21,824 research outputs found
Harnack Inequality for a Subelliptic PDE in nondivergence form
We consider subelliptic equations in non divergence form of the type , where are the Grushin vector fields, and the
matrix coefficient is uniformly elliptic. We obtain a scale invariant Harnack's
inequality on the 's CC balls for nonnegative solutions under the only
assumption that the ratio between the maximum and minimum eigenvalues of the
coefficient matrix is bounded. In the paper we first prove a weighted
Aleksandrov Bakelman Pucci estimate, and then we show a critical density
estimate, the double ball property and the power decay property. Once this is
established, Harnack's inequality follows directly from the axiomatic theory
developed by Di Fazio, Gutierrez and Lanconelli in [6]
Tight bounds for LDPC and LDGM codes under MAP decoding
A new method for analyzing low density parity check (LDPC) codes and low
density generator matrix (LDGM) codes under bit maximum a posteriori
probability (MAP) decoding is introduced. The method is based on a rigorous
approach to spin glasses developed by Francesco Guerra. It allows to construct
lower bounds on the entropy of the transmitted message conditional to the
received one. Based on heuristic statistical mechanics calculations, we
conjecture such bounds to be tight. The result holds for standard irregular
ensembles when used over binary input output symmetric channels. The method is
first developed for Tanner graph ensembles with Poisson left degree
distribution. It is then generalized to `multi-Poisson' graphs, and, by a
completion procedure, to arbitrary degree distribution.Comment: 28 pages, 9 eps figures; Second version contains a generalization of
the previous resul
Estimating Random Variables from Random Sparse Observations
Let X_1,...., X_n be a collection of iid discrete random variables, and
Y_1,..., Y_m a set of noisy observations of such variables. Assume each
observation Y_a to be a random function of some a random subset of the X_i's,
and consider the conditional distribution of X_i given the observations, namely
\mu_i(x_i)\equiv\prob\{X_i=x_i|Y\} (a posteriori probability).
We establish a general relation between the distribution of \mu_i, and the
fixed points of the associated density evolution operator. Such relation holds
asymptotically in the large system limit, provided the average number of
variables an observation depends on is bounded. We discuss the relevance of our
result to a number of applications, ranging from sparse graph codes, to
multi-user detection, to group testing.Comment: 22 pages, 1 eps figures, invited paper for European Transactions on
Telecommunication
Turbo codes: the phase transition
Turbo codes are a very efficient method for communicating reliably through a
noisy channel. There is no theoretical understanding of their effectiveness. In
[1] they are mapped onto a class of disordered spin models. The analytical
calculations concerning these models are reported here. We prove the existence
of a no-error phase and compute its local stability threshold. As a byproduct,
we gain some insight into the dynamics of the decoding algorithm.Comment: 26 pages, 3 eps figure
Scale versus heterogeneity: how the economy affects public support for the EU
This paper proposes a simple political economic model of public opinion support for the EU, drawing on the recent economic literature on integration processes. The basic element is the existence of a trade-off between the benefits of centralisation and the costs of harmonising policies in the presence of heterogeneous preferences among countries. Subsequently we test the model with panel data on the EU member countries. The findings broadly confirm that economic benefits and costs do consistently shape citizensâ attitude towards EU membership. Our analysis may thus shed some light also on the awkward process of ratification of the European Constitution.Economic integration; European Union; Panel Data; Political Economy; Public Opinion
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