Random fractal strings: their zeta functions, complex dimensions and spectral asymptotics

In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that using a random recursive self-similar construction it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary

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

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 $(3.02 \pm 0.27 (stat.) \pm 0.31 (syst.)) \times 10^{-5}$ for hydrogen and $(1.42 \pm ^{0.09}_{0.12} (stat.) \pm 0.11 (syst.)) \times 10^{-5}$ for deuterium, and relative branching ratios of double radiative capture to single radiative capture of $(7.68 \pm 0.69(stat.) \pm 0.79(syst.)) \times 10^{-5}$ for hydrogen and $(5.44 \pm^{0.34}_{0.46}(stat.) \pm 0.42(syst.)) \times 10^{-5}$ 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 $\pi^-$ on the $\pi^+$ 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.

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

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

Generalized sum rules of the nucleon in the constituent quark model

We study the generalized sum rules and polarizabilities of the nucleon in the framework of the hypercentral constituent quark model. We include in the calculation all the well known $3^*$ and $4^*$ resonances and consider all the generalized sum rules for which there are data available. To test the model dependence of the calculation, we compare our results to the results obtained in the harmonic oscillator CQM. We furthermore confront our results to the model-independent sum rules values and to the predictions of the phenomenological MAID model. The CQM calculations provide a good description of most of the presented generalized sum rules in the intermediate $Q^2$ region (above $\sim0.2$ GeV$^2$) while they encounter difficulties in describing these observables at low $Q^2$, where the effects of the pion cloud, not included in the present calculation, are expected to be important.Comment: 26 pages, 10 figure