3,241 research outputs found
The Symmetrical -Semiclassical Orthogonal Polynomials of Class One
We investigate the quadratic decomposition and duality to classify
symmetrical -semiclassical orthogonal -polynomials of class one where
is the Hahn's operator. For any canonical situation, the recurrence
coefficients, the -analog of the distributional equation of Pearson type,
the moments and integral or discrete representations are given
Fractal Tube Formulas for Compact Sets and Relative Fractal Drums: Oscillations, Complex Dimensions and Fractality
We establish pointwise and distributional fractal tube formulas for a large
class of relative fractal drums in Euclidean spaces of arbitrary dimensions. A
relative fractal drum (or RFD, in short) is an ordered pair of
subsets of the Euclidean space (under some mild assumptions) which generalizes
the notion of a (compact) subset and that of a fractal string. By a fractal
tube formula for an RFD , we mean an explicit expression for the
volume of the -neighborhood of intersected by as a sum of
residues of a suitable meromorphic function (here, a fractal zeta function)
over the complex dimensions of the RFD . The complex dimensions of
an RFD are defined as the poles of its meromorphically continued fractal zeta
function (namely, the distance or the tube zeta function), which generalizes
the well-known geometric zeta function for fractal strings. These fractal tube
formulas generalize in a significant way to higher dimensions the corresponding
ones previously obtained for fractal strings by the first author and van
Frankenhuijsen and later on, by the first author, Pearse and Winter in the case
of fractal sprays. They are illustrated by several interesting examples. These
examples include fractal strings, the Sierpi\'nski gasket and the 3-dimensional
carpet, fractal nests and geometric chirps, as well as self-similar fractal
sprays. We also propose a new definition of fractality according to which a
bounded set (or RFD) is considered to be fractal if it possesses at least one
nonreal complex dimension or if its fractal zeta function possesses a natural
boundary. This definition, which extends to RFDs and arbitrary bounded subsets
of the previous one introduced in the context of fractal
strings, is illustrated by the Cantor graph (or devil's staircase) RFD, which
is shown to be `subcritically fractal'.Comment: 90 pages (because of different style file), 5 figures, corrected
typos, updated reference
Renormalized Volume
We develop a universal distributional calculus for regulated volumes of
metrics that are singular along hypersurfaces. When the hypersurface is a
conformal infinity we give simple integrated distribution expressions for the
divergences and anomaly of the regulated volume functional valid for any choice
of regulator. For closed hypersurfaces or conformally compact geometries,
methods from a previously developed boundary calculus for conformally compact
manifolds can be applied to give explicit holographic formulae for the
divergences and anomaly expressed as hypersurface integrals over local
quantities (the method also extends to non-closed hypersurfaces). The resulting
anomaly does not depend on any particular choice of regulator, while the
regulator dependence of the divergences is precisely captured by these
formulae. Conformal hypersurface invariants can be studied by demanding that
the singular metric obey, smoothly and formally to a suitable order, a Yamabe
type problem with boundary data along the conformal infinity. We prove that the
volume anomaly for these singular Yamabe solutions is a conformally invariant
integral of a local Q-curvature that generalizes the Branson Q-curvature by
including data of the embedding. In each dimension this canonically defines a
higher dimensional generalization of the Willmore energy/rigid string action.
Recently Graham proved that the first variation of the volume anomaly recovers
the density obstructing smooth solutions to this singular Yamabe problem; we
give a new proof of this result employing our boundary calculus. Physical
applications of our results include studies of quantum corrections to
entanglement entropies.Comment: 31 pages, LaTeX, 5 figures, anomaly formula generalized to any bulk
geometry, improved discussion of hypersurfaces with boundar
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