A general method to determine covariant Lyapunov vectors in both discrete-and continuous-time dynamical systems is introduced. This allows us to address fundamental questions such as the degree of hyperbolicity, which can be quantified in terms of the transversality of these intrinsic vectors. For spatially extended systems, the covariant Lyapunov vectors have localization properties and spatial Fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the Lyapunov exponents. DOI: 10.1103/PhysRevLett.99.130601 PACS numbers: 05.70.Ln, 05.90.+m, 45.70.ÿn, 87.18.Ed Measuring Lyapunov exponents (LEs) is a central issue in the investigation of chaotic dynamical systems because they are intrinsic observables that allow us to quantify a number of different physical properties such as sensitivity to initial conditions, local entropy production and attractor dimension [1]. Moreover, in the context of spatiotemporal chaos, the very existence of a well-defined Lyapunov spectrum in the thermodynamic limit is a proof of the extensivity of chaos [2], and it has been speculated that the small exponents contain information on the ''hydrodynamic'' modes of the dynamics (e.g., see [3] and references therein).In this latter perspective, a growing interest has been devoted not only to the LEs but also to some corresponding vectors, with the motivation that they could contribute to identifying both the real-space structure of collective modes [4] and the regions characterized by stronger or weaker instabilities [5]. However, the only available approach so far is based on the vectors yielded by the standard procedure used to calculate the LEs [6]. This allows us to identify the most expanding subspaces, but has the drawback that these vectors-that we shall call GramSchmidt vectors (GSV) after the procedure used-are, by construction, orthogonal, even where stable and unstable manifolds are nearly tangent. Moreover, GSV are not invariant under time reversal, and they are not covariant; i.e., the GSV at a given phase-space point are not mapped by the linearized dynamics into the GSV of the forward images of this point.While the existence, for invertible dynamics, of a coordinate-independent, local decomposition of phase space into covariant Lyapunov directions -the so-called Oseledec splitting [1]-has been discussed by Ruelle long ago [7], it received almost no attention in the literature, because of the absence of algorithms to practically determine it. In this Letter, we propose an innovative approach based on both forward and backward iterations of the tangent dynamics, which allows determining a set of directions at each point of phase space that are invariant under time reversal and covariant with the dynamics. We argue that, for any invertible dynamical system, the intrinsic tangent-space decomposition introduced by these covariant Lyapunov vectors (CLV) coincides with the Oseledec splitting.As a first important and general application of the CLV...

One of the main goals of the feasibility study MOSE (MOdelling ESO Sites) is to evaluate the performances of a method conceived to forecast the optical turbulence above the ESO sites of the Very Large Telescope and the European-Extremely Large Telescope in Chile. The method implied the use of a dedicated code conceived for the optical turbulence (OT) called Astro-Meso-Nh. In this paper we present results we obtained at conclusion of this project concerning the performances of this method in forecasting the most relevant parameters related to the optical turbulence (C 2 N , seeing ε, isoplanatic angle θ 0 and wavefront coherence time τ 0 ). Numerical predictions related to a very rich statistical sample of nights uniformly distributed along a solar year and belonging to different years have been compared to observations and different statistical operators have been analyzed such as the classical bias, RMSE, σ and more sophisticated statistical operators derived by the contingency tables that are able to quantify the score of success of a predictive method such as the percentage of correct detection (PC) and the probability to detect a parameter within a specific range of values (POD). The main conclusions of the study tell us that the Astro-Meso-Nh model provides performances that are already very good to definitely guarantee a not negligible positive impact on the Service Mode of top-class telescopes and ELTs. A demonstrator for an automatic and operational version of the Astro-Meso-Nh model will be soon implemented on the sites of VLT and E-ELT.

According to thermodynamics, the specific heat of Boltzmannian short-range interacting systems is a positive quantity. Less intuitive properties are instead displayed by systems characterized by long-range interactions. In that case, the sign of specific heat depends on the considered statistical ensemble: Negative specific heat can be found in isolated systems, which are studied in the framework of the microcanonical ensemble; on the other hand, it is generally recognized that a positive specific heat should always be measured in systems in contact with a thermal bath, for which the canonical ensemble is the appropriate one. We demonstrate that the latter assumption is not generally true: One can, in principle, measure negative specific heat also in the canonical ensemble if the system under scrutiny is non-Boltzmannian and/or out-of-equilibrium.

The efficiency of the management of top-class ground-based astronomical facilities supported by Adaptive Optics (AO) relies on our ability to forecast the optical turbulence (OT) and a set of relevant atmospheric parameters. Indeed, in spite of the fact that the AO is able to achieve, at present, excellent levels of wavefront corrections (a Strehl Ratio up to 90% in H band), its performances strongly depend on the atmospheric conditions. Knowing in advance the atmospheric turbulence conditions allows an optimization of the AO use. It has already been proven that it is possible to provide reliable forecasts of the optical turbulence (C 2 N profiles and integrated astroclimatic parameters such as seeing, isoplanantic angle, wavefront coherence time, ...) for the next night. In this paper we prove that it is possible to improve the forecast performances on shorter time scales (order of one or two hours) with consistent gains (order of 2 to 8) employing filtering techniques that make use of real-time measurements. This has permitted us to achieve forecasts accuracies never obtained before and reach a fundamental milestone for the astronomical applications. The time scale of one or two hours is the most critical one for an efficient management of the ground-based telescopes supported by AO. We implemented this method in the operational forecast system of the Large Binocular Telescope, named ALTA Center that is, at our knowledge, the first operational system providing forecasts of turbulence and atmospheric parameters at short time scales to support science operations.

Context. In the present-day panorama of large spectroscopic surveys, the amount, diversity, and complexity of the available data continuously increase. The overarching goal of studying the formation and evolution of our Galaxy is hampered by the heterogeneity of instruments, selection functions, analysis methods, and measured quantities. Aims. We present a comprehensive catalogue, the Survey of Surveys (SoS), built by homogeneously merging the radial velocity (RV) determinations of the largest ground-based spectroscopic surveys to date, such as APOGEE, GALAH, Gaia-ESO, RAVE, and LAMOST, using Gaia as a reference. This pilot study serves to prove the concept and to test the methodology that we plan to apply in the future to the stellar parameters and abundance ratios as well. Methods. We have devised a multi-staged procedure that includes: i) the cross match between Gaia and the spectroscopic surveys using the official Gaia cross-match algorithm, ii) the normalisation of uncertainties using repeated measurements or the three-cornered hat method, iii) the cross calibration of the RVs as a function of the main parameters on which depend (magnitude, effective temperature, surface gravity, metallicity, and signal-to-noise ratio) to remove trends and zero point offsets, and iv) the comparison with external high-resolution samples, such as the Gaia RV standards and the Geneva-Copenhagen survey, to validate the homogenisation procedure and to calibrate the RV zero-point of the SoS catalogue. Results. We provide the largest homogenised RV catalogue to date, containing almost 11 million stars, of which about half come exclusively from Gaia and half in combination with the ground-based surveys. We estimate the accuracy of the RV zero-point to be about 0.16-0.31 km s −1 and the RV precision to be in the range 0.05-1.50 km s −1 depending on the type of star and on its survey provenance. We validate the SoS RVs with open clusters from a high resolution homogeneous samples and provide the systemic velocity of 55 individual open clusters. Additionally, we provide median RVs for 532 clusters recently discovered by Gaia data. Conclusions. The SoS is publicly available and ready to be applied to various research projects, such as the study of star clusters, Galactic archaeology, stellar streams, or the characterisation of planet-hosting stars, to name a few. We also plan to include survey updates and more data sources in future versions of the SoS.