36,972 research outputs found
Growth Tight Actions of Product Groups
A group action on a metric space is called growth tight if the exponential
growth rate of the group with respect to the induced pseudo-metric is strictly
greater than that of its quotients. A prototypical example is the action of a
free group on its Cayley graph with respect to a free generating set. More
generally, with Arzhantseva we have shown that group actions with strongly
contracting elements are growth tight.
Examples of non-growth tight actions are product groups acting on the
products of Cayley graphs of the factors.
In this paper we consider actions of product groups on product spaces, where
each factor group acts with a strongly contracting element on its respective
factor space. We show that this action is growth tight with respect to the
metric on the product space, for all . In particular, the
metric on a product of Cayley graphs corresponds to a word metric on
the product group. This gives the first examples of groups that are growth
tight with respect to an action on one of their Cayley graphs and non-growth
tight with respect to an action on another, answering a question of Grigorchuk
and de la Harpe.Comment: 13 pages v2 15 pages, minor changes, to appear in Groups, Geometry,
and Dynamic
Asymptotic normality of extreme value estimators on
Consider i.i.d. random elements on . We show that, under an
appropriate strengthening of the domain of attraction condition, natural
estimators of the extreme-value index, which is now a continuous function, and
the normalizing functions have a Gaussian process as limiting distribution. A
key tool is the weak convergence of a weighted tail empirical process, which
makes it possible to obtain the results uniformly on . Detailed examples
are also presented.Comment: Published at http://dx.doi.org/10.1214/009053605000000831 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Game-Theoretic Framework for Medium Access Control
In this paper, we generalize the random access game model, and show that it provides a general game-theoretic framework for designing contention based medium access control. We extend the random access game model to the network with multiple contention measure signals, study the design of random access games, and analyze different distributed algorithms achieving their equilibria. As examples, a series of utility functions is proposed for games achieving the maximum throughput in a network of homogeneous nodes. In a network with n traffic classes, an N-signal game model is proposed which achieves the maximum throughput under the fairness constraint among different traffic classes. In addition, the convergence of different dynamic algorithms such as best response, gradient play and Jacobi play under propagation delay and estimation error is established. Simulation results show that game model based protocols can achieve superior performance over the standard IEEE 802.11 DCF, and comparable performance as existing protocols with the best performance in literature
Improved transfer matrix method without numerical instability
A new improved transfer matrix method (TMM) is presented. It is shown that
the method not only overcomes the numerical instability found in the original
TMM, but also greatly improves the scalability of computation. The new improved
TMM has no extra cost of computing time as the length of homogeneous scattering
region becomes large. The comparison between the scattering matrix method(SMM)
and our new TMM is given. It clearly shows that our new method is much faster
than SMM.Comment: 5 pages,3 figure
Growth Tight Actions
We introduce and systematically study the concept of a growth tight action.
This generalizes growth tightness for word metrics as initiated by Grigorchuk
and de la Harpe. Given a finitely generated, non-elementary group acting on
a --space , we prove that if contains a strongly
contracting element and if is not too badly distorted in ,
then the action of on is a growth tight action. It follows
that if is a cocompact, relatively hyperbolic --space, then
the action of on is a growth tight action. This generalizes
all previously known results for growth tightness of cocompact actions: every
already known example of a group that admits a growth tight action and has some
infinite, infinite index normal subgroups is relatively hyperbolic, and,
conversely, relatively hyperbolic groups admit growth tight actions. This also
allows us to prove that many CAT(0) groups, including flip-graph-manifold
groups and many Right Angled Artin Groups, and snowflake groups admit
cocompact, growth tight actions. These provide first examples of non-relatively
hyperbolic groups admitting interesting growth tight actions. Our main result
applies as well to cusp uniform actions on hyperbolic spaces and to the action
of the mapping class group on Teichmueller space with the Teichmueller metric.
Towards the proof of our main result, we give equivalent characterizations of
strongly contracting elements and produce new examples of group actions with
strongly contracting elements.Comment: 29 pages, 4 figures v2 added references v3 40 pages, 6 figures,
expanded preliminary sections to make paper more self-contained, other minor
improvements v4 updated bibliography, to appear in Pacific Journal of
Mathematic
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