Recurrence and algorithmic information

In this paper we initiate a somewhat detailed investigation of the relationships between quantitative recurrence indicators and algorithmic complexity of orbits in weakly chaotic dynamical systems. We mainly focus on examples.Comment: 26 pages, no figure

A two-band approach to nλ\lambda phase error corrections with LBTI's PHASECam

PHASECam is the Large Binocular Telescope Interferometer's (LBTI) phase sensor, a near-infrared camera which is used to measure tip/tilt and phase variations between the two AO-corrected apertures of the Large Binocular Telescope (LBT). Tip/tilt and phase sensing are currently performed in the H (1.65 $\mu$m) and K (2.2 $\mu$m) bands at 1 kHz, and the K band phase telemetry is used to send tip/tilt and Optical Path Difference (OPD) corrections to the system. However, phase variations outside the range [-$\pi$, $\pi$] are not sensed, and thus are not fully corrected during closed-loop operation. PHASECam's phase unwrapping algorithm, which attempts to mitigate this issue, still occasionally fails in the case of fast, large phase variations. This can cause a fringe jump, in which case the unwrapped phase will be incorrect by a wavelength or more. This can currently be manually corrected by the observer, but this is inefficient. A more reliable and automated solution is desired, especially as the LBTI begins to commission further modes which require robust, active phase control, including controlled multi-axial (Fizeau) interferometry and dual-aperture non-redundant aperture masking interferometry. We present a multi-wavelength method of fringe jump capture and correction which involves direct comparison between the K band and currently unused H band phase telemetry.Comment: 17 pages, 10 figure

When do finite sample effects significantly affect entropy estimates ?

An expression is proposed for determining the error caused on entropy estimates by finite sample effects. This expression is based on the Ansatz that the ranked distribution of probabilities tends to follow an empirical Zipf law.Comment: 10 pages, 2 figure

Occam's Quantum Strop: Synchronizing and Compressing Classical Cryptic Processes via a Quantum Channel

A stochastic process's statistical complexity stands out as a fundamental property: the minimum information required to synchronize one process generator to another. How much information is required, though, when synchronizing over a quantum channel? Recent work demonstrated that representing causal similarity as quantum state-indistinguishability provides a quantum advantage. We generalize this to synchronization and offer a sequence of constructions that exploit extended causal structures, finding substantial increase of the quantum advantage. We demonstrate that maximum compression is determined by the process's cryptic order---a classical, topological property closely allied to Markov order, itself a measure of historical dependence. We introduce an efficient algorithm that computes the quantum advantage and close noting that the advantage comes at a cost---one trades off prediction for generation complexity.Comment: 10 pages, 6 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/oqs.ht

A "metric" complexity for weakly chaotic systems

We consider the number of Bowen sets which are necessary to cover a large measure subset of the phase space. This introduce some complexity indicator characterizing different kind of (weakly) chaotic dynamics. Since in many systems its value is given by a sort of local entropy, this indicator is quite simple to be calculated. We give some example of calculation in nontrivial systems (interval exchanges, piecewise isometries e.g.) and a formula similar to the Ruelle-Pesin one, relating the complexity indicator to some initial condition sensitivity indicators playing the role of positive Lyapunov exponents.Comment: 15 pages, no figures. Articl

Source-Channel Diversity for Parallel Channels

We consider transmitting a source across a pair of independent, non-ergodic channels with random states (e.g., slow fading channels) so as to minimize the average distortion. The general problem is unsolved. Hence, we focus on comparing two commonly used source and channel encoding systems which correspond to exploiting diversity either at the physical layer through parallel channel coding or at the application layer through multiple description source coding. For on-off channel models, source coding diversity offers better performance. For channels with a continuous range of reception quality, we show the reverse is true. Specifically, we introduce a new figure of merit called the distortion exponent which measures how fast the average distortion decays with SNR. For continuous-state models such as additive white Gaussian noise channels with multiplicative Rayleigh fading, optimal channel coding diversity at the physical layer is more efficient than source coding diversity at the application layer in that the former achieves a better distortion exponent. Finally, we consider a third decoding architecture: multiple description encoding with a joint source-channel decoding. We show that this architecture achieves the same distortion exponent as systems with optimal channel coding diversity for continuous-state channels, and maintains the the advantages of multiple description systems for on-off channels. Thus, the multiple description system with joint decoding achieves the best performance, from among the three architectures considered, on both continuous-state and on-off channels.Comment: 48 pages, 14 figure