Complexity of Leading Digit Sequences

Let $S_{a,b}$ denote the sequence of leading digits of $a^n$ in base $b$. It is well known that if $a$ is not a rational power of $b$, then the sequence $S_{a,b}$ satisfies Benford's Law; that is, digit $d$ occurs in $S_{a,b}$ with frequency $\log_{b}(1+1/d)$, for $d=1,2,\dots,b-1$. In this paper, we investigate the \emph{complexity} of such sequences. We focus mainly on the \emph{block complexity}, $p_{a,b}(n)$, defined as the number of distinct blocks of length $n$ appearing in $S_{a,b}$. In our main result we determine $p_{a,b}(n)$ for all squarefree bases $b\ge 5$ and all rational numbers $a>0$ that are not integral powers of $b$. In particular, we show that, for all such pairs $(a,b)$, the complexity function $p_{a,b}(n)$ is \emph{affine}, i.e., satisfies $p_{a,b}(n)=c_{a,b} n + d_{a,b}$ for all $n\ge1$, with coefficients $c_{a,b}\ge1$ and $d_{a,b}\ge0$, given explicitly in terms of $a$ and $b$. We also show that the requirement that $b$ be squarefree cannot be dropped: If $b$ is not squarefree, then there exist integers $a$ with $1<a<b$ for which $p_{a,b}(n)$ is not of the above form. We use this result to obtain sharp upper and lower bounds for $p_{a,b}(n)$, and to determine the asymptotic behavior of this function as $b\to\infty$ through squarefree values. We also consider the question which linear functions $p(n)=cn+d$ arise as the complexity function $p_{a,b}(n)$ of some leading digit sequence $S_{a,b}$. We conclude with a discussion of other complexity measures for the sequences $S_{a,b}$ and some open problems

Spin currents in superconductors

It is argued that experiments on rotating superconductors provide evidence for the existence of macroscopic spin currents in superconductors in the absence of applied external fields. Furthermore it is shown that the model of hole superconductivity predicts the existence of such currents in all superconductors. In addition it is pointed out that spin currents are required within a related macroscopic (London-like) electrodynamic description of superconductors recently proposed. The spin current arises through an intrinsic spin Hall effect when negative charge is expelled from the interior of the metal upon the transition to the superconducting state

Observation of infinite-range intensity correlations above, at and below the 3D Anderson localization transition

We investigate long-range intensity correlations on both sides of the Anderson transition of classical waves in a three-dimensional (3D) disordered material. Our ultrasonic experiments are designed to unambiguously detect a recently predicted infinite-range C0 contribution, due to local density of states fluctuations near the source. We find that these C0 correlations, in addition to C2 and C3 contributions, are significantly enhanced near mobility edges. Separate measurements of the inverse participation ratio reveal a link between C0 and the anomalous dimension \Delta_2, implying that C0 may also be used to explore the critical regime of the Anderson transition.Comment: 13 pages, 11 figures (main text plus supplemental information). Updated version includes an improved introductory paragraph, minor text revisions, a revised title and additional supplemental information on the experimental detail

Far infrared maps of the ridge between OMC-1 and OMC-2

Dust continuum emission from a 6 ft x 20 ft region surrounding OMC-1 and OMC-2 were mapped at 55 and 125 microns with 4 ft resolution. The dominant features of the maps are a strong peak at OMC-1 and a ridge of lower surface brightness between OMC-1 and OMC-2. Along the ridge the infrared flux densities and the color temperature decreases smoothly from OMC-1 to OMC-2. OMC-1 is heated primarily by several optical and infrared stars situated within or just at the boundary of the cloud. At the region of minimum column density between OMC-1 and OMC-2 the nearby B0.5 V star NU Ori may contribute significantly to the dust heating. Near OMC-2 dust column densities are large enough so that, in addition to the OMC-2 infrared cluster, the nonlocal infrared sources associated with OMC-1 and NU Ori can contribute to the heating

The Venus Balloon Project

On June 11 and 15, 1985, two instrumental balloons were released from the Soviet VEGA 1 and VEGA 2 spacecraft and deployed in the atmosphere of Venus. The VEGA probes flew by the planet on their way to a rendezvous with comet Halley in March 1986. Drifting with the wind at altitudes of 54 km, the balloons traveled one-third of the way around the planet during their 46-hour lifetimes. Sensors on-board the gondolas made periodic measurements of pressure, temperature, vertical wind velocity, cloud particle density, ambient light level, and frequency of lightning. The data were transmitted to Earth and received at the Deep Space Network (DSN) 64-m stations and at several large antennas in the USSR. Approximately 95 percent of the telemetry data were successfully decoded at the DSN complexes and in the Soviet Union, and were provided to the international science team for analysis. Very Long Baseline Interferometry (VLBI) data were acquired by 20 radio observatories around the world for the purpose of monitoring the Venus winds. The DSN 64-m subnet was part of a 15-station VLBI network organized by the Centre National d'Etudes Spatiales (CNES) of France. In addition, five antennas of the Soviet network participated. VLBI data from the CNES network are currently being processed at the Jet Propulsion Laboratory

Accurate Transfer Maps for Realistic Beamline Elements: Part I, Straight Elements

The behavior of orbits in charged-particle beam transport systems, including both linear and circular accelerators as well as final focus sections and spectrometers, can depend sensitively on nonlinear fringe-field and high-order-multipole effects in the various beam-line elements. The inclusion of these effects requires a detailed and realistic model of the interior and fringe fields, including their high spatial derivatives. A collection of surface fitting methods has been developed for extracting this information accurately from 3-dimensional field data on a grid, as provided by various 3-dimensional finite-element field codes. Based on these realistic field models, Lie or other methods may be used to compute accurate design orbits and accurate transfer maps about these orbits. Part I of this work presents a treatment of straight-axis magnetic elements, while Part II will treat bending dipoles with large sagitta. An exactly-soluble but numerically challenging model field is used to provide a rigorous collection of performance benchmarks.Comment: Accepted to PRST-AB. Changes: minor figure modifications, reference added, typos corrected

Polarization of Thermal Emission from Aligned Dust Grains Under an Anisotropic Radiation Field

If aspherical dust grains are immersed in an anisotropic radiation field, their temperature depends on the cross-sections projected in the direction of the anisotropy.It was shown that the temperature difference produces polarized thermal emission even without alignment, if the observer looks at the grains from a direction different from the anisotropic radiation. When the dust grains are aligned, the anisotropy in the radiation makes various effects on the polarization of the thermal emission, depending on the relative angle between the anisotropy and alignment directions. If the both directions are parallel, the anisotropy produces a steep increase in the polarization degree at short wavelengths. If they are perpendicular, the polarization reversal occurs at a wavelength shorter than the emission peak. The effect of the anisotropic radiation will make a change of more than a few % in the polarization degree for short wavelengths and the effect must be taken into account in the interpretation of the polarization in the thermal emission. The anisotropy in the radiation field produces a strong spectral dependence of the polarization degree and position angle, which is not seen under isotropic radiation. The dependence changes with the grain shape to a detectable level and thus it will provide a new tool to investigate the shape of dust grains. This paper presents examples of numerical calculations of the effects and demonstrates the importance of anisotropic radiation field on the polarized thermal emission.Comment: 13pages, 7figure