2,743 research outputs found
Fractal Strings and Multifractal Zeta Functions
For a Borel measure on the unit interval and a sequence of scales that tend
to zero, we define a one-parameter family of zeta functions called multifractal
zeta functions. These functions are a first attempt to associate a zeta
function to certain multifractal measures. However, we primarily show that they
associate a new zeta function, the topological zeta function, to a fractal
string in order to take into account the topology of its fractal boundary. This
expands upon the geometric information garnered by the traditional geometric
zeta function of a fractal string in the theory of complex dimensions. In
particular, one can distinguish between a fractal string whose boundary is the
classical Cantor set, and one whose boundary has a single limit point but has
the same sequence of lengths as the complement of the Cantor set. Later work
will address related, but somewhat different, approaches to multifractals
themselves, via zeta functions, partly motivated by the present paper.Comment: 32 pages, 9 figures. This revised version contains new sections and
figures illustrating the main results of this paper and recent results from
others. Sections 0, 2, and 6 have been significantly rewritte
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.
Analysis of ultrasonic transducers with fractal architecture
Ultrasonic transducers composed of a periodic piezoelectric composite are generally accepted as the design of choice in many applications. Their architecture is normally very regular and this is due to manufacturing constraints rather than performance optimisation. Many of these manufacturing restrictions no longer hold due to new production methods such as computer controlled, laser cutting, and so there is now freedom to investigate new types of geometry. In this paper, the plane wave expansion model is utilised to investigate the behaviour of a transducer with a self-similar architecture. The Cantor set is utilised to design a 2-2 conguration, and a 1-3 conguration is investigated with a Sierpinski Carpet geometry
Q^2 Evolution of Generalized Baldin Sum Rule for the Proton
The generalized Baldin sum rule for virtual photon scattering, the
unpolarized analogy of the generalized Gerasimov-Drell-Hearn integral, provides
an important way to investigate the transition between perturbative QCD and
hadronic descriptions of nucleon structure. This sum rule requires integration
of the nucleon structure function F_1, which until recently had not been
measured at low Q^2 and large x, i.e. in the nucleon resonance region. This
work uses new data from inclusive electron-proton scattering in the resonance
region obtained at Jefferson Lab, in combination with SLAC deep inelastic
scattering data, to present first precision measurements of the generalized
Baldin integral for the proton in the Q^2 range of 0.3 to 4.0 GeV^2.Comment: 4 pages, 3 figures, one table; text added, one figure replace
Multifractal analysis via scaling zeta functions and recursive structure of lattice strings
The multifractal structure underlying a self-similar measure stems directly
from the weighted self-similar system (or weighted iterated function system)
which is used to construct the measure. This follows much in the way that the
dimension of a self-similar set, be it the Hausdorff, Minkowski, or similarity
dimension, is determined by the scaling ratios of the corresponding
self-similar system via Moran's theorem. The multifractal structure allows for
our definition of scaling regularity and scaling zeta functions motivated by
geometric zeta functions and, in particular, partition zeta functions. Some of
the results of this paper consolidate and partially extend the results
regarding a multifractal analysis for certain self-similar measures supported
on compact subsets of a Euclidean space based on partition zeta functions.
Specifically, scaling zeta functions generalize partition zeta functions when
the choice of the family of partitions is given by the natural family of
partitions determined by the self-similar system in question. Moreover, in
certain cases, self-similar measures can be shown to exhibit lattice or
nonlattice structure with respect to specified scaling regularity values.
Additionally, in the context provided by generalized fractal strings viewed as
measures, we define generalized self-similar strings, allowing for the
examination of many of the results presented here in a specific overarching
context and for a connection to the results regarding the corresponding complex
dimensions as roots of Dirichlet polynomials. Furthermore, generalized lattice
strings and recursive strings are defined and shown to be very closely related.Comment: 33 pages, no figures, in pres
Sum Rules for Magnetic Moments and Polarizabilities in QED and Chiral Effective-Field Theory
We elaborate on a recently proposed extension of the Gerasimov-Drell-Hearn
(GDH) sum rule which is achieved by taking derivatives with respect to the
anomalous magnetic moment. The new sum rule features a {\it linear} relation
between the anomalous magnetic moment and the dispersion integral over a
cross-section quantity. We find some analogy of the linearized form of the GDH
sum rule with the `sideways dispersion relations'. As an example, we apply the
linear sum rule to reproduce the famous Schwinger's correction to the magnetic
moment in QED from a tree-level cross-section calculation and outline the
procedure for computing the two-loop correction from a one-loop cross-section
calculation. The polarizabilities of the electron in QED are considered as well
by using the other forward-Compton-scattering sum rules. We also employ the sum
rules to study the magnetic moment and polarizabilities of the nucleon in a
relativistic chiral EFT framework. In particular we investigate the chiral
extrapolation of these quantities.Comment: 24 pages, 7 figures; several additions, published versio
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