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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
Spinning Braid Group Representation and the Fractional Quantum Hall Effect
The path integral approach to representing braid group is generalized for
particles with spin. Introducing the notion of {\em charged} winding number in
the super-plane, we represent the braid group generators as homotopically
constrained Feynman kernels. In this framework, super Knizhnik-Zamolodchikov
operators appear naturally in the Hamiltonian, suggesting the possibility of
{\em spinning nonabelian} anyons. We then apply our formulation to the study of
fractional quantum Hall effect (FQHE). A systematic discussion of the ground
states and their quasi-hole excitations is given. We obtain Laughlin, Halperin
and Moore-Read states as {\em exact} ground state solutions to the respective
Hamiltonians associated to the braid group representations. The energy gap of
the quasi-excitation is also obtainable from this approach.Comment: (36 pages) e-mail [email protected]
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