46 research outputs found
The Inviscid, Compressible and Rotational, 2D Isotropic Burgers and Pressureless Euler-Coriolis Fluids; Solvable models with illustrations
The coupling between dilatation and vorticity, two coexisting and fundamental
processes in fluid dynamics is investigated here, in the simplest cases of
inviscid 2D isotropic Burgers and pressureless Euler-Coriolis fluids
respectively modeled by single vortices confined in compressible, local,
inertial and global, rotating, environments. The field equations are
established, inductively, starting from the equations of the characteristics
solved with an initial Helmholtz decomposition of the velocity fields namely a
vorticity free and a divergence free part and, deductively, by means of a
canonical Hamiltonian Clebsch like formalism, implying two pairs of conjugate
variables. Two vector valued fields are constants of the motion: the velocity
field in the Burgers case and the momentum field per unit mass in the
Euler-Coriolis one. Taking advantage of this property, a class of solutions for
the mass densities of the fluids is given by the Jacobian of their sum with
respect to the actual coordinates. Implementation of the isotropy hypothesis
results in the cancellation of the dilatation-rotational cross terms in the
Jacobian. A simple expression is obtained for all the radially symmetric
Jacobians occurring in the theory. Representative examples of regular and
singular solutions are shown and the competition between dilatation and
vorticity is illustrated. Inspired by thermodynamical, mean field theoretical
analogies, a genuine variational formula is proposed which yields unique
measure solutions for the radially symmetric fluid densities investigated. We
stress that this variational formula, unlike the Hopf-Lax formula, enables us
to treat systems which are both compressible and rotational. Moreover in the
one-dimensional case, we show for an interesting application that both
variational formulas are equivalent
Polymer Expansions for Cycle LDPC Codes
We prove that the Bethe expression for the conditional input-output entropy
of cycle LDPC codes on binary symmetric channels above the MAP threshold is
exact in the large block length limit. The analysis relies on methods from
statistical physics. The finite size corrections to the Bethe expression are
expressed through a polymer expansion which is controlled thanks to expander
and counting arguments
Approaching the Rate-Distortion Limit with Spatial Coupling, Belief propagation and Decimation
We investigate an encoding scheme for lossy compression of a binary symmetric
source based on simple spatially coupled Low-Density Generator-Matrix codes.
The degree of the check nodes is regular and the one of code-bits is Poisson
distributed with an average depending on the compression rate. The performance
of a low complexity Belief Propagation Guided Decimation algorithm is
excellent. The algorithmic rate-distortion curve approaches the optimal curve
of the ensemble as the width of the coupling window grows. Moreover, as the
check degree grows both curves approach the ultimate Shannon rate-distortion
limit. The Belief Propagation Guided Decimation encoder is based on the
posterior measure of a binary symmetric test-channel. This measure can be
interpreted as a random Gibbs measure at a "temperature" directly related to
the "noise level of the test-channel". We investigate the links between the
algorithmic performance of the Belief Propagation Guided Decimation encoder and
the phase diagram of this Gibbs measure. The phase diagram is investigated
thanks to the cavity method of spin glass theory which predicts a number of
phase transition thresholds. In particular the dynamical and condensation
"phase transition temperatures" (equivalently test-channel noise thresholds)
are computed. We observe that: (i) the dynamical temperature of the spatially
coupled construction saturates towards the condensation temperature; (ii) for
large degrees the condensation temperature approaches the temperature (i.e.
noise level) related to the information theoretic Shannon test-channel noise
parameter of rate-distortion theory. This provides heuristic insight into the
excellent performance of the Belief Propagation Guided Decimation algorithm.
The paper contains an introduction to the cavity method