63,835 research outputs found
Distributed Decision Through Self-Synchronizing Sensor Networks in the Presence of Propagation Delays and Nonreciprocal Channels
In this paper we propose and analyze a distributed algorithm for achieving
globally optimal decisions, either estimation or detection, through a
self-synchronization mechanism among linearly coupled integrators initialized
with local measurements. We model the interaction among the nodes as a directed
graph with weights dependent on the radio interface and we pose special
attention to the effect of the propagation delays occurring in the exchange of
data among sensors, as a function of the network geometry. We derive necessary
and sufficient conditions for the proposed system to reach a consensus on
globally optimal decision statistics. One of the major results proved in this
work is that a consensus is achieved for any bounded delay condition if and
only if the directed graph is quasi-strongly connected. We also provide a
closed form expression for the global consensus, showing that the effect of
delays is, in general, to introduce a bias in the final decision. The closed
form expression is also useful to modify the consensus mechanism in order to
get rid of the bias with minimum extra complexity.Comment: Conference paper. Journal version submitted to IEEE Transactions on
Signal Processing, January 10, 2007. Paper accepted for the publication on
the VIII IEEE Workshop on Signal Processing Advances in Wireless
Communications, (SPAWC 2007), January 22, 200
Elements of a Theory of Simulation
Unlike computation or the numerical analysis of differential equations,
simulation does not have a well established conceptual and mathematical
foundation. Simulation is an arguable unique union of modeling and computation.
However, simulation also qualifies as a separate species of system
representation with its own motivations, characteristics, and implications.
This work outlines how simulation can be rooted in mathematics and shows which
properties some of the elements of such a mathematical framework has. The
properties of simulation are described and analyzed in terms of properties of
dynamical systems. It is shown how and why a simulation produces emergent
behavior and why the analysis of the dynamics of the system being simulated
always is an analysis of emergent phenomena. A notion of a universal simulator
and the definition of simulatability is proposed. This allows a description of
conditions under which simulations can distribute update functions over system
components, thereby determining simulatability. The connection between the
notion of simulatability and the notion of computability is defined and the
concepts are distinguished. The basis of practical detection methods for
determining effectively non-simulatable systems in practice is presented. The
conceptual framework is illustrated through examples from molecular
self-assembly end engineering.Comment: Also available via http://studguppy.tsasa.lanl.gov/research_team/
Keywords: simulatability, computability, dynamics, emergence, system
representation, universal simulato
Repetitive Delone Sets and Quasicrystals
This paper considers the problem of characterizing the simplest discrete
point sets that are aperiodic, using invariants based on topological dynamics.
A Delone set whose patch-counting function N(T), for radius T, is finite for
all T is called repetitive if there is a function M(T) such that every ball of
radius M(T)+T contains a copy of each kind of patch of radius T that occurs in
the set. This is equivalent to the minimality of an associated topological
dynamical system with R^n-action. There is a lower bound for M(T) in terms of
N(T), namely N(T) = O(M(T)^n), but no general upper bound.
The complexity of a repetitive Delone set can be measured by the growth rate
of its repetitivity function M(T). For example, M(T) is bounded if and only if
the set is a crystal. A set is called is linearly repetitive if M(T) = O(T) and
densely repetitive if M(T) = O(N(T))^{1/n}). We show that linearly repetitive
sets and densely repetitive sets have strict uniform patch frequencies, i.e.
the associated topological dynamical system is strictly ergodic. It follows
that such sets are diffractive. In the reverse direction, we construct a
repetitive Delone set in R^n which has
M(T) = O(T(log T)^{2/n}(log log log T)^{4/n}), but does not have uniform
patch frequencies. Aperiodic linearly repetitive sets have many claims to be
the simplest class of aperiodic sets, and we propose considering them as a
notion of "perfectly ordered quasicrystal".Comment: To appear in "Ergodic Theory and Dynamical Systems" vol.23 (2003). 37
pages. Uses packages latexsym, ifthen, cite and files amssym.def, amssym.te
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