773,829 research outputs found
Completeness of Lyapunov Abstraction
In this work, we continue our study on discrete abstractions of dynamical
systems. To this end, we use a family of partitioning functions to generate an
abstraction. The intersection of sub-level sets of the partitioning functions
defines cells, which are regarded as discrete objects. The union of cells makes
up the state space of the dynamical systems. Our construction gives rise to a
combinatorial object - a timed automaton. We examine sound and complete
abstractions. An abstraction is said to be sound when the flow of the time
automata covers the flow lines of the dynamical systems. If the dynamics of the
dynamical system and the time automaton are equivalent, the abstraction is
complete.
The commonly accepted paradigm for partitioning functions is that they ought
to be transversal to the studied vector field. We show that there is no
complete partitioning with transversal functions, even for particular dynamical
systems whose critical sets are isolated critical points. Therefore, we allow
the directional derivative along the vector field to be non-positive in this
work. This considerably complicates the abstraction technique. For
understanding dynamical systems, it is vital to study stable and unstable
manifolds and their intersections. These objects appear naturally in this work.
Indeed, we show that for an abstraction to be complete, the set of critical
points of an abstraction function shall contain either the stable or unstable
manifold of the dynamical system.Comment: In Proceedings HAS 2013, arXiv:1308.490
Algebraic totality, towards completeness
Finiteness spaces constitute a categorical model of Linear Logic (LL) whose
objects can be seen as linearly topologised spaces, (a class of topological
vector spaces introduced by Lefschetz in 1942) and morphisms as continuous
linear maps. First, we recall definitions of finiteness spaces and describe
their basic properties deduced from the general theory of linearly topologised
spaces. Then we give an interpretation of LL based on linear algebra. Second,
thanks to separation properties, we can introduce an algebraic notion of
totality candidate in the framework of linearly topologised spaces: a totality
candidate is a closed affine subspace which does not contain 0. We show that
finiteness spaces with totality candidates constitute a model of classical LL.
Finally, we give a barycentric simply typed lambda-calculus, with booleans
and a conditional operator, which can be interpreted in this
model. We prove completeness at type for
every n by an algebraic method
Global Hyperbolicity and Completeness
We prove global hyperbolicity of spacetimes under generic regularity
conditions on the metric. We then show that these spacetimes are timelike and
null geodesically complete if the gradient of the lapse and the extrinsic
curvature are integrable. This last condition is required only for the
tracefree part of if the universe is expanding.Comment: 7 pages. Accepted for publication in the Journal of Geometry and
Physics. v2: Minor typos correcte
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