2,699 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
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
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
On the Gerasimov-Drell-Hearn sum rule for the deuteron
The Gerasimov-Drell-Hearn sum rule is evaluated for the deuteron by explicit
integration up to 550 MeV including contributions from the photodisintegration
channel and from coherent and incoherent single pion production as well. The
photodisintegration channel converges fast enough in this energy range and
gives a large negative contribution, essentially from the resonant
state near threshold. Its absolute value is about the same size as the sum of
proton and neutron GDH values. It is only partially cancelled by the single
pion production contribution. But the incoherent channel has not reached
convergence at 550 MeV.Comment: 6 pages latex including 3 postscript figures, talk at the 15th Int.
Conf. on Few-Body Problems in Physics, Groningen, Netherlands, 22-26 July
1997. To be published in Nucl. Phys.
The Long Term Stability of Oscillations During Thermonuclear X-ray Bursts: Constraining the Binary X-ray Mass Function
We report on the long term stability of the millisecond oscillations observed
with the Rossi X-ray Timing Explorer (RXTE) during thermonuclear X-ray bursts
from the low mass X-ray binaries (LMXB) 4U 1728-34 and 4U 1636-53. We show that
bursts from 4U 1728-34 spanning more than 1.5 years have observed asymptotic
oscillation periods which are within 0.2 microsec. of each other, well within
the magnitude which could be produced by the orbital motion of the neutron star
in a typical LMXB. This stability implies a timescale to change the oscillation
period of > 23,000 years, suggesting a highly stable process such as stellar
rotation as the oscillation mechanism. We show that period offsets in three
distinct bursts from 4U 1636-53 can be plausibly interpreted as due to orbital
motion of the neutron star in this 3.8 hour binary system. We discuss the
constraints on the mass function which can in principle be derived using this
technique.Comment: 11 pages, 4 figures. AASTeX, to be published in the Astrophysical
Journal Letter
Double radiative pion capture on hydrogen and deuterium and the nucleon's pion cloud
We report measurements of double radiative capture in pionic hydrogen and
pionic deuterium. The measurements were performed with the RMC spectrometer at
the TRIUMF cyclotron by recording photon pairs from pion stops in liquid
hydrogen and deuterium targets. We obtained absolute branching ratios of for hydrogen and for deuterium, and
relative branching ratios of double radiative capture to single radiative
capture of for hydrogen
and for
deuterium. For hydrogen, the measured branching ratio and photon energy-angle
distributions are in fair agreement with a reaction mechanism involving the
annihilation of the incident on the cloud of the target proton.
For deuterium, the measured branching ratio and energy-angle distributions are
qualitatively consistent with simple arguments for the expected role of the
spectator neutron. A comparison between our hydrogen and deuterium data and
earlier beryllium and carbon data reveals substantial changes in the relative
branching ratios and the energy-angle distributions and is in agreement with
the expected evolution of the reaction dynamics from an annihilation process in
S-state capture to a bremsstrahlung process in P-state capture. Lastly, we
comment on the relevance of the double radiative process to the investigation
of the charged pion polarizability and the in-medium pion field.Comment: 44 pages, 7 tables, 13 figures, submitted to Phys. Rev.
A multifractal zeta function for cookie cutter sets
Starting with the work of Lapidus and van Frankenhuysen a number of papers
have introduced zeta functions as a way of capturing multifractal information.
In this paper we propose a new multifractal zeta function and show that under
certain conditions the abscissa of convergence yields the Hausdorff
multifractal spectrum for a class of measures
Electromagnetic Calorimeter for HADES
We propose to build the Electromagnetic calorimeter for the HADES di-lepton
spectrometer. It will enable to measure the data on neutral meson production
from nucleus-nucleus collisions, which are essential for interpretation of
dilepton data, but are unknown in the energy range of planned experiments (2-10
GeV per nucleon). The calorimeter will improve the electron-hadron separation,
and will be used for detection of photons from strange resonances in elementary
and HI reactions.
Detailed description of the detector layout, the support structure, the
electronic readout and its performance studied via Monte Carlo simulations and
series of dedicated test experiments is presented.
The device will cover the total area of about 8 m^2 at polar angles between
12 and 45 degrees with almost full azimuthal coverage. The photon and electron
energy resolution achieved in test experiments amounts to 5-6%/sqrt(E[GeV])
which is sufficient for the eta meson reconstruction with S/B ratio of 0.4% in
Ni+Ni collisions at 8 AGeV. A purity of the identified leptons after the hadron
rejection, resulting from simulations based on the test measurements, is better
than 80% at momenta above 500 MeV/c, where time-of-flight cannot be used.Comment: 40 pages, 38 figures version2 - the time schedule added, information
about PMTs in Sec.III update
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