56,402 research outputs found
Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures
This arXived paper has two independant parts, that are improved and corrected
versions of different parts of a single paper once named "On equations in
relatively hyperbolic groups".
The first part is entitled "Existential questions in (relatively) hyperbolic
groups". We study there the existential theory of torsion free hyperbolic and
relatively hyperbolic groups, in particular those with virtually abelian
parabolic subgroups. We show that the satisfiability of systems of equations
and inequations is decidable in these groups.
In the second part, called "Finding relative hyperbolic structures", we
provide a general algorithm that recognizes the class of groups that are
hyperbolic relative to abelian subgroups.Comment: Two independant parts 23p + 9p, revised. To appear separately in
Israel J. Math, and Bull. London Math. Soc. respectivel
Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility
Word equations are a crucial element in the theoretical foundation of
constraint solving over strings, which have received a lot of attention in
recent years. A word equation relates two words over string variables and
constants. Its solution amounts to a function mapping variables to constant
strings that equate the left and right hand sides of the equation. While the
problem of solving word equations is decidable, the decidability of the problem
of solving a word equation with a length constraint (i.e., a constraint
relating the lengths of words in the word equation) has remained a
long-standing open problem. In this paper, we focus on the subclass of
quadratic word equations, i.e., in which each variable occurs at most twice. We
first show that the length abstractions of solutions to quadratic word
equations are in general not Presburger-definable. We then describe a class of
counter systems with Presburger transition relations which capture the length
abstraction of a quadratic word equation with regular constraints. We provide
an encoding of the effect of a simple loop of the counter systems in the theory
of existential Presburger Arithmetic with divisibility (PAD). Since PAD is
decidable, we get a decision procedure for quadratic words equations with
length constraints for which the associated counter system is \emph{flat}
(i.e., all nodes belong to at most one cycle). We show a decidability result
(in fact, also an NP algorithm with a PAD oracle) for a recently proposed
NP-complete fragment of word equations called regular-oriented word equations,
together with length constraints. Decidability holds when the constraints are
additionally extended with regular constraints with a 1-weak control structure.Comment: 18 page
Topological fluid mechanics of point vortex motions
Topological techniques are used to study the motions of systems of point
vortices in the infinite plane, in singly-periodic arrays, and in
doubly-periodic lattices. The reduction of each system using its symmetries is
described in detail. Restricting to three vortices with zero net circulation,
each reduced system is described by a one degree of freedom Hamiltonian. The
phase portrait of this reduced system is subdivided into regimes using the
separatrix motions, and a braid representing the topology of all vortex motions
in each regime is computed. This braid also describes the isotopy class of the
advection homeomorphism induced by the vortex motion. The Thurston-Nielsen
theory is then used to analyse these isotopy classes, and in certain cases
strong conclusions about the dynamics of the advection can be made
Context, spacetime loops, and the interpretation of quantum mechanics
Three postulates are discussed: first that well-defined properties cannot be
assigned to an isolated system, secondly that quantum unitary evolution is
atemporal, and thirdly that some physical processes are never reversed. It is
argued that these give useful insight into quantum behaviour. The first
postulate emphasizes the fundamental role in physics of interactions and
correlations, as opposed to internal properties of systems. Statements about
physical interactions can only be framed in a context of further interactions.
This undermines the possibility of objectivity in physics. However, quantum
mechanics retains objectivity through the combination of the second and third
postulates. A rule is given for determining the circumstances in which physical
evolution is non-unitary. This rule appeals to the absence of spacetime loops
in the future evolution of a set of interacting systems. A single universe
undergoing non-unitary evolution is a viable interpretation.Comment: 19 pages. For special issue of J.Phys.A, "The Quantum Universe", on
the occasion of 70th birthday of Professor Giancarlo Ghirard
Implementing Multi-Periodic Critical Systems: from Design to Code Generation
This article presents a complete scheme for the development of Critical
Embedded Systems with Multiple Real-Time Constraints. The system is programmed
with a language that extends the synchronous approach with high-level real-time
primitives. It enables to assemble in a modular and hierarchical manner several
locally mono-periodic synchronous systems into a globally multi-periodic
synchronous system. It also allows to specify flow latency constraints. A
program is translated into a set of real-time tasks. The generated code (\C\
code) can be executed on a simple real-time platform with a dynamic-priority
scheduler (EDF). The compilation process (each algorithm of the process, not
the compiler itself) is formally proved correct, meaning that the generated
code respects the real-time semantics of the original program (respect of
periods, deadlines, release dates and precedences) as well as its functional
semantics (respect of variable consumption).Comment: 15 pages, published in Workshop on Formal Methods for Aerospace
(FMA'09), part of Formal Methods Week 2009
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