A Tale of Two Fractals: The Hofstadter Butterfly and The Integral Apollonian Gaskets

This paper unveils a mapping between a quantum fractal that describes a physical phenomena, and an abstract geometrical fractal. The quantum fractal is the Hofstadter butterfly discovered in 1976 in an iconic condensed matter problem of electrons moving in a two-dimensional lattice in a transverse magnetic field. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles. Both of these fractals are made up of integers. In the Hofstadter butterfly, these integers encode the topological quantum numbers of quantum Hall conductivity. In the Apollonian gaskets an infinite number of mutually tangent circles are nested inside each other, where each circle has integer curvature. The mapping between these two fractals reveals a hidden threefold symmetry embedded in the kaleidoscopic images that describe the asymptotic scaling properties of the butterfly. This paper also serves as a mini review of these fractals, emphasizing their hierarchical aspects in terms of Farey fractions

Effects of Anomalous Magnetic Moment in the Quantum Motion of Neutral Particle in Magnetic and Electric Fields Produced by a Linear Source in a Conical Spacetime

In this paper we analyse the effect of the anomalous magnetic moment on the non-relativistic quantum motion of a neutral particle in magnetic and electric fields produced by linear sources of constant current and charge density, respectively.Comment: 17 pages, no figur

Effect of Chaotic Noise on Multistable Systems

In a recent letter [Phys.Rev.Lett. {\bf 30}, 3269 (1995), chao-dyn/9510011], we reported that a macroscopic chaotic determinism emerges in a multistable system: the unidirectional motion of a dissipative particle subject to an apparently symmetric chaotic noise occurs even if the particle is in a spatially symmetric potential. In this paper, we study the global dynamics of a dissipative particle by investigating the barrier crossing probability of the particle between two basins of the multistable potential. We derive analytically an expression of the barrier crossing probability of the particle subject to a chaotic noise generated by a general piecewise linear map. We also show that the obtained analytical barrier crossing probability is applicable to a chaotic noise generated not only by a piecewise linear map with a uniform invariant density but also by a non-piecewise linear map with non-uniform invariant density. We claim, from the viewpoint of the noise induced motion in a multistable system, that chaotic noise is a first realization of the effect of {\em dynamical asymmetry} of general noise which induces the symmetry breaking dynamics.Comment: 14 pages, 9 figures, to appear in Phys.Rev.

Conformal linear gravity in de Sitter space II

From the group theoretical point of view, it is proved that the theory of linear conformal gravity should be written in terms of a tensor field of rank-3 and mixed symmetry [Binegar, et al, Phys. Rev. D 27, (1983) 2249]. We obtained such a field equation in de Sitter space [Takook, et al, J. Math. Phys. 51, (2010) 032503]. In this paper, a proper solution to this equation is obtained as a product of a generalized polarization tensor and a massless scalar field and then the conformally invariant two-point function is calculated. This two-point function is de Sitter invariant and free of any pathological large-distance behavior.Comment: 16 pages, no figure, published versio

Naked Singularity Formation In f(R) Gravity

We study the gravitational collapse of a star with barotropic equation of state $p=w\rho$ in the context of $f({\mathcal R})$ theories of gravity. Utilizing the metric formalism, we rewrite the field equations as those of Brans-Dicke theory with vanishing coupling parameter. By choosing the functionality of Ricci scalar as $f({\mathcal R})=\alpha{\mathcal R}^{m}$, we show that for an appropriate initial value of the energy density, if $\alpha$ and $m$ satisfy certain conditions, the resulting singularity would be naked, violating the cosmic censorship conjecture. These conditions are the ratio of the mass function to the area radius of the collapsing ball, negativity of the effective pressure, and the time behavior of the Kretschmann scalar. Also, as long as parameter $\alpha$ obeys certain conditions, the satisfaction of the weak energy condition is guaranteed by the collapsing configuration.Comment: 15 pages, 4 figures, to appear in GR

Production and Decay of D_1(2420)^0 and D_2^*(2460)^0

We have investigated $D^{+}\pi^{-}$ and $D^{*+}\pi^{-}$ final states and observed the two established $L=1$ charmed mesons, the $D_1(2420)^0$ with mass $2421^{+1+2}_{-2-2}$ MeV/c$^{2}$ and width $20^{+6+3}_{-5-3}$ MeV/c$^{2}$ and the $D_2^*(2460)^0$ with mass $2465 \pm 3 \pm 3$ MeV/c$^{2}$ and width $28^{+8+6}_{-7-6}$ MeV/c$^{2}$. Properties of these final states, including their decay angular distributions and spin-parity assignments, have been studied. We identify these two mesons as the $j_{light}=3/2$ doublet predicted by HQET. We also obtain constraints on {\footnotesize $\Gamma_S/(\Gamma_S + \Gamma_D)$} as a function of the cosine of the relative phase of the two amplitudes in the $D_1(2420)^0$ decay.Comment: 15 pages in REVTEX format. hardcopies with figures can be obtained by sending mail to: [email protected]

Measurement of the branching fraction for Υ(1S)→τ+τ−\Upsilon (1S) \to \tau^+ \tau^-

We have studied the leptonic decay of the $\Upsilon (1S)$ resonance into tau pairs using the CLEO II detector. A clean sample of tau pair events is identified via events containing two charged particles where exactly one of the particles is an identified electron. We find $B(\Upsilon(1S) \to \tau^+ \tau^-) = (2.61~\pm~0.12~{+0.09\atop{-0.13}})$. The result is consistent with expectations from lepton universality.Comment: 9 pages, RevTeX, two Postscript figures available upon request, CLNS 94/1297, CLEO 94-20 (submitted to Physics Letters B

The Kuiper Belt and Other Debris Disks

We discuss the current knowledge of the Solar system, focusing on bodies in the outer regions, on the information they provide concerning Solar system formation, and on the possible relationships that may exist between our system and the debris disks of other stars. Beyond the domains of the Terrestrial and giant planets, the comets in the Kuiper belt and the Oort cloud preserve some of our most pristine materials. The Kuiper belt, in particular, is a collisional dust source and a scientific bridge to the dusty "debris disks" observed around many nearby main-sequence stars. Study of the Solar system provides a level of detail that we cannot discern in the distant disks while observations of the disks may help to set the Solar system in proper context.Comment: 50 pages, 25 Figures. To appear in conference proceedings book "Astrophysics in the Next Decade