1,393 research outputs found
Assouad-Nagata dimension and gap for ordered metric spaces
We prove that all spaces of finite Assouad-Nagata dimension admit a good
order for Travelling Salesman Problem, and provide sufficient conditions under
which the converse is true. We formulate a conjectural characterisation of
spaces of finite -dimension, which would yield a gap statement for the
efficiency of orders on metric spaces. Under assumption of doubling, we prove a
stronger gap phenomenon about all orders on a given metric space.Comment: 27 pages. This paper appeared originally as the second part of first
versions of arXiv:2011.01732. Now the paper is split it two parts, the first
one "Spaces that can be ordered effectively: virtually free groups and
hyperbolicity" arXiv:2011.01732, and the second part her
Detection of an anomalous cluster in a network
We consider the problem of detecting whether or not, in a given sensor
network, there is a cluster of sensors which exhibit an "unusual behavior."
Formally, suppose we are given a set of nodes and attach a random variable to
each node. We observe a realization of this process and want to decide between
the following two hypotheses: under the null, the variables are i.i.d. standard
normal; under the alternative, there is a cluster of variables that are i.i.d.
normal with positive mean and unit variance, while the rest are i.i.d. standard
normal. We also address surveillance settings where each sensor in the network
collects information over time. The resulting model is similar, now with a time
series attached to each node. We again observe the process over time and want
to decide between the null, where all the variables are i.i.d. standard normal,
and the alternative, where there is an emerging cluster of i.i.d. normal
variables with positive mean and unit variance. The growth models used to
represent the emerging cluster are quite general and, in particular, include
cellular automata used in modeling epidemics. In both settings, we consider
classes of clusters that are quite general, for which we obtain a lower bound
on their respective minimax detection rate and show that some form of scan
statistic, by far the most popular method in practice, achieves that same rate
to within a logarithmic factor. Our results are not limited to the normal
location model, but generalize to any one-parameter exponential family when the
anomalous clusters are large enough.Comment: Published in at http://dx.doi.org/10.1214/10-AOS839 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The Brownian map is the scaling limit of uniform random plane quadrangulations
76 pages, 7 figures, improved versionWe prove that uniform random quadrangulations of the sphere with n faces, endowed with the usual graph distance and renormalized by n −1/4 , converge as n → ∞ in distribution for the Gromov-Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called Brownian map, which was introduced by Marckert & Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere
Hyperbolicity, Assouad-Nagata dimension and orders on metric spaces
We study asymptotic invariants of metric spaces, defined in terms of the
travelling salesman problem, and our goal is to classify groups and spaces
depending on how well they can be ordered in this context. We characterize
virtually free groups as those admitting an order which has some efficiency on
-point subsets. We show that all -hyperbolic spaces can be ordered
extremely efficiently, for the question when the number of points of a subset
tends to . We prove that all spaces of finite Assouad-Nagata dimension
admit a good order for the above mentioned problem, and under an additional
hypothesis we prove the converse. Despite travelling salesman terminology, our
paper does not aim at applications in computer science. Our goal is to study
new properties of groups and metric spaces, and describe their connection with
more traditional invariants, such as hyperbolicity, dimension, number of ends
and doubling.Comment: title and abstract revised. minor correction
Extended affine Weyl groups and Frobenius manifolds
We define certain extensions of affine Weyl groups (distinct from these
considered by K. Saito [S1] in the theory of extended affine root systems),
prove an analogue of Chevalley theorem for their invariants, and construct a
Frobenius structure on their orbit spaces. This produces solutions of WDVV equations of associativity polynomial in .Comment: 69 pages, amslatex, some references added, position of Table 1 is
corrected. Revised version for Compositio Mathematic
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