5,317 research outputs found
Observable Graphs
An edge-colored directed graph is \emph{observable} if an agent that moves
along its edges is able to determine his position in the graph after a
sufficiently long observation of the edge colors. When the agent is able to
determine his position only from time to time, the graph is said to be
\emph{partly observable}. Observability in graphs is desirable in situations
where autonomous agents are moving on a network and one wants to localize them
(or the agent wants to localize himself) with limited information. In this
paper, we completely characterize observable and partly observable graphs and
show how these concepts relate to observable discrete event systems and to
local automata. Based on these characterizations, we provide polynomial time
algorithms to decide observability, to decide partial observability, and to
compute the minimal number of observations necessary for finding the position
of an agent. In particular we prove that in the worst case this minimal number
of observations increases quadratically with the number of nodes in the graph.
From this it follows that it may be necessary for an agent to pass through
the same node several times before he is finally able to determine his position
in the graph. We then consider the more difficult question of assigning colors
to a graph so as to make it observable and we prove that two different versions
of this problem are NP-complete.Comment: 15 pages, 8 figure
On the Finiteness Property for Rational Matrices
We analyze the periodicity of optimal long products of matrices. A set of
matrices is said to have the finiteness property if the maximal rate of growth
of long products of matrices taken from the set can be obtained by a periodic
product. It was conjectured a decade ago that all finite sets of real matrices
have the finiteness property. This conjecture, known as the ``finiteness
conjecture", is now known to be false but no explicit counterexample to the
conjecture is available and in particular it is unclear if a counterexample is
possible whose matrices have rational or binary entries. In this paper, we
prove that finite sets of nonnegative rational matrices have the finiteness
property if and only if \emph{pairs} of \emph{binary} matrices do. We also show
that all {pairs} of binary matrices have the finiteness property.
These results have direct implications for the stability problem for sets of
matrices. Stability is algorithmically decidable for sets of matrices that have
the finiteness property and so it follows from our results that if all pairs of
binary matrices have the finiteness property then stability is decidable for
sets of nonnegative rational matrices. This would be in sharp contrast with the
fact that the related problem of boundedness is known to be undecidable for
sets of nonnegative rational matrices.Comment: 12 pages, 1 figur
On Primitivity of Sets of Matrices
A nonnegative matrix is called primitive if is positive for some
integer . A generalization of this concept to finite sets of matrices is
as follows: a set of matrices is
primitive if is positive for some indices
. The concept of primitive sets of matrices comes up in a
number of problems within the study of discrete-time switched systems. In this
paper, we analyze the computational complexity of deciding if a given set of
matrices is primitive and we derive bounds on the length of the shortest
positive product.
We show that while primitivity is algorithmically decidable, unless it
is not possible to decide primitivity of a matrix set in polynomial time.
Moreover, we show that the length of the shortest positive sequence can be
superpolynomial in the dimension of the matrices. On the other hand, defining
to be the set of matrices with no zero rows or columns, we give
a simple combinatorial proof of a previously-known characterization of
primitivity for matrices in which can be tested in polynomial
time. This latter observation is related to the well-known 1964 conjecture of
Cerny on synchronizing automata; in fact, any bound on the minimal length of a
synchronizing word for synchronizing automata immediately translates into a
bound on the length of the shortest positive product of a primitive set of
matrices in . In particular, any primitive set of
matrices in has a positive product of length
Continuous-time average-preserving opinion dynamics with opinion-dependent communications
We study a simple continuous-time multi-agent system related to Krause's
model of opinion dynamics: each agent holds a real value, and this value is
continuously attracted by every other value differing from it by less than 1,
with an intensity proportional to the difference.
We prove convergence to a set of clusters, with the agents in each cluster
sharing a common value, and provide a lower bound on the distance between
clusters at a stable equilibrium, under a suitable notion of multi-agent system
stability.
To better understand the behavior of the system for a large number of agents,
we introduce a variant involving a continuum of agents. We prove, under some
conditions, the existence of a solution to the system dynamics, convergence to
clusters, and a non-trivial lower bound on the distance between clusters.
Finally, we establish that the continuum model accurately represents the
asymptotic behavior of a system with a finite but large number of agents.Comment: 25 pages, 2 figures, 11 tex files and 2 eps file
Searches for Clean Anomalous Gauge Couplings effects at present and future colliders
We consider the virtual effects of a general type of Anomalous (triple) Gauge
Couplings on various experimental observables in the process of
electron-positron annihilation into a final fermion-antifermion state. We show
that the use of a recently proposed "-peak subtracted" theoretical
description of the process allows to reduce substantially the number of
relevant parameters of the model, so that a calculation of observability limits
can be performed in a rather simple way. As an illustration of our approach, we
discuss the cases of future measurements at LEP2 and at a new 500 GeV linear
collider.Comment: 23 pages incl. 5 figures (e-mail [email protected]
Physics opportunities with future proton accelerators at CERN
We analyze the physics opportunities that would be made possible by upgrades
of CERN's proton accelerator complex. These include the new physics possible
with luminosity or energy upgrades of the LHC, options for a possible future
neutrino complex at CERN, and opportunities in other physics including rare
kaon decays, other fixed-target experiments, nuclear physics and antiproton
physics, among other possibilities. We stress the importance of inputs from
initial LHC running and planned neutrino experiments, and summarize the
principal detector R&D issues.Comment: 39 page, word document, full resolution version available from
http://cern.ch/pofpa/POFPA-arXive.pd
The Continuous Skolem-Pisot Problem: On the Complexity of Reachability for Linear Ordinary Differential Equations
We study decidability and complexity questions related to a continuous
analogue of the Skolem-Pisot problem concerning the zeros and nonnegativity of
a linear recurrent sequence. In particular, we show that the continuous version
of the nonnegativity problem is NP-hard in general and we show that the
presence of a zero is decidable for several subcases, including instances of
depth two or less, although the decidability in general is left open. The
problems may also be stated as reachability problems related to real zeros of
exponential polynomials or solutions to initial value problems of linear
differential equations, which are interesting problems in their own right.Comment: 14 pages, no figur
Reachability problems for PAMs
Piecewise affine maps (PAMs) are frequently used as a reference model to show
the openness of the reachability questions in other systems. The reachability
problem for one-dimentional PAM is still open even if we define it with only
two intervals. As the main contribution of this paper we introduce new
techniques for solving reachability problems based on p-adic norms and weights
as well as showing decidability for two classes of maps. Then we show the
connections between topological properties for PAM's orbits, reachability
problems and representation of numbers in a rational base system. Finally we
show a particular instance where the uniform distribution of the original orbit
may not remain uniform or even dense after making regular shifts and taking a
fractional part in that sequence.Comment: 16 page
- …