616 research outputs found
Sofic groups and direct finiteness
We construct an analogue of Neumann's affiliated algebras for sofic group
algebras over arbitrary fields. Consequently, we settle Kaplansky's direct
finiteness conjecture for sofic groups.Comment: 9 pages, a shortened proof of the main resul
Matchings on infinite graphs
Elek and Lippner (2010) showed that the convergence of a sequence of
bounded-degree graphs implies the existence of a limit for the proportion of
vertices covered by a maximum matching. We provide a characterization of the
limiting parameter via a local recursion defined directly on the limit of the
graph sequence. Interestingly, the recursion may admit multiple solutions,
implying non-trivial long-range dependencies between the covered vertices. We
overcome this lack of correlation decay by introducing a perturbative parameter
(temperature), which we let progressively go to zero. This allows us to
uniquely identify the correct solution. In the important case where the graph
limit is a unimodular Galton-Watson tree, the recursion simplifies into a
distributional equation that can be solved explicitly, leading to a new
asymptotic formula that considerably extends the well-known one by Karp and
Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page
Expansive actions on uniform spaces and surjunctive maps
We present a uniform version of a result of M. Gromov on the surjunctivity of
maps commuting with expansive group actions and discuss several applications.
We prove in particular that for any group and any field \K, the
space of -marked groups such that the group algebra \K[G] is
stably finite is compact.Comment: 21 page
On Turing dynamical systems and the Atiyah problem
Main theorems of the article concern the problem of M. Atiyah on possible
values of l^2-Betti numbers. It is shown that all non-negative real numbers are
l^2-Betti numbers, and that "many" (for example all non-negative algebraic)
real numbers are l^2-Betti numbers of simply connected manifolds with respect
to a free cocompact action. Also an explicit example is constructed which leads
to a simply connected manifold with a transcendental l^2-Betti number with
respect to an action of the threefold direct product of the lamplighter group
Z/2 wr Z. The main new idea is embedding Turing machines into integral group
rings. The main tool developed generalizes known techniques of spectral
computations for certain random walk operators to arbitrary operators in
groupoid rings of discrete measured groupoids.Comment: 35 pages; essentially identical to the published versio
Identification of Young Stellar Object candidates in the DR2 x AllWISE catalogue with machine learning methods
The second Data Release (DR2) contains astrometric and photometric
data for more than 1.6 billion objects with mean magnitude 20.7,
including many Young Stellar Objects (YSOs) in different evolutionary stages.
In order to explore the YSO population of the Milky Way, we combined the
DR2 database with WISE and Planck measurements and made an all-sky
probabilistic catalogue of YSOs using machine learning techniques, such as
Support Vector Machines, Random Forests, or Neural Networks. Our input
catalogue contains 103 million objects from the DR2xAllWISE cross-match table.
We classified each object into four main classes: YSOs, extragalactic objects,
main-sequence stars and evolved stars. At a 90% probability threshold we
identified 1,129,295 YSO candidates. To demonstrate the quality and potential
of our YSO catalogue, here we present two applications of it. (1) We explore
the 3D structure of the Orion A star forming complex and show that the spatial
distribution of the YSOs classified by our procedure is in agreement with
recent results from the literature. (2) We use our catalogue to classify
published Science Alerts. As measures the sources at multiple
epochs, it can efficiently discover transient events, including sudden
brightness changes of YSOs caused by dynamic processes of their circumstellar
disk. However, in many cases the physical nature of the published alert sources
are not known. A cross-check with our new catalogue shows that about 30% more
of the published alerts can most likely be attributed to YSO activity.
The catalogue can be also useful to identify YSOs among future alerts.Comment: 19 pages, 12 figures, 3 table
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