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Existence and uniqueness of rotating fluid bodies in equilibrium is still poorly understood in General Relativity (GR). Apart from the limiting case of infinitely thin disks, the only known global results in the stationary rotating case (Heilig [14] and Makino [21] [arXiv:1705.07392]) show existence in GR nearby a Newtonian configuration (under suitable additional restrictions). In this work we prove existence and uniqueness of rigidly (slowly) rotating fluid bodies in equilibrium to second order in perturbation theory in GR. The most widely used perturbation framework to describe slowly rigidly rotating stars in the strong field regime is the Hartle-Thorne model. The model involves a number of hypotheses, some explicit, like equatorial symmetry or that the perturbation parameter is proportional to the rotation, but some implicit, particularly on the structure and regularity of the perturbation tensors and the conditions of their matching at the surface. In this work, with basis on the gauge results obtained in [25], the Hartle-Thorne model is fully derived from first principles and only assuming that the perturbations describe a rigidly rotating finite perfect fluid ball (with no layer at the surface) with the same barotropic equation of state as the static ball. Rigidly rotating fluid balls are analyzed consistently in second order perturbation theory by imposing only basic differentiability requirements and boundedness. Our results prove in particular that, at this level of approximation, the spacetime must be indeed equatorially symmetric and is fully determined by two parameters, namely the central pressure and the uniform angular velocity of the fluid., Comment: 81 pages, no figures

Perturbation theory in geometric theories of gravitation is a gauge theory of symmetric tensors defined on a Lorentzian manifold (the background spacetime). The gauge freedom makes uniqueness problems in perturbation theory particularly hard as one needs to understand in depth the process of gauge fixing before attempting any uniqueness proof. This is the first paper of a series of two aimed at deriving an existence and uniqueness result for rigidly rotating stars to second order in perturbation theory in General Relativity. A necessary step is to show the existence of a suitable choice of gauge and to understand the differentiability and regularity properties of the resulting gauge tensors in some "canonical form", particularly at the centre of the star. With a wider range of applications in mind, in this paper we analyse the fixing and regularity problem in a more general setting. In particular we tackle the problem of the Hodge-type decomposition into scalar, vector and tensor components on spheres of symmetric and axially symmetric tensors with finite differentiability down to the origin, exploiting a strategy in which the loss of differentiability is as low as possible. Our primary interest, and main result, is to show that stationary and axially symmetric second order perturbations around static and spherically symmetric background configurations can indeed be rendered in the usual "canonical form" used in the literature while loosing only one degree of differentiability and keeping all relevant quantities bounded near the origin., Comment: 60 pages, no figures

We use observations from the APOGEE survey to explore the relationship between stellar parameters and multiplicity. We combine high-resolution repeat spectroscopy for 41,363 dwarf and subgiant stars with abundance measurements from the APOGEE pipeline and distances and stellar parameters derived using \textit{Gaia} DR2 parallaxes from \cite{Sanders2018} to identify and characterise stellar multiples with periods below 30 years, corresponding to \drvm$\gtrsim$ 3 \kms, where \drvm\ is the maximum APOGEE-detected shift in the radial velocities. Chemical composition is responsible for most of the variation in the close binary fraction in our sample, with stellar parameters like mass and age playing a secondary role. In addition to the previously identified strong anti-correlation between the close binary fraction and \feh\, we find that high abundances of $\alpha$ elements also suppress multiplicity at most values of \feh\ sampled by APOGEE. The anti-correlation between $\alpha$ abundances and multiplicity is substantially steeper than that observed for Fe, suggesting C, O, and Si in the form of dust and ices dominate the opacity of primordial protostellar disks and their propensity for fragmentation via gravitational stability. Near \feh{} = 0 dex, the bias-corrected close binary fraction ($a<10$ au) decreases from $\approx$ 100 per cent at \alh{} = $-$0.2 dex to $\approx$ 15 per cent near \alh{} = 0.08 dex, with a suggestive turn-up to $\approx$20 per cent near \alh{} = 0.2. We conclude that the relationship between stellar multiplicity and chemical composition for sun-like dwarf stars in the field of the Milky Way is complex, and that this complexity should be accounted for in future studies of interacting binaries., Comment: 15 pages, 10 figures, plus appendices; accepted to MNRAS

Differentially Private Stochastic Gradient Descent (DP-SGD) forms a fundamental building block in many applications for learning over sensitive data. Two standard approaches, privacy amplification by subsampling, and privacy amplification by shuffling, permit adding lower noise in DP-SGD than via na\"{\i}ve schemes. A key assumption in both these approaches is that the elements in the data set can be uniformly sampled, or be uniformly permuted -- constraints that may become prohibitive when the data is processed in a decentralized or distributed fashion. In this paper, we focus on conducting iterative methods like DP-SGD in the setting of federated learning (FL) wherein the data is distributed among many devices (clients). Our main contribution is the \emph{random check-in} distributed protocol, which crucially relies only on randomized participation decisions made locally and independently by each client. It has privacy/accuracy trade-offs similar to privacy amplification by subsampling/shuffling. However, our method does not require server-initiated communication, or even knowledge of the population size. To our knowledge, this is the first privacy amplification tailored for a distributed learning framework, and it may have broader applicability beyond FL. Along the way, we extend privacy amplification by shuffling to incorporate $(\epsilon,\delta)$-DP local randomizers, and exponentially improve its guarantees. In practical regimes, this improvement allows for similar privacy and utility using data from an order of magnitude fewer users., Comment: Updated proof for $(\epsilon_0, \delta_0)$-DP local randomizers

This work introduces a neuro-symbolic agent that combines deep reinforcement learning (DRL) with temporal logic (TL) to achieve systematic zero-shot, i.e., never-seen-before, generalisation of formally specified instructions. In particular, we present a neuro-symbolic framework where a symbolic module transforms TL specifications into a form that helps the training of a DRL agent targeting generalisation, while a neural module learns systematically to solve the given tasks. We study the emergence of systematic learning in different settings and find that the architecture of the convolutional layers is key when generalising to new instructions. We also provide evidence that systematic learning can emerge with abstract operators such as negation when learning from a few training examples, which previous research have struggled with.

In this article, we propose a new variational approach to learn private and/or fair representations. This approach is based on the Lagrangians of a new formulation of the privacy and fairness optimization problems that we propose. In this formulation, we aim to generate representations of the data that keep a prescribed level of the relevant information that is not shared by the private or sensitive data, while minimizing the remaining information they keep. The proposed approach (i) exhibits the similarities of the privacy and fairness problems, (ii) allows us to control the trade-off between utility and privacy or fairness through the Lagrange multiplier parameter, and (iii) can be comfortably incorporated to common representation learning algorithms such as the VAE, the $\beta$-VAE, the VIB, or the nonlinear IB., Comment: Accepted at the ITW 2021 conference. Previously presented at the PPAI-21 workshop from the AAAI-21 conference. Content distribution: 5 pages of main content + 2 pages of references + 11 pages of supplementary material

In the context of the global energy ecosystem transformation, we introduce a new approach to reduce the carbon emissions of the cloud-computing sector and, at the same time, foster the deployment of small-scale private photovoltaic plants. We consider the opportunity cost of moving some cloud services to private, distributed, solar-powered computing facilities. To this end, we compare the potential revenue of leasing computing resources to a cloud pool with the revenue obtained by selling the surplus energy to the grid. We first estimate the consumption of virtualized cloud computing instances, establishing a metric of computational efficiency per nominal photovoltaic power installed. Based on this metric and characterizing the site's annual solar production, we estimate the total return and payback. The results show that the model is economically viable and technically feasible. We finally depict the still many questions open, such as security, and the fundamental barriers to address, mainly related with a cloud model ruled by a few big players.

The stellar velocity distribution function (DF) in the solar vicinity is re-examined using data from the SDSS APOGEE survey's DR16 and \emph{Gaia} DR2. By exploiting APOGEE's ability to chemically discriminate with great reliability the thin disk, thick disk and (accreted) halo populations, we can, for the first time, derive the three-dimensional velocity DFs for these chemically-separated populations. We employ this smaller, but more data-rich APOGEE+{\it Gaia} sample to build a \emph{data-driven model} of the local stellar population velocity DFs, and use these as basis vectors for assessing the relative density proportions of these populations over 5 $<$ $R$ $<$ 12 kpc, and $-1.5$ $<$ $z$ $<$ 2.5 kpc range as derived from the larger, more complete (i.e., all-sky, magnitude-limited) {\it Gaia} database. We find that 81.9 $\pm$ 3.1$\%$ of the objects in the selected \emph{Gaia} data-set are thin-disk stars, 16.6 $\pm$ 3.2$\%$ are thick-disk stars, and 1.5 $\pm$ 0.1$\%$ belong to the Milky Way stellar halo. We also find the local thick-to-thin-disk density normalization to be $\rho_{T}(R_{\odot})$/$\rho_{t}(R_{\odot})$ = 2.1 $\pm$ 0.2$\%$, a result consistent with, but determined in a completely different way than, typical starcount/density analyses. Using the same methodology, the local halo-to-disk density normalization is found to be $\rho_{H}(R_{\odot})$/($\rho_{T}(R_{\odot})$ + $\rho_{t}(R_{\odot})$) = 1.2 $\pm$ 0.6$\%$, a value that may be inflated due to chemical overlap of halo and metal-weak thick disk stars., Comment: 15 pages, 12 figures, accepted for publication in ApJ

Photon shot noise, arising from the quantum-mechanical nature of the light, currently limits the sensitivity of all the gravitational wave observatories at frequencies above one kilohertz. We report a successful application of squeezed vacuum states of light at the GEO\,600 observatory and demonstrate for the first time a reduction of quantum noise up to $6.03 \pm 0.02$ dB in a kilometer-scale interferometer. This is equivalent at high frequencies to increasing the laser power circulating in the interferometer by a factor of four. Achieving this milestone, a key goal for the upgrades of the advanced detectors, required a better understanding of the noise sources and losses, and implementation of robust control schemes to mitigate their contributions. In particular, we address the optical losses from beam propagation, phase noise from the squeezing ellipse, and backscattered light from the squeezed light source. The expertise gained from this work carried out at GEO 600 provides insight towards the implementation of 10 dB of squeezing envisioned for third-generation gravitational wave detectors.

In this work, we study the generalization capability of algorithms from an information-theoretic perspective. It has been shown that the expected generalization error of an algorithm is bounded from above by a function of the relative entropy between the conditional probability distribution of the algorithm's output hypothesis, given the dataset with which it was trained, and its marginal probability distribution. We build upon this fact and introduce a mathematical formulation to obtain upper bounds on this relative entropy. Assuming that the data is discrete, we then develop a strategy using this formulation, based on the method of types and typicality, to find explicit upper bounds on the generalization error of stable algorithms, i.e., algorithms that produce similar output hypotheses given similar input datasets. In particular, we show the bounds obtained with this strategy for the case of $\epsilon$-DP and $\mu$-GDP algorithms., Comment: 18 pages (double column), 4 figures, accepted at IEEE Transactions on Information Theory

In this extended abstract, we propose a tuning approach for nonlinear mechanical systems to modify the behavior of the closed-loop system, where we are particularly interested in attenuating oscillations from the transient response. Towards this end, we inject damping into the system, and we provide two tuning methods to select the gains that are appropriate for our purposes. Furthermore, we apply these tuning rules to a 2DoF planar manipulator and present its simulation results., Comment: This work is accepted as a Late-Breaking result at the IFAC World Congress 2020 Version 2 changes: -Corrected minor typos -Corrected equation 21 and 27

The development of synthetic strategies to engineer \pi-conjugated polymers is of paramount importance in modern chemistry and materials science. Here we introduce a theoretical and experimental synthetic paradigm based on the search for specific vibrational modes through an appropriate tailoring of the \pi-conjugation of the precursors, in order to increase the attempt frequency of a chemical reaction. First, we on-surface design a 1D \pi-conjugated polymer with specific \pi-topology, which is based on bisanthene monomers linked by cumulene bridges that tune specific vibrational modes. In a second step, upon further annealing, such vibrational modes steer the two-fold cyclization reaction between adjacent bisanthene moieties, which gives rise to a long and free-defect pentalene-bridged conjugated ladder polymer featuring a low band gap. In addition, high resolution atomic force microscopy allows us to identify by atomistic insights the resonant form of the polymer, thus confirming the validity of the Glidewell and Lloyd's rules for aromaticity. This on-surface synthetic strategy may stimulate exploiting previously precluded reactions towards novel pi-conjugated polymers with specific structures and properties.

The detection of binary neutron star mergers represents one of the most important astrophysical discoveries of the recent years. Due to the extreme matter and gravity conditions and the rich dynamics developed, it becomes a tremendous challenge to accurately simulate numerically all the scales present during the collision. Here we present how to study such systems by using large eddy simulations with a self-consistent subgrid-scale gradient model, that we generalized to the special relativistic case in a previous work and now extend to the general relativistic case. Adapted from nonrelativistic scenarios, the so-called gradient model allows to capture part of the effects of the hidden dynamics on the resolved scales, by means of a physically-agnostic, mathematically-based Taylor expansion of the nonlinear terms in the conservative evolution equations' fluxes. We assess the validity of this approach in bounding-box simulations of the magnetic Kelvin-Helmholtz instability. Several resolutions and a broad range of scenarios are considered in order to carefully test the performance of the model under three crucial aspects: (i) highly curved backgrounds, (ii) jumps on the fluid density profiles and (iii) strong shocks. The results suggest our extension of the gradient subgrid-scale model to general relativistic magnetohydrodynamics is a promising approach for studying binary neutron stars mergers, and potentially to other relevant astrophysical scenarios., Comment: 15 pages, 7 figures Submitted to PRD