89,713 research outputs found
Entropy of nonautonomous dynamical systems
Different notions of entropy play a fundamental role in the classical theory
of dynamical systems. Unlike many other concepts used to analyze autonomous
dynamics, both measure-theoretic and topological entropy can be extended quite
naturally to discrete-time nonautonomous dynamical systems given in the process
formulation. This paper provides an overview of the author's work on this
subject. Also an example is presented that has not appeared before in the
literature
Tameness in generalized metric structures
We broaden the framework of metric abstract elementary classes (mAECs) in
several essential ways, chiefly by allowing the metric to take values in a
well-behaved quantale. As a proof of concept we show that the result of Boney
and Zambrano on (metric) tameness under a large cardinal assumption holds in
this more general context. We briefly consider a further generalization to
partial metric spaces, and hint at connections to classes of fuzzy structures,
and structures on sheaves
Logarithm laws and shrinking target properties
We survey some of the recent developments in the study of logarithm laws and
shrinking target properties for various families of dynamical systems. We
discuss connections to geometry, diophantine approximation, and probability
theory.Comment: This is a survey paper written following the Conference on Measures
and Dyanmics on groups and homogeneous spaces, at TIFR, Mumbai, in Dec. 2007.
It is in honor of Prof. S.G. Dani's 60th Birthda
Labelled transition systems as a Stone space
A fully abstract and universal domain model for modal transition systems and
refinement is shown to be a maximal-points space model for the bisimulation
quotient of labelled transition systems over a finite set of events. In this
domain model we prove that this quotient is a Stone space whose compact,
zero-dimensional, and ultra-metrizable Hausdorff topology measures the degree
of bisimilarity such that image-finite labelled transition systems are dense.
Using this compactness we show that the set of labelled transition systems that
refine a modal transition system, its ''set of implementations'', is compact
and derive a compactness theorem for Hennessy-Milner logic on such
implementation sets. These results extend to systems that also have partially
specified state propositions, unify existing denotational, operational, and
metric semantics on partial processes, render robust consistency measures for
modal transition systems, and yield an abstract interpretation of compact sets
of labelled transition systems as Scott-closed sets of modal transition
systems.Comment: Changes since v2: Metadata updat
The Einstein-Vlasov system
Rigorous results on solutions of the Einstein-Vlasov system are surveyed.
After an introduction to this system of equations and the reasons for studying
it, a general discussion of various classes of solutions is given. The emphasis
is on presenting important conceptual ideas, while avoiding entering into
technical details. Topics covered include spatially homogenous models, static
solutions, spherically symmetric collapse and isotropic singularities.Comment: Lecture notes from Cargese worksho
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