3,840 research outputs found
Direct topological factorization for topological flows
This paper considers the general question of when a topological action of a
countable group can be factored into a direct product of a nontrivial actions.
In the early 1980's D. Lind considered such questions for -shifts
of finite type. We study in particular direct factorizations of subshifts of
finite type over and other groups, and -subshifts
which are not of finite type. The main results concern direct factors of the
multidimensional full -shift, the multidimensional -colored chessboard
and the Dyck shift over a prime alphabet.
A direct factorization of an expansive -action must be finite,
but a example is provided of a non-expansive -action for which
there is no finite direct prime factorization. The question about existence of
direct prime factorization of expansive actions remains open, even for
.Comment: 21 pages, some changes and remarks added in response to suggestions
by the referee. To appear in ETD
On the reachability and observability of path and cycle graphs
In this paper we investigate the reachability and observability properties of
a network system, running a Laplacian based average consensus algorithm, when
the communication graph is a path or a cycle. More in detail, we provide
necessary and sufficient conditions, based on simple algebraic rules from
number theory, to characterize all and only the nodes from which the network
system is reachable (respectively observable). Interesting immediate
corollaries of our results are: (i) a path graph is reachable (observable) from
any single node if and only if the number of nodes of the graph is a power of
two, , and (ii) a cycle is reachable (observable) from
any pair of nodes if and only if is a prime number. For any set of control
(observation) nodes, we provide a closed form expression for the (unreachable)
unobservable eigenvalues and for the eigenvectors of the (unreachable)
unobservable subsystem
Model reduction of networked passive systems through clustering
In this paper, a model reduction procedure for a network of interconnected
identical passive subsystems is presented. Here, rather than performing model
reduction on the subsystems, adjacent subsystems are clustered, leading to a
reduced-order networked system that allows for a convenient physical
interpretation. The identification of the subsystems to be clustered is
performed through controllability and observability analysis of an associated
edge system and it is shown that the property of synchronization (i.e., the
convergence of trajectories of the subsystems to each other) is preserved
during reduction. The results are illustrated by means of an example.Comment: 7 pages, 2 figures; minor changes in the final version, as accepted
for publication at the 13th European Control Conference, Strasbourg, Franc
Consciousness as a State of Matter
We examine the hypothesis that consciousness can be understood as a state of
matter, "perceptronium", with distinctive information processing abilities. We
explore five basic principles that may distinguish conscious matter from other
physical systems such as solids, liquids and gases: the information,
integration, independence, dynamics and utility principles. If such principles
can identify conscious entities, then they can help solve the quantum
factorization problem: why do conscious observers like us perceive the
particular Hilbert space factorization corresponding to classical space (rather
than Fourier space, say), and more generally, why do we perceive the world
around us as a dynamic hierarchy of objects that are strongly integrated and
relatively independent? Tensor factorization of matrices is found to play a
central role, and our technical results include a theorem about Hamiltonian
separability (defined using Hilbert-Schmidt superoperators) being maximized in
the energy eigenbasis. Our approach generalizes Giulio Tononi's integrated
information framework for neural-network-based consciousness to arbitrary
quantum systems, and we find interesting links to error-correcting codes,
condensed matter criticality, and the Quantum Darwinism program, as well as an
interesting connection between the emergence of consciousness and the emergence
of time.Comment: Replaced to match accepted CSF version; discussion improved, typos
corrected. 36 pages, 15 fig
Superadditivity of quantum relative entropy for general states
The property of superadditivity of the quantum relative entropy states that,
in a bipartite system ,
for every density operator one has . In
this work, we provide an extension of this inequality for arbitrary density
operators . More specifically, we prove that holds for all bipartite states
and , where .Comment: 14 pages. v3: Final version. The main theorem has been improved,
adding a fourth step to its proof and also some remarks. v2: There was a flaw
in the proof of the previous version. This has been corrected in this
version. The constant appearing in the main Theorem has changed accordingl
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