4,979 research outputs found
On the dynamics of a family of renormalization transformations
We study the family of renormalization transformations of the generalized
--dimensional diamond hierarchical Potts model in statistical mechanic and
prove that their Julia sets and non-escaping loci are always connected, where
. In particular, we prove that their Julia sets can never be a
Sierpi\'{n}ski carpet if the parameter is real. We show that the Julia set is a
quasicircle if and only if the parameter lies in the unbounded capture domain
of these models. Moreover, the asymptotic formula of the Hausdorff dimension of
the Julia set is calculated as the parameter tends to infinity.Comment: 21 pages, 5 figures, to appear in J. Math. Anal. App
New Hamiltonian constraint operator for loop quantum gravity
A new symmetric Hamiltonian constraint operator is proposed for loop quantum
gravity, which is well defined in the Hilbert space of diffeomorphism invariant
states up to non-planar vertices with valence higher than three. It inherits
the advantage of the original regularization method, so that its regulated
version in the kinematical Hilbert space is diffeomorphism covariant and
creates new vertices to the spin networks. The quantum algebra of this
Hamiltonian is anomaly-free on shell, and there is less ambiguity in its
construction in comparison with the original method. The regularization
procedure for this Hamiltonian constraint operator can also be applied to the
symmetric model of loop quantum cosmology, which leads to a new quantum
dynamics of the cosmological model.Comment: 5 pages; a few modification
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