20,478 research outputs found
Tube formulas and complex dimensions of self-similar tilings
We use the self-similar tilings constructed by the second author in
"Canonical self-affine tilings by iterated function systems" to define a
generating function for the geometry of a self-similar set in Euclidean space.
This tubular zeta function encodes scaling and curvature properties related to
the complement of the fractal set, and the associated system of mappings. This
allows one to obtain the complex dimensions of the self-similar tiling as the
poles of the tubular zeta function and hence develop a tube formula for
self-similar tilings in \. The resulting power series in
is a fractal extension of Steiner's classical tube formula for
convex bodies K \ci \bRd. Our sum has coefficients related to the curvatures
of the tiling, and contains terms for each integer , just as
Steiner's does. However, our formula also contains terms for each complex
dimension. This provides further justification for the term "complex
dimension". It also extends several aspects of the theory of fractal strings to
higher dimensions and sheds new light on the tube formula for fractals strings
obtained in "Fractal Geometry and Complex Dimensions" by the first author and
Machiel van Frankenhuijsen.Comment: 41 pages, 6 figures, incorporates referee comments and references to
new result
Clumpy Disc and Bulge Formation
We present a set of hydrodynamical/Nbody controlled simulations of isolated
gas rich galaxies that self-consistently include SN feedback and a detailed
chemical evolution model, both tested in cosmological simulations. The initial
conditions are motivated by the observed star forming galaxies at z ~ 2-3. We
find that the presence of a multiphase interstellar media in our models
promotes the growth of disc instability favouring the formation of clumps which
in general, are not easily disrupted on timescales compared to the migration
time. We show that stellar clumps migrate towards the central region and
contribute to form a classical-like bulge with a Sersic index, n > 2. Our
physically-motivated Supernova feedback has a mild influence on clump survival
and evolution, partially limiting the mass growth of clumps as the energy
released per Supernova event is increased, with the consequent flattening of
the bulge profile. This regulation does not prevent the building of a
classical-like bulge even for the most energetic feedback tested. Our Supernova
feedback model is able to establish a self-regulated star formation, producing
mass-loaded outflows and stellar age spreads comparable to observations. We
find that the bulge formation by clumps may coexit with other channels of bulge
assembly such as bar and mergers. Our results suggest that galactic bulges
could be interpreted as composite systems with structural components and
stellar populations storing archaeological information of the dynamical history
of their galaxy.Comment: Accepted for publication in MNRAS - Aug. 20, 201
Minkowski measurability results for self-similar tilings and fractals with monophase generators
In a previous paper [arXiv:1006.3807], the authors obtained tube formulas for
certain fractals under rather general conditions. Based on these formulas, we
give here a characterization of Minkowski measurability of a certain class of
self-similar tilings and self-similar sets. Under appropriate hypotheses,
self-similar tilings with simple generators (more precisely, monophase
generators) are shown to be Minkowski measurable if and only if the associated
scaling zeta function is of nonlattice type. Under a natural geometric
condition on the tiling, the result is transferred to the associated
self-similar set (i.e., the fractal itself). Also, the latter is shown to be
Minkowski measurable if and only if the associated scaling zeta function is of
nonlattice type.Comment: 18 pages, 1 figur
Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators
In a previous paper by the first two authors, a tube formula for fractal
sprays was obtained which also applies to a certain class of self-similar
fractals. The proof of this formula uses distributional techniques and requires
fairly strong conditions on the geometry of the tiling (specifically, the inner
tube formula for each generator of the fractal spray is required to be
polynomial). Now we extend and strengthen the tube formula by removing the
conditions on the geometry of the generators, and also by giving a proof which
holds pointwise, rather than distributionally.
Hence, our results for fractal sprays extend to higher dimensions the
pointwise tube formula for (1-dimensional) fractal strings obtained earlier by
Lapidus and van Frankenhuijsen.
Our pointwise tube formulas are expressed as a sum of the residues of the
"tubular zeta function" of the fractal spray in . This sum ranges
over the complex dimensions of the spray, that is, over the poles of the
geometric zeta function of the underlying fractal string and the integers
. The resulting "fractal tube formulas" are applied to the important
special case of self-similar tilings, but are also illustrated in other
geometrically natural situations. Our tube formulas may also be seen as fractal
analogues of the classical Steiner formula.Comment: 43 pages, 13 figures. To appear: Advances in Mathematic
Geometric invariance of mass-like asymptotic invariants
We study coordinate-invariance of some asymptotic invariants such as the ADM
mass or the Chru\'sciel-Herzlich momentum, given by an integral over a
"boundary at infinity". When changing the coordinates at infinity, some terms
in the change of integrand do not decay fast enough to have a vanishing
integral at infinity; but they may be gathered in a divergence, thus having
vanishing integral over any closed hypersurface. This fact could only be
checked after direct calculation (and was called a "curious cancellation"). We
give a conceptual explanation thereof.Comment: 13 page
Effects of Hyperbolic Rotation in Minkowski Space on the Modeling of Plasma Accelerators in a Lorentz Boosted Frame
Laser driven plasma accelerators promise much shorter particle accelerators
but their development requires detailed simulations that challenge or exceed
current capabilities. We report the first direct simulations of stages up to 1
TeV from simulations using a Lorentz boosted calculation frame resulting in a
million times speedup, thanks to a frame boost as high as gamma=1300. Effects
of the hyperbolic rotation in Minkowski space resulting from the frame boost on
the laser propagation in the plasma is shown to be key in the mitigation of a
numerical instability that was limiting previous attempts
Observation of noise correlated by the Hawking effect in a water tank
We measured the power spectrum and two-point correlation function for the
randomly fluctuating free surface on the downstream side of a stationary flow
with a maximum Froude number reached above a
localised obstacle. On such a flow the scattering of incident long wavelength
modes is analogous to that responsible for black hole radiation (the Hawking
effect). Our measurements of the noise show a clear correlation between pairs
of modes of opposite energies. We also measure the scattering coefficients by
applying the same analysis of correlations to waves produced by a wave maker.Comment: 11 pages, 14 figures. several points clarified; two new subsections
in the Supplemental Material on the wave equation and the links with
experiments in BEC
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