8,493 research outputs found
Multistability in dynamical systems
In neuroscience, optics and condensed matter there is ample physical evidence
for multistable dynamical systems, that is, systems with a large number of
attractors. The known mathematical mechanisms that lead to multiple attractors
are homoclinic tangencies and stabilization, by small perturbations or by
coupling, of systems possessing a large number of unstable invariant sets. A
short review of the existent results is presented, as well as two new results
concerning the existence of a large number of stable periodic orbits in a
perturbed marginally stable dissipative map and an infinite number of such
orbits in two coupled quadratic maps working on the Feigenbaum accumulation
point.Comment: 11 pages Latex, to appear in Dynamical Systems: From Crystal to
Chaos, World Scientific, 199
A consistent measure for lattice Yang-Mills
The construction of a consistent measure for Yang-Mills is a precondition for
an accurate formulation of non-perturbative approaches to QCD, both analytical
and numerical. Using projective limits as subsets of Cartesian products of
homomorphisms from a lattice to the structure group, a consistent interaction
measure and an infinite-dimensional calculus has been constructed for a theory
of non-abelian generalized connections on a hypercubic lattice. Here, after
reviewing and clarifying past work, new results are obtained for the mass gap
when the structure group is compact.Comment: 15 pages Latex, 2 figures. arXiv admin note: substantial text overlap
with arXiv:1504.0779
Current algebra, statistical mechanics and quantum models
Results obtained in the past for free boson systems at zero and nonzero
temperature are revisited to clarify the physical meaning of current algebra
reducible functionals which are associated to systems with density
fluctuations, leading to observable effects on phase transitions. To use
current algebra as a tool for the formulation of quantum statistical mechanics
amounts to the construction of unitary representations of diffeomorphism
groups. Two mathematical equivalent procedures exist for this purpose. One
searches for quasi-invariant measures on configuration spaces, the other for a
cyclic vector in Hilbert space. Here, one argues that the second approach is
closer to the physical intuition when modelling complex systems. An example of
application of the current algebra methodology to the pairing phenomenon in
two-dimensional fermion systems is discussed.Comment: 30 pages Latex, 2 figure
Quantum collision states for positive charges in an octahedral cage
One-electron energy levels are studied for a configuration of two positive
charges inside an octahedral cage, the vertices of the cage being occupied by
atoms with a partially filled shell. Although ground states correspond to large
separations, there are relatively low-lying states with large collision
probabilities. Electromagnetic radiation fields used to excite the quantum
collisional levels may provide a means to control nuclear reactions. However,
given the scale of the excitation energies involved, this mechanism cannot
provide an explanation for the unexplained ``cold fusion'' events.Comment: 12 pages Latex, 5 eps figure
Quantum games and social norms. The quantum ultimatum game
The noncooperative Nash equilibrium solution of classical games corresponds
to a rational expectations attitude on the part of the players. However, in
many cases, games played by human players have outcomes very different from
Nash equilibria. A restricted version of quantum games is proposed to
implement, in mathematical games, the interplay of self-interest and
internalized social norms that rules human behavior.Comment: 6 pages Late
The geometry of noncommutative space-time
Stabilization, by deformation, of the Poincar\'{e}-Heisenberg algebra
requires both the introduction of a fundamental lentgh and the noncommutativity
of translations which is associated to the gravitational field. The
noncommutative geometry structure that follows from the deformed algebra is
studied both for the non-commutative tangent space and the full space with
gravity. The contact points of this approach with the work of David Finkelstein
are emphasized.Comment: 12 pages Late
Active control of ionized boundary layers
The challenging problems, in the field of control of chaos or of transition
to chaos, lie in the domain of infinite-dimensional systems. Access to all
variables being impossible in this case and the controlling action being
limited to a few collective variables, it will not in general be possible to
drive the whole system to the desired behaviour. A paradigmatic problem of this
type is the control of the transition to turbulence in the boundary layer of
fluid motion. By analysing a boundary layer flow for an ionized fluid near an
airfoil, one concludes that active control of the transition amounts to the
resolution of an generalized integro-differential eigenvalue problem. To cope
with the required response times and phase accuracy, electromagnetic control,
whenever possible, seems more appropriate than mechanical control by
microactuators.Comment: 10 pages Latex, 5 ps-figure
An extended Dirac equation in noncommutative space-time
Stabilizing, by deformation, the algebra of relativistic quantum mechanics a
non-commutative space-time geometry is obtained. The exterior algebra of this
geometry leads to an extended massless Dirac equation which has both a massless
and a large mass solution. The nature of the solutions is discussed, as well as
the effects of coupling the two solutions.Comment: 9 pages Late
Space-times over normed division algebras, revisited
Normed division and Clifford algebras have been extensively used in the past
as a mathematical framework to accommodate the structures of the standard model
and grand unified theories. Less discussed has been the question of why such
algebraic structures appear in Nature. One possibility could be an intrinsic
complex, quaternionic or octonionic nature of the spacetime manifold. Then, an
obvious question is why spacetime appears nevertheless to be simply
parametrized by the real numbers. How the real slices of an higher dimensional
spacetime manifold might be almost independent from each other is discussed
here. This comes about as a result of the different nature of the
representations of the real kinematical groups and those of the extended
spaces. Some of the internal symmetry transformations might however appear as
representations on homogeneous spaces of the extended group transformations
that cannot be implemented on the elementary states.Comment: 21 pages Late
Dynamics of networks and applications
A survey is made of several aspects of the dynamics of networks, with special
emphasis on unsupervised learning processes, non-Gaussian data analysis and
pattern recognition in networks with complex nodes.Comment: Talk at the Heraeus Seminar "Scientific Applications of Neural Nets",
to appear in the proceedings (Springer LNP
- …