128 research outputs found
An example of a non-Borel locally-connected finite-dimensional topological group
Answering a question posed by S.Maillot in MathOverFlow, for every
we construct a locally connected subgroup of dimension , which is not locally compact.Comment: 2 page
The coarse classification of countable abelian groups
We prove that two countable locally finite-by-abelian groups G,H endowed with
proper left-invariant metrics are coarsely equivalent if and only if their
asymptotic dimensions coincide and the groups are either both
finitely-generated or both are infinitely generated. On the other hand, we show
that each countable group G that coarsely embeds into a countable abelian group
is locally nilpotent-by-finite. Moreover, the group G is locally
abelian-by-finite if and only if G is undistorted in the sense that G can be
written as the union of countably many finitely generated subgroups G_n such
that each G_n is undistorted in G_{n+1} (which means that the identity
inclusion from G_n to G_{n+1} is a quasi-isometric embedding with respect to
word metrics).Comment: 25 pages. Longer version with new results about FCC groups, locally
finite-by-abelian groups, locally nilpotent-by-finite groups
On metric spaces with the properties of de Groot and Nagata in dimension one
A metric space has the de Groot property if for any points
there are positive indices such that
and . If, in addition, then
is said to have the Nagata property . It is known that a compact
metrizable space has dimension iff has an admissible
-metric iff has an admissible -metric.
We prove that an embedding of the interval into a
locally connected metric space with property (resp. ) is open
provided is an isometric embedding (resp. has distortion
Dist(f)=\|f\|_\Lip\cdot\|f^{-1}\|_\Lip<2). This implies that the Euclidean
metric cannot be extended from the interval to an admissible
-metric on the triode . Another corollary says that a
topologically homogeneous -space cannot contain an isometric copy of the
interval and a topological copy of the triode simultaneously. Also
we prove that a -metric space containing an isometric copy of each
compact -metric space has density not less than continuum.Comment: 10 page
Constructing balleans
A ballean is a set endowed with a coarse structure.We introduce and explore three constructions of balleans from a pregiven family of balleans: bornological products, bouquets, and combs. We analyze also the smallest and largest coarse structures on a set X compatible with a given bornology on X
Photon Distribution Function for Long-Distance Propagation of Partially Coherent Beams through the Turbulent Atmosphere
The photon density operator function is used to calculate light beam
propagation through turbulent atmosphere. A kinetic equation for the photon
distribution function is derived and solved using the method of
characteristics. Optical wave correlations are described in terms of photon
trajectories that depend on fluctuations of the refractive index. It is shown
that both linear and quadratic disturbances produce sizable effects for
long-distance propagation. The quadratic terms are shown to suppress the
correlation of waves with different wave vectors. We examine the intensity
fluctuations of partially coherent beams (beams whose initial spatial coherence
is partially destroyed). Our calculations show that it is possible to
significantly reduce the intensity fluctuations by using a partially coherent
beam. The physical mechanism responsible for this pronounced reduction is
similar to that of the Hanbury-Braun, Twiss effect.Comment: 28 pages, 4 figure
Partitions of groups and matroids into independent subsets
Can the set R∖{0} be covered by countably many linearly (algebraically) independent subsets over the field Q? We use a matroid approach to show that an answer is ``Yes'' under the Continuum Hypothesis, and ``No'' under its negation
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