114,756 research outputs found
Toward an Algebraic Theory of Systems
We propose the concept of a system algebra with a parallel composition
operation and an interface connection operation, and formalize
composition-order invariance, which postulates that the order of composing and
connecting systems is irrelevant, a generalized form of associativity.
Composition-order invariance explicitly captures a common property that is
implicit in any context where one can draw a figure (hiding the drawing order)
of several connected systems, which appears in many scientific contexts. This
abstract algebra captures settings where one is interested in the behavior of a
composed system in an environment and wants to abstract away anything internal
not relevant for the behavior. This may include physical systems, electronic
circuits, or interacting distributed systems.
One specific such setting, of special interest in computer science, are
functional system algebras, which capture, in the most general sense, any type
of system that takes inputs and produces outputs depending on the inputs, and
where the output of a system can be the input to another system. The behavior
of such a system is uniquely determined by the function mapping inputs to
outputs. We consider several instantiations of this very general concept. In
particular, we show that Kahn networks form a functional system algebra and
prove their composition-order invariance.
Moreover, we define a functional system algebra of causal systems,
characterized by the property that inputs can only influence future outputs,
where an abstract partial order relation captures the notion of "later". This
system algebra is also shown to be composition-order invariant and appropriate
instantiations thereof allow to model and analyze systems that depend on time
Further properties of causal relationship: causal structure stability, new criteria for isocausality and counterexamples
Recently ({\em Class. Quant. Grav.} {\bf 20} 625-664) the concept of {\em
causal mapping} between spacetimes --essentially equivalent in this context to
the {\em chronological map} one in abstract chronological spaces--, and the
related notion of {\em causal structure}, have been introduced as new tools to
study causality in Lorentzian geometry. In the present paper, these tools are
further developed in several directions such as: (i) causal mappings --and,
thus, abstract chronological ones-- do not preserve two levels of the standard
hierarchy of causality conditions (however, they preserve the remaining levels
as shown in the above reference), (ii) even though global hyperbolicity is a
stable property (in the set of all time-oriented Lorentzian metrics on a fixed
manifold), the causal structure of a globally hyperbolic spacetime can be
unstable against perturbations; in fact, we show that the causal structures of
Minkowski and Einstein static spacetimes remain stable, whereas that of de
Sitter becomes unstable, (iii) general criteria allow us to discriminate
different causal structures in some general spacetimes (e.g. globally
hyperbolic, stationary standard); in particular, there are infinitely many
different globally hyperbolic causal structures (and thus, different conformal
ones) on , (iv) plane waves with the same number of positive eigenvalues
in the frequency matrix share the same causal structure and, thus, they have
equal causal extensions and causal boundaries.Comment: 33 pages, 9 figures, final version (the paper title has been
changed). To appear in Classical and Quantum Gravit
A causal multifractal stochastic equation and its statistical properties
Multiplicative cascades have been introduced in turbulence to generate random
or deterministic fields having intermittent values and long-range power-law
correlations. Generally this is done using discrete construction rules leading
to discrete cascades. Here a causal log-normal stochastic process is
introduced; its multifractal properties are demonstrated together with other
properties such as the composition rule for scale dependence and stochastic
differential equations for time and scale evolutions. This multifractal
stochastic process is continuous in scale ratio and in time. It has a simple
generating equation and can be used to generate sequentially time series of any
length.Comment: Eur. Phys. J. B (in press
On Lorentzian causality with continuous metrics
We present a systematic study of causality theory on Lorentzian manifolds
with continuous metrics. Examples are given which show that some standard facts
in smooth Lorentzian geometry, such as light-cones being hypersurfaces, are
wrong when metrics which are merely continuous are considered. We show that
existence of time functions remains true on domains of dependence with
continuous metrics, and that differentiability of the metric suffices
for many key results of the smooth causality theory.Comment: Minor changes. Version to appear in Classical and Quantum Gravit
Geometry of Schroedinger Space-Times II: Particle and Field Probes of the Causal Structure
We continue our study of the global properties of the z=2 Schroedinger
space-time. In particular, we provide a codimension 2 isometric embedding which
naturally gives rise to the previously introduced global coordinates.
Furthermore, we study the causal structure by probing the space-time with point
particles as well as with scalar fields. We show that, even though there is no
global time function in the technical sense (Schroedinger space-time being
non-distinguishing), the time coordinate of the global Schroedinger coordinate
system is, in a precise way, the closest one can get to having such a time
function. In spite of this and the corresponding strongly Galilean and almost
pathological causal structure of this space-time, it is nevertheless possible
to define a Hilbert space of normalisable scalar modes with a well-defined
time-evolution. We also discuss how the Galilean causal structure is reflected
and encoded in the scalar Wightman functions and the bulk-to-bulk propagator.Comment: 32 page
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