2,221 research outputs found
Stability and control synthesis for discrete-time linear systems subject to actuator saturation by output feedback
This paper presents sufficient conditions of asymptotic stability
for discrete-time linear systems subject to actuator saturations
with an output feedback law. The derived stability results are
given in terms of LMIs. A new proof is presented to
obtain previous conditions of asymptotic stability. A numerical
example is used to illustrate this technique by using a linear
optimization problem subject to LMI constraints
Unimodular measures on the space of all Riemannian manifolds
We study unimodular measures on the space of all pointed
Riemannian -manifolds. Examples can be constructed from finite volume
manifolds, from measured foliations with Riemannian leaves, and from invariant
random subgroups of Lie groups. Unimodularity is preserved under weak* limits,
and under certain geometric constraints (e.g. bounded geometry) unimodular
measures can be used to compactify sets of finite volume manifolds. One can
then understand the geometry of manifolds with large, finite volume by
passing to unimodular limits.
We develop a structure theory for unimodular measures on ,
characterizing them via invariance under a certain geodesic flow, and showing
that they correspond to transverse measures on a foliated `desingularization'
of . We also give a geometric proof of a compactness theorem for
unimodular measures on the space of pointed manifolds with pinched negative
curvature, and characterize unimodular measures supported on hyperbolic
-manifolds with finitely generated fundamental group.Comment: 81 page
Symmetric Gibbs measures
We prove that certain Gibbs measures on subshifts of finite type are
nonsingular and ergodic for certain countable equivalence relations, including
the orbit relation of the adic transformation (the same as equality after a
permutation of finitely many coordinates). The relations we consider are
defined by cocycles taking values in groups, including some nonabelian ones.
This generalizes (half of) the identification of the invariant ergodic
probability measures for the Pascal adic transformation as exactly the
Bernoulli measures---a version of de Finetti's Theorem. Generalizing the other
half, we characterize the measures on subshifts of finite type that are
invariant under both the adic and the shift as the Gibbs measures whose
potential functions depend on only a single coordinate. There are connections
with and implications for exchangeability, ratio limit theorems for transient
Markov chains, interval splitting procedures, `canonical' Gibbs states, and the
triviality of remote sigma-fields finer than the usual tail field
Robust permanence for interacting structured populations
The dynamics of interacting structured populations can be modeled by
where , , and
are matrices with non-negative off-diagonal entries. These models are
permanent if there exists a positive global attractor and are robustly
permanent if they remain permanent following perturbations of .
Necessary and sufficient conditions for robust permanence are derived using
dominant Lyapunov exponents of the with respect to
invariant measures . The necessary condition requires for all ergodic measures with support in the boundary of the
non-negative cone. The sufficient condition requires that the boundary admits a
Morse decomposition such that for all invariant
measures supported by a component of the Morse decomposition. When the
Morse components are Axiom A, uniquely ergodic, or support all but one
population, the necessary and sufficient conditions are equivalent.
Applications to spatial ecology, epidemiology, and gene networks are given
Basic Types of Coarse-Graining
We consider two basic types of coarse-graining: the Ehrenfests'
coarse-graining and its extension to a general principle of non-equilibrium
thermodynamics, and the coarse-graining based on uncertainty of dynamical
models and Epsilon-motions (orbits). Non-technical discussion of basic notions
and main coarse-graining theorems are presented: the theorem about entropy
overproduction for the Ehrenfests' coarse-graining and its generalizations,
both for conservative and for dissipative systems, and the theorems about
stable properties and the Smale order for Epsilon-motions of general dynamical
systems including structurally unstable systems. Computational kinetic models
of macroscopic dynamics are considered. We construct a theoretical basis for
these kinetic models using generalizations of the Ehrenfests' coarse-graining.
General theory of reversible regularization and filtering semigroups in
kinetics is presented, both for linear and non-linear filters. We obtain
explicit expressions and entropic stability conditions for filtered equations.
A brief discussion of coarse-graining by rounding and by small noise is also
presented.Comment: 60 pgs, 11 figs., includes new analysis of coarse-graining by
filtering. A talk given at the research workshop: "Model Reduction and
Coarse-Graining Approaches for Multiscale Phenomena," University of
Leicester, UK, August 24-26, 200
Entropic Uncertainty Relations in Quantum Physics
Uncertainty relations have become the trademark of quantum theory since they
were formulated by Bohr and Heisenberg. This review covers various
generalizations and extensions of the uncertainty relations in quantum theory
that involve the R\'enyi and the Shannon entropies. The advantages of these
entropic uncertainty relations are pointed out and their more direct connection
to the observed phenomena is emphasized. Several remaining open problems are
mentionedComment: 35 pages, review pape
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