18,294 research outputs found
Linear Connections in Non-Commutative Geometry
A construction is proposed for linear connections on non-commutative
algebras. The construction relies on a generalisation of the Leibnitz rules of
commutative geometry and uses the bimodule structure of . A special
role is played by the extension to the framework of non-commutative geometry of
the permutation of two copies of . The construction of the linear
connection as well as the definition of torsion and curvature is first proposed
in the setting of the derivations based differential calculus of Dubois-
Violette and then a generalisation to the framework proposed by Connes as well
as other non-commutative differential calculi is suggested. The covariant
derivative obtained admits an extension to the tensor product of several copies
of . These constructions are illustrated with the example of the
algebra of matrices.Comment: 15 pages, LMPM ../94 (uses phyzzx
Tuning the electronic transport properties of graphene through functionalisation with fluorine
Engineering the electronic properties of graphene has triggered great
interest for potential applications in electronics and opto-electronics. Here
we demonstrate the possibility to tune the electronic transport properties of
graphene monolayers and multilayers by functionalisation with fluorine. We show
that by adjusting the fluorine content different electronic transport regimes
can be accessed. For monolayer samples, with increasing the fluorine content,
we observe a transition from electronic transport through Mott variable range
hopping in two dimensions to Efros - Shklovskii variable range hopping.
Multilayer fluorinated graphene with high concentration of fluorine show
two-dimensional Mott variable range hopping transport, whereas CF0.28
multilayer flakes have a band gap of 0.25eV and exhibit thermally activated
transport. Our experimental findings demonstrate that the ability to control
the degree of functionalisation of graphene is instrumental to engineer
different electronic properties in graphene materials.Comment: 6 pages, 5 figure
Linear Connections on Fuzzy Manifolds
Linear connections are introduced on a series of noncommutative geometries
which have commutative limits. Quasicommutative corrections are calculated.Comment: 10 pages PlainTex; LPTHE Orsay 95/42; ESI Vienna 23
A Scalable Byzantine Grid
Modern networks assemble an ever growing number of nodes. However, it remains
difficult to increase the number of channels per node, thus the maximal degree
of the network may be bounded. This is typically the case in grid topology
networks, where each node has at most four neighbors. In this paper, we address
the following issue: if each node is likely to fail in an unpredictable manner,
how can we preserve some global reliability guarantees when the number of nodes
keeps increasing unboundedly ? To be more specific, we consider the problem or
reliably broadcasting information on an asynchronous grid in the presence of
Byzantine failures -- that is, some nodes may have an arbitrary and potentially
malicious behavior. Our requirement is that a constant fraction of correct
nodes remain able to achieve reliable communication. Existing solutions can
only tolerate a fixed number of Byzantine failures if they adopt a worst-case
placement scheme. Besides, if we assume a constant Byzantine ratio (each node
has the same probability to be Byzantine), the probability to have a fatal
placement approaches 1 when the number of nodes increases, and reliability
guarantees collapse. In this paper, we propose the first broadcast protocol
that overcomes these difficulties. First, the number of Byzantine failures that
can be tolerated (if they adopt the worst-case placement) now increases with
the number of nodes. Second, we are able to tolerate a constant Byzantine
ratio, however large the grid may be. In other words, the grid becomes
scalable. This result has important security applications in ultra-large
networks, where each node has a given probability to misbehave.Comment: 17 page
A Quantum is a Complex Structure on Classical Phase Space
Duality transformations within the quantum mechanics of a finite number of
degrees of freedom can be regarded as the dependence of the notion of a
quantum, i.e., an elementary excitation of the vacuum, on the observer on
classical phase space. Under an observer we understand, as in general
relativity, a local coordinate chart. While classical mechanics can be
formulated using a symplectic structure on classical phase space, quantum
mechanics requires a complex-differentiable structure on that same space.
Complex-differentiable structures on a given real manifold are often not
unique. This article is devoted to analysing the dependence of the notion of a
quantum on the complex-differentiable structure chosen on classical phase
space
AGN Feedback Compared: Jets versus Radiation
Feedback by Active Galactic Nuclei is often divided into quasar and radio
mode, powered by radiation or radio jets, respectively. Both are fundamental in
galaxy evolution, especially in late-type galaxies, as shown by cosmological
simulations and observations of jet-ISM interactions in these systems. We
compare AGN feedback by radiation and by collimated jets through a suite of
simulations, in which a central AGN interacts with a clumpy, fractal galactic
disc. We test AGN of and erg/s, considering jets
perpendicular or parallel to the disc. Mechanical jets drive the more powerful
outflows, exhibiting stronger mass and momentum coupling with the dense gas,
while radiation heats and rarifies the gas more. Radiation and perpendicular
jets evolve to be quite similar in outflow properties and effect on the cold
ISM, while inclined jets interact more efficiently with all the disc gas,
removing the densest in Myr, and thereby reducing the amount of
cold gas available for star formation. All simulations show small-scale inflows
of M/yr, which can easily reach down to the Bondi radius of
the central supermassive black hole (especially for radiation and perpendicular
jets), implying that AGN modulate their own duty cycle in a feedback/feeding
cycle.Comment: 21 pages, 15 figures, 2 table
Almost commutative Riemannian geometry: wave operators
Associated to any (pseudo)-Riemannian manifold of dimension is an
-dimensional noncommutative differential structure (\Omega^1,\extd) on
the manifold, with the extra dimension encoding the classical Laplacian as a
noncommutative `vector field'. We use the classical connection, Ricci tensor
and Hodge Laplacian to construct (\Omega^2,\extd) and a natural
noncommutative torsion free connection on . We show
that its generalised braiding \sigma:\Omega^1\tens\Omega^1\to
\Omega^1\tens\Omega^1 obeys the quantum Yang-Baxter or braid relations only
when the original is flat, i.e their failure is governed by the Riemann
curvature, and that \sigma^2=\id only when is Einstein. We show that if
has a conformal Killing vector field then the cross product algebra
viewed as a noncommutative analogue of has a
natural -dimensional calculus extending and a natural spacetime
Laplacian now directly defined by the extra dimension. The case
recovers the Majid-Ruegg bicrossproduct flat spacetime model and the
wave-operator used in its variable speed of light preduction, but now as an
example of a general construction. As an application we construct the wave
operator on a noncommutative Schwarzschild black hole and take a first look at
its features. It appears that the infinite classical redshift/time dilation
factor at the event horizon is made finite.Comment: 39 pages, 4 pdf images. Removed previous Sections 5.1-5.2 to a
separate paper (now ArXived) to meet referee length requirements.
Corresponding slight restructure but no change to remaining conten
On Byzantine Broadcast in Loosely Connected Networks
We consider the problem of reliably broadcasting information in a multihop
asynchronous network that is subject to Byzantine failures. Most existing
approaches give conditions for perfect reliable broadcast (all correct nodes
deliver the authentic message and nothing else), but they require a highly
connected network. An approach giving only probabilistic guarantees (correct
nodes deliver the authentic message with high probability) was recently
proposed for loosely connected networks, such as grids and tori. Yet, the
proposed solution requires a specific initialization (that includes global
knowledge) of each node, which may be difficult or impossible to guarantee in
self-organizing networks - for instance, a wireless sensor network, especially
if they are prone to Byzantine failures. In this paper, we propose a new
protocol offering guarantees for loosely connected networks that does not
require such global knowledge dependent initialization. In more details, we
give a methodology to determine whether a set of nodes will always deliver the
authentic message, in any execution. Then, we give conditions for perfect
reliable broadcast in a torus network. Finally, we provide experimental
evaluation for our solution, and determine the number of randomly distributed
Byzantine failures than can be tolerated, for a given correct broadcast
probability.Comment: 1
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