1,114 research outputs found
Mass Partitions via Equivariant Sections of Stiefel Bundles
We consider a geometric combinatorial problem naturally associated to the
geometric topology of certain spherical space forms. Given a collection of
mass distributions on , the existence of affinely independent
regular -fans, each of which equipartitions each of the measures, can in
many cases be deduced from the existence of a -equivariant
section of the Stiefel bundle over , where
is the Stiefel manifold of all orthonormal -frames in
or , and
is the corresponding unit sphere. For example, the
parallelizability of when , or implies that any
two masses on can be simultaneously bisected by each of
pairwise-orthogonal hyperplanes, while when or 4, the triviality of the
circle bundle over the standard Lens Spaces
yields that for any mass on , there exist a pair of
complex orthogonal regular -fans, each of which equipartitions the mass.Comment: 11 pages, final versio
Tropical Limits of Probability Spaces, Part I: The Intrinsic Kolmogorov-Sinai Distance and the Asymptotic Equipartition Property for Configurations
The entropy of a finite probability space measures the observable
cardinality of large independent products of the probability
space. If two probability spaces and have the same entropy, there is an
almost measure-preserving bijection between large parts of and
. In this way, and are asymptotically equivalent.
It turns out to be challenging to generalize this notion of asymptotic
equivalence to configurations of probability spaces, which are collections of
probability spaces with measure-preserving maps between some of them.
In this article we introduce the intrinsic Kolmogorov-Sinai distance on the
space of configurations of probability spaces. Concentrating on the large-scale
geometry we pass to the asymptotic Kolmogorov-Sinai distance. It induces an
asymptotic equivalence relation on sequences of configurations of probability
spaces. We will call the equivalence classes \emph{tropical probability
spaces}.
In this context we prove an Asymptotic Equipartition Property for
configurations. It states that tropical configurations can always be
approximated by homogeneous configurations. In addition, we show that the
solutions to certain Information-Optimization problems are
Lipschitz-con\-tinuous with respect to the asymptotic Kolmogorov-Sinai
distance. It follows from these two statements that in order to solve an
Information-Optimization problem, it suffices to consider homogeneous
configurations.
Finally, we show that spaces of trajectories of length of certain
stochastic processes, in particular stationary Markov chains, have a tropical
limit.Comment: Comment to version 2: Fixed typos, a calculation mistake in Lemma 5.1
and its consequences in Proposition 5.2 and Theorem 6.
Magnetism in the spiral galaxy NGC 6946: magnetic arms, depolarization rings, dynamo modes and helical fields
The spiral galaxy NGC 6946 was observed in total intensity and linear
polarization in five radio bands between 3cm and 21cm. At the inner edge of the
inner gas spiral arm the ordered magnetic field is only mildly compressed and
turns smoothly, to become aligned along the gas arm. Hence the field is not
shocked and is probably connected to the warm, diffuse gas. At larger radii,
two bright magnetic arms between the optical arms are visible in polarized
intensity. The field in the northern magnetic arm is almost totally aligned.
Faraday rotation measures (RM) in these arms are consistent with the
superposition of two low azimuthal dynamo modes. Three more magnetic arms are
discovered in the outer galaxy, located between HI arms. Due to strong Faraday
depolarization the galaxy is not transparent to polarized waves at 18cm and
20cm. The large-scale asymmetry in depolarization with respect to the major
axis may be another indication of large-scale helical fields. Three
depolarization rings of almost zero polarization seen at 20cm are probably
generated by differential Faraday rotation in HII complexes in NGC 6946 of
300-500 pc size. - In the gas/optical spiral arms, the total (mostly turbulent)
magnetic field is amplified to \simeq 20\muG. Its energy density is \simeq 10
times larger than that of the ionized gas and is similar to that of the
turbulent gas motions in the inner galaxy. The magnetic energy exceeds that of
the turbulent energy in the outer galaxy.Comment: 18 pages, 28 figures. Accepted for publication in A&A. Corrected typo
in Sect. 3.1 04/06/200
Balanced Islands in Two Colored Point Sets in the Plane
Let be a set of points in general position in the plane, of which
are red and of which are blue. In this paper we prove that there exist: for
every , a convex set containing
exactly red points and exactly
blue points of ; a convex set containing exactly red points and exactly blue points of . Furthermore, we present
polynomial time algorithms to find these convex sets. In the first case we
provide an time algorithm and an time algorithm in the
second case. Finally, if is
small, that is, not much larger than , we improve the running
time to
- …