12,940 research outputs found
Characterization of Information Channels for Asymptotic Mean Stationarity and Stochastic Stability of Non-stationary/Unstable Linear Systems
Stabilization of non-stationary linear systems over noisy communication
channels is considered. Stochastically stable sources, and unstable but
noise-free or bounded-noise systems have been extensively studied in
information theory and control theory literature since 1970s, with a renewed
interest in the past decade. There have also been studies on non-causal and
causal coding of unstable/non-stationary linear Gaussian sources. In this
paper, tight necessary and sufficient conditions for stochastic stabilizability
of unstable (non-stationary) possibly multi-dimensional linear systems driven
by Gaussian noise over discrete channels (possibly with memory and feedback)
are presented. Stochastic stability notions include recurrence, asymptotic mean
stationarity and sample path ergodicity, and the existence of finite second
moments. Our constructive proof uses random-time state-dependent stochastic
drift criteria for stabilization of Markov chains. For asymptotic mean
stationarity (and thus sample path ergodicity), it is sufficient that the
capacity of a channel is (strictly) greater than the sum of the logarithms of
the unstable pole magnitudes for memoryless channels and a class of channels
with memory. This condition is also necessary under a mild technical condition.
Sufficient conditions for the existence of finite average second moments for
such systems driven by unbounded noise are provided.Comment: To appear in IEEE Transactions on Information Theor
Criticality and Bifurcation in the Gravitational Collapse of a Self-Coupled Scalar Field
We examine the gravitational collapse of a non-linear sigma model in
spherical symmetry. There exists a family of continuously self-similar
solutions parameterized by the coupling constant of the theory. These solutions
are calculated together with the critical exponents for black hole formation of
these collapse models. We also find that the sequence of solutions exhibits a
Hopf-type bifurcation as the continuously self-similar solutions become
unstable to perturbations away from self-similarity.Comment: 18 pages; one figure, uuencoded postscript; figure is also available
at http://www.physics.ucsb.edu/people/eric_hirschman
Do All Integrable Evolution Equations Have the Painlev\'e Property?
We examine whether the Painleve property is necessary for the integrability
of partial differential equations (PDEs). We show that in analogy to what
happens in the case of ordinary differential equations (ODEs) there exists a
class of PDEs, integrable through linearisation, which do not possess the
Painleve property. The same question is addressed in a discrete setting where
we show that there exist linearisable lattice equations which do not possess
the singularity confinement property (again in analogy to the one-dimensional
case).Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
On the Implications of Discrete Symmetries for the Beta Function of Quantum Hall Systems
We argue that the large discrete symmetry group of quantum Hall systems is
insufficient in itself to determine the complete beta function for the scaling
of the conductivities, and . We illustrate this
point by showing that a recent ansatz for this function is one of a
many-parameter family. A clean prediction for the delocalization exponents for
these systems therefore requires the specification of more information, such as
past proposals that the beta function is either holomorphic or
quasi-holomorphic in the variable .Comment: Minor typographical errors corrected. 6 pages, LaTeX, no figure
Random Network Models and Quantum Phase Transitions in Two Dimensions
An overview of the random network model invented by Chalker and Coddington,
and its generalizations, is provided. After a short introduction into the
physics of the Integer Quantum Hall Effect, which historically has been the
motivation for introducing the network model, the percolation model for
electrons in spatial dimension 2 in a strong perpendicular magnetic field and a
spatially correlated random potential is described. Based on this, the network
model is established, using the concepts of percolating probability amplitude
and tunneling. Its localization properties and its behavior at the critical
point are discussed including a short survey on the statistics of energy levels
and wave function amplitudes. Magneto-transport is reviewed with emphasis on
some new results on conductance distributions. Generalizations are performed by
establishing equivalent Hamiltonians. In particular, the significance of
mappings to the Dirac model and the two dimensional Ising model are discussed.
A description of renormalization group treatments is given. The classification
of two dimensional random systems according to their symmetries is outlined.
This provides access to the complete set of quantum phase transitions like the
thermal Hall transition and the spin quantum Hall transition in two dimension.
The supersymmetric effective field theory for the critical properties of
network models is formulated. The network model is extended to higher
dimensions including remarks on the chiral metal phase at the surface of a
multi-layer quantum Hall system.Comment: 176 pages, final version, references correcte
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