16,731 research outputs found
A common generalization of the Fr\"olicher-Nijenhuis bracket and the Schouten bracket for symmetry multi vector fields
There is a canonical mapping from the space of sections of the bundle to . It is shown that
this is a homomorphism on \Gamma(ST\ M)\Omega(T^\ast M;T(T^\ast M))$.Comment: 14 pages, AMSTEX, LPTHE-ORSAY 94/05 and ESI 70 (1994
Noncommutative generalization of SU(n)-principal fiber bundles: a review
This is an extended version of a communication made at the international
conference ``Noncommutative Geometry and Physics'' held at Orsay in april 2007.
In this proceeding, we make a review of some noncommutative constructions
connected to the ordinary fiber bundle theory. The noncommutative algebra is
the endomorphism algebra of a SU(n)-vector bundle, and its differential
calculus is based on its Lie algebra of derivations. It is shown that this
noncommutative geometry contains some of the most important constructions
introduced and used in the theory of connections on vector bundles, in
particular, what is needed to introduce gauge models in physics, and it also
contains naturally the essential aspects of the Higgs fields and its associated
mechanics of mass generation. It permits one also to extend some previous
constructions, as for instance symmetric reduction of (here noncommutative)
connections. From a mathematical point of view, these geometrico-algebraic
considerations highlight some new point on view, in particular we introduce a
new construction of the Chern characteristic classes
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
Properties of Phase transitions of a Higher Order
The following is a thermodynamic analysis of a III order (and some aspects of
a IV order) phase transition. Such a transition can occur in a superconductor
if the normal state is a diamagnet. The equation for a phase boundary in an H-T
(H is the magnetic field, T, the temperature) plane is derived. by considering
two possible forms of the gradient energy, it is possible to construct a field
theory which describes a III or a IV order transition and permits a study of
thermal fluctuations and inhomogeneous order parameters.Comment: 13 pages, revtex, no figure
The Origin of Chiral Anomaly and the Noncommutative Geometry
We describe the scalar and spinor fields on noncommutative sphere starting
from canonical realizations of the enveloping algebra . The gauge extension of a free spinor model, the Schwinger model on
a noncommutative sphere, is defined and the model is quantized. The
noncommutative version of the model contains only a finite number of dynamical
modes and is non-perturbatively UV-regular. An exact expresion for the chiral
anomaly is found. In the commutative limit the standard formula is recovered.Comment: 30 page
Chirality and Dirac Operator on Noncommutative Sphere
We give a derivation of the Dirac operator on the noncommutative -sphere
within the framework of the bosonic fuzzy sphere and define Connes' triple. It
turns out that there are two different types of spectra of the Dirac operator
and correspondingly there are two classes of quantized algebras. As a result we
obtain a new restriction on the Planck constant in Berezin's quantization. The
map to the local frame in noncommutative geometry is also discussed.Comment: 24 pages, latex, no figure
Evidence Propagation and Consensus Formation in Noisy Environments
We study the effectiveness of consensus formation in multi-agent systems
where there is both belief updating based on direct evidence and also belief
combination between agents. In particular, we consider the scenario in which a
population of agents collaborate on the best-of-n problem where the aim is to
reach a consensus about which is the best (alternatively, true) state from
amongst a set of states, each with a different quality value (or level of
evidence). Agents' beliefs are represented within Dempster-Shafer theory by
mass functions and we investigate the macro-level properties of four well-known
belief combination operators for this multi-agent consensus formation problem:
Dempster's rule, Yager's rule, Dubois & Prade's operator and the averaging
operator. The convergence properties of the operators are considered and
simulation experiments are conducted for different evidence rates and noise
levels. Results show that a combination of updating on direct evidence and
belief combination between agents results in better consensus to the best state
than does evidence updating alone. We also find that in this framework the
operators are robust to noise. Broadly, Yager's rule is shown to be the better
operator under various parameter values, i.e. convergence to the best state,
robustness to noise, and scalability.Comment: 13th international conference on Scalable Uncertainty Managemen
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