the field . We apply the criterions for
groups generated by reflections that act cocompactly on irreducible properly
convex open subdomains of the -dimensional projective sphere. This gives a
method for constructing injective group homomorphisms from such Coxeter groups
to \SL^{\pm}(n+1,\mathbb{Z}). Finally we provide some examples of
\SL^{\pm}(n+1,\mathbb{Z})-representations of such Coxeter groups. In
particular, we consider simplicial reflection groups that are isomorphic to
hyperbolic simplicial groups and classify all the conjugacy classes of the
reflection subgroups in \SL^{\pm}(n+1,\mathbb{R}) that are definable over
. These were known by Goldman, Benoist, and so on previously.Comment: 31 pages, 8 figure
A switch convergence for a small perturbation of a linear recurrence equation
In this article we study a small random perturbation of a linear recurrence
equation. If all the roots of its corresponding characteristic equation have
modulus strictly less than one, the random linear recurrence goes exponentially
fast to its limiting distribution in the total variation distance as time
increases. By assuming that all the roots of its corresponding characteristic
equation have modulus strictly less than one and some suitable conditions, we
prove that this convergence happens as a switch-type, i.e., there is a sharp
transition in the convergence to its limiting distribution. This fact is known
as a cut-off phenomenon in the context of stochastic processes.Comment: 19 pages. Brazilian Journal of Probability and Statistics 2020
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