Block Edit Errors with Transpositions: Deterministic Document Exchange Protocols and Almost Optimal Binary Codes

Document exchange and error correcting codes are two fundamental problems regarding communications. In the first problem, Alice and Bob each holds a string, and the goal is for Alice to send a short sketch to Bob, so that Bob can recover Alice\u27s string. In the second problem, Alice sends a message with some redundant information to Bob through a channel that can add adversarial errors, and the goal is for Bob to correctly recover the message despite the errors. In both problems, an upper bound is placed on the number of errors between the two strings or that the channel can add, and a major goal is to minimize the size of the sketch or the redundant information. In this paper we focus on deterministic document exchange protocols and binary error correcting codes. Both problems have been studied extensively. In the case of Hamming errors (i.e., bit substitutions) and bit erasures, we have explicit constructions with asymptotically optimal parameters. However, other error types are still rather poorly understood. In a recent work [Kuan Cheng et al., 2018], the authors constructed explicit deterministic document exchange protocols and binary error correcting codes for edit errors with almost optimal parameters. Unfortunately, the constructions in [Kuan Cheng et al., 2018] do not work for other common errors such as block transpositions. In this paper, we generalize the constructions in [Kuan Cheng et al., 2018] to handle a much larger class of errors. These include bursts of insertions and deletions, as well as block transpositions. Specifically, we consider document exchange and error correcting codes where the total number of block insertions, block deletions, and block transpositions is at most k <= alpha n/log n for some constant 0<alpha<1. In addition, the total number of bits inserted and deleted by the first two kinds of operations is at most t <= beta n for some constant 0<beta<1, where n is the length of Alice\u27s string or message. We construct explicit, deterministic document exchange protocols with sketch size O((k log n +t) log^2 n/{k log n + t}) and explicit binary error correcting code with O(k log n log log log n+t) redundant bits. As a comparison, the information-theoretic optimum for both problems is Theta(k log n+t). As far as we know, previously there are no known explicit deterministic document exchange protocols in this case, and the best known binary code needs Omega(n) redundant bits even to correct just one block transposition [L. J. Schulman and D. Zuckerman, 1999]

Power-Law Decay of Standing Waves on the Surface of Topological Insulators

We propose a general theory on the standing waves (quasiparticle interference pattern) caused by the scattering of surface states off step edges in topological insulators, in which the extremal points on the constant energy contour of surface band play the dominant role. Experimentally we image the interference patterns on both Bi$_2$Te$_3$ and Bi$_2$Se$_3$ films by measuring the local density of states using a scanning tunneling microscope. The observed decay indices of the standing waves agree excellently with the theoretical prediction: In Bi$_2$Se$_3$, only a single decay index of -3/2 exists; while in Bi$_2$Te$_3$ with strongly warped surface band, it varies from -3/2 to -1/2 and finally to -1 as the energy increases. The -1/2 decay indicates that the suppression of backscattering due to time-reversal symmetry does not necessarily lead to a spatial decay rate faster than that in the conventional two-dimensional electron system. Our formalism can also explain the characteristic scattering wave vectors of the standing wave caused by non-magnetic impurities on Bi$_2$Te$_3$.Comment: 4 pages, 3 figure

Supermassive Black Holes with High Accretion Rates in Active Galactic Nuclei. IV. Hβ\beta Time Lags and Implications for Super-Eddington Accretion

We have completed two years of photometric and spectroscopic monitoring of a large number of active galactic nuclei (AGNs) with very high accretion rates. In this paper, we report on the result of the second phase of the campaign, during 2013--2014, and the measurements of five new H$\beta$ time lags out of eight monitored AGNs. All five objects were identified as super-Eddington accreting massive black holes (SEAMBHs). The highest measured accretion rates for the objects in this campaign are $\dot{\mathscr{M}}\gtrsim 200$, where $\dot{\mathscr{M}}= \dot{M}_{\bullet}/L_{\rm Edd}c^{-2}$, $\dot{M}_{\bullet}$ is the mass accretion rates, $L_{\rm Edd}$ is the Eddington luminosity and $c$ is the speed of light. We find that the H$\beta$ time lags in SEAMBHs are significantly shorter than those measured in sub-Eddington AGNs, and the deviations increase with increasing accretion rates. Thus, the relationship between broad-line region size ($R_{_{\rm H\beta}}$) and optical luminosity at 5100\AA, $R_{_{\rm H\beta}}-L_{5100}$, requires accretion rate as an additional parameter. We propose that much of the effect may be due to the strong anisotropy of the emitted slim-disk radiation. Scaling $R_{_{\rm H\beta}}$ by the gravitational radius of the black hole, we define a new radius-mass parameter ($Y$) and show that it saturates at a critical accretion rate of $\dot{\mathscr{M}}_c=6\sim 30$, indicating a transition from thin to slim accretion disk and a saturated luminosity of the slim disks. The parameter $Y$ is a very useful probe for understanding the various types of accretion onto massive black holes. We briefly comment on implications to the general population of super-Eddington AGNs in the universe and applications to cosmology.Comment: 53 pages, 12 figures, 7 tables, accepted for publication in The Astrophysical Journa

Simultaneous Conversion of Polarization and Frequency via Time‐Division‐Multiplexing Metasurfaces

AbstractMetasurfaces are artificially engineered two‐dimensional materials composed of sub‐wavelength meta‐atoms, which have shown unprecedented capabilities in manipulating the amplitude, phase, frequency, and polarization states of electromagnetic waves. Specifically, polarization control can be attained via suitable anisotropic, linear, and time‐invariant designs, while frequency conversion is realized via nonlinear or time‐varying platforms. Simultaneous manipulations of polarization and frequency would be of considerable practical interest in many application scenarios, but remain unattainable with current approaches. Here, a time‐division‐multiplexing metasurface is proposed to realize the simultaneous conversion of polarization and frequency. The platform relies on time‐modulated polarization switches and, by varying the duty cycle and time delays of the polarization channels, can arbitrarily rotate the polarization at the central frequency of operation, and synthesize various polarization states at selected harmonic frequencies. Theoretical predictions are validated via measurements on a prototype operating at microwave frequencies, providing the first experimental evidence of simultaneous polarization and frequency conversions via time‐division‐multiplexing metasurfaces. The outcomes open a new pathway in manipulating the electromagnetic waves via time‐varying metasurfaces, and may be of interest for a broad variety of applications in scenarios ranging from polarization imaging to quantum optics