2,036 research outputs found
Robust exponential decay of correlations for singular-flows
We construct open sets of Ck (k bigger or equal to 2) vector fields with
singularities that have robust exponential decay of correlations with respect
to the unique physical measure. In particular we prove that the geometric
Lorenz attractor has exponential decay of correlations with respect to the
unique physical measure.Comment: Final version accepted for publication with added corrections (not in
official published version) after O. Butterley pointed out to the authors
that the last estimate in the argument in Subsection 4.2.3 of the previous
version is not enough to guarantee the uniform non-integrability condition
claimed. We have modified the argument and present it here in the same
Subsection. 3 figures, 34 page
Statistical properties of Lorenz like flows, recent developments and perspectives
We comment on mathematical results about the statistical behavior of Lorenz
equations an its attractor, and more generally to the class of singular
hyperbolic systems. The mathematical theory of such kind of systems turned out
to be surprisingly difficult. It is remarkable that a rigorous proof of the
existence of the Lorenz attractor was presented only around the year 2000 with
a computer assisted proof together with an extension of the hyperbolic theory
developed to encompass attractors robustly containing equilibria. We present
some of the main results on the statisitcal behavior of such systems. We show
that for attractors of three-dimensional flows, robust chaotic behavior is
equivalent to the existence of certain hyperbolic structures, known as
singular-hyperbolicity. These structures, in turn, are associated to the
existence of physical measures: \emph{in low dimensions, robust chaotic
behavior for flows ensures the existence of a physical measure}. We then give
more details on recent results on the dynamics of singular-hyperbolic
(Lorenz-like) attractors.Comment: 40 pages; 10 figures; Keywords: sensitive dependence on initial
conditions, physical measure, singular-hyperbolicity, expansiveness, robust
attractor, robust chaotic flow, positive Lyapunov exponent, large deviations,
hitting and recurrence times. Minor typos corrected and precise
acknowledgments of financial support added. To appear in Int J of Bif and
Chaos in App Sciences and Engineerin
Unique Information via Dependency Constraints
The partial information decomposition (PID) is perhaps the leading proposal
for resolving information shared between a set of sources and a target into
redundant, synergistic, and unique constituents. Unfortunately, the PID
framework has been hindered by a lack of a generally agreed-upon, multivariate
method of quantifying the constituents. Here, we take a step toward rectifying
this by developing a decomposition based on a new method that quantifies unique
information. We first develop a broadly applicable method---the dependency
decomposition---that delineates how statistical dependencies influence the
structure of a joint distribution. The dependency decomposition then allows us
to define a measure of the information about a target that can be uniquely
attributed to a particular source as the least amount which the source-target
statistical dependency can influence the information shared between the sources
and the target. The result is the first measure that satisfies the core axioms
of the PID framework while not satisfying the Blackwell relation, which depends
on a particular interpretation of how the variables are related. This makes a
key step forward to a practical PID.Comment: 15 pages, 7 figures, 2 tables, 3 appendices;
http://csc.ucdavis.edu/~cmg/compmech/pubs/idep.ht
Model testing for causal models
Finding cause-effect relationships is the central aim of many studies in the physical, behavioral, social and biological sciences. We consider two well-known mathematical causal models: Structural equation models and causal Bayesian networks. When we hypothesize a causal model, that model often imposes constraints on the statistics of the data collected. These constraints enable us to test or falsify the hypothesized causal model. The goal of our research is to develop efficient and reliable methods to test a causal model or distinguish between causal models using various types of constraints.
For linear structural equation models, we investigate the problem of generating a small number of constraints in the form of zero partial correlations, providing an efficient way to test hypothesized models. We study linear structural equation models with correlated errors focusing on the graphical aspects of the models. We provide a set of local Markov properties and prove that they are equivalent to the global Markov property.
For causal Bayesian networks, we study equality and inequality constraints imposed on data and investigate a way to use these constraints for model testing and selection. For equality constraints, we formulate an implicitization problem and show how we may reduce the complexity of the problem. We also study the algebraic structure of the equality constraints. For inequality constraints, we present a class of inequality constraints on both nonexperimental and interventional distributions
Abelian networks III. The critical group
The critical group of an abelian network is a finite abelian group that
governs the behavior of the network on large inputs. It generalizes the
sandpile group of a graph. We show that the critical group of an irreducible
abelian network acts freely and transitively on recurrent states of the
network. We exhibit the critical group as a quotient of a free abelian group by
a subgroup containing the image of the Laplacian, with equality in the case
that the network is rectangular. We generalize Dhar's burning algorithm to
abelian networks, and estimate the running time of an abelian network on an
arbitrary input up to a constant additive error.Comment: supersedes sections 7 and 8 of arXiv:1309.3445v1. To appear in the
Journal of Algebraic Combinatoric
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