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In planar analytic vector fields, a monodromic singularity can be distinguished between a focus or a center by means of the Lyapunov coefficients, which are given in terms of the power series coefficients of the first-return map defined around the singularity. In this paper, we are interested in an analogous problem for monodromic tangential singularities of piecewise analytic vector fields $Z=(Z^+ ,Z^-)$. First, we prove that the first-return map, defined in a neighborhood of a monodromic tangential singularity, is analytic, which allows the definition of the Lyapunov coefficients. Then, as a consequence of a general property for pair of involutions, we obtain that the index of the first non-vanishing Lyapunov coefficient is always even. In addition, a general recursive formula together with a Mathematica algorithm for computing the Lyapunov coefficients is obtained. We also provide results regarding limit cycles bifurcating from monodromic tangential singularities. Several examples are analyzed.

Since thin disc stars are younger than thick disc stars on average, the thin disc is predicted by some models to start forming after the thick disc had formed, around 10 Gyr ago. Accordingly, no significant old thin disc population should exist. Using 6-D coordinates from Gaia-DR2 and age estimates from Sanders & Das (2018), we select $\sim 24000$ old stars (${\tau > 10}$ Gyr, with uncertainties $\lesssim 15\%$) within 2 kpc from the Sun (full sample). A cross-match with APOGEE-DR16 ($\sim 1000$ stars) reveals comparable fractions of old chemically defined thin/thick disc stars. We show that the full sample pericenter radius ($r_\mathrm{per}$) distribution has three peaks, one associated with the stellar halo and the other two having contributions from the thin/thick discs. Using a high-resolution $N$-body+Smooth Particle Hydrodynamics simulation, we demonstrate that one peak, at $r_\mathrm{per}\approx 7.1$ kpc, is produced by stars from both discs which were born in the inner Galaxy and migrated to the Solar Neighbourhood. In the Solar Neighbourhood, $\sim 1/2$ ($\sim 1/3$) of the old thin (thick) disc stars are classified as migrators. Our results suggest that thin/thick discs are affected differently by radial migration inasmuch as they have different eccentricity distributions, regardless of vertical scale heights. We interpret the existence of a significant old thin disc population as evidence for an early co-formation of thin/thick discs, arguing that clump instabilities in the early disc offer a compelling explanation for the observed trends., Comment: 12+2 pages, 10 figures, minor changes to match published version

A wave energy converter (WEC) similar to the CETO system developed by Carnegie Clean Energy is considered for design optimisation. This WEC is able to absorb power from heave, surge and pitch motion modes, making the optimisation problem nontrivial. The WEC dynamics is simulated using the spectral-domain model taking into account hydrodynamic forces, viscous drag, and power take-off forces. The design parameters for optimisation include the buoy radius, buoy height, tether inclination angles, and control variables (damping and stiffness). The WEC design is optimised for the wave climate at Albany test site in Western Australia considering unidirectional irregular waves. Two objective functions are considered: (i) maximisation of the annual average power output, and (ii) minimisation of the levelised cost of energy (LCoE) for a given sea site. The LCoE calculation is approximated as a ratio of the produced energy to the significant mass of the system that includes the mass of the buoy and anchor system. Six different heuristic optimisation methods are applied in order to evaluate and compare the performance of the best known evolutionary algorithms, a swarm intelligence technique and a numerical optimisation approach. The results demonstrate that if we are interested in maximising energy production without taking into account the cost of manufacturing such a system, the buoy should be built as large as possible (20 m radius and 30 m height). However, if we want the system that produces cheap energy, then the radius of the buoy should be approximately 11-14~m while the height should be as low as possible. These results coincide with the overall design that Carnegie Clean Energy has selected for its CETO 6 multi-moored unit. However, it should be noted that this study is not informed by them, so this can be seen as an independent validation of the design choices.

The Milky Way's progenitor experienced several merger events which left their imprints on the stellar halo, including the Gaia-Sausage/Enceladus. Recently, it has been proposed that this event perturbed the proto-disk and gave rise to a metal rich ([Fe/H] $>-1$) low angular momentum ($v_{\phi} < 100$ km/s) stellar population. These stars have dynamical and chemical properties different from the accreted stellar halo, but are continuous with the canonical thick disk. In this letter, we use a hydrodynamical simulation of an isolated galaxy which develops clumps that produce a bimodal thin$+$thick disk chemistry to explore whether it forms such a population. We demonstrate clump scattering forms a metal-rich, low angular momentum population, without the need for a major merger. We show that, in the simulation, these stars have chemistry, kinematics and density distribution in good agreement with those in the Milky Way., Comment: Accepted for publication in ApJL

A scenario for the formation of the bi-modality in the chemical space [$\alpha$/Fe] vs [Fe/H] of the Milky Way was recently proposed in which $\alpha$-enhanced stars are produced early and quickly in clumps. Besides accelerating the enrichment of the medium with $\alpha$-elements, these clumps scatter the old stars, converting in-plane to vertical motion, forming a geometric thick disc. In this paper, by means of a detailed analysis of the data from smooth particle hydrodynamical simulations, we investigate the geometric properties (in particular of the chemical thick disc) produced in this scenario. For mono-age populations we show that the surface radial density profiles of high-[$\alpha$/Fe] stars are well described by single exponentials, while that of low-[$\alpha$/Fe] stars require broken exponentials. This break is sharp for young populations and broadens for older ones. The position of the break does not depend significantly on age. The vertical density profiles of mono-age populations are characterized by single exponentials, which flare significantly for low-[$\alpha$/Fe] stars but only weakly (or not at all) for high-[$\alpha$/Fe] stars. For low-[$\alpha$/Fe] stars, the flaring level decreases with age, while for high-[$\alpha$/Fe] stars it weakly increases with age (although with large uncertainties). All these properties are in agreement with observational results recently reported for the Milky Way, making this a plausible scenario for the formation of the Galactic thick disc., Comment: 11 pages, 12 figures; Updated to published (MNRAS) version - inclusion of extra references and minor changes suggested by the referee

We introduce the study of box/peanut (B/P) bulges in the action space of the initial axisymmetric system. We explore where populations with different actions end up once a bar forms and a B/P bulge develops. We find that the density bimodality due to the B/P bulge (the X-shape) is better traced by populations with low radial, JR,0, or vertical, Jz,0, actions, or high azimuthal action, J{\phi},0. Generally populations separated by JR,0 have a greater variation in bar strength and vertical heating than those separated by Jz,0. While the bar substantially weakens the initial vertical gradient of Jz,0, it also drives a strikingly monotonic vertical profile of JR,0. We then use these results to guide us in assigning metallicity to star particles in a pure N-body model. Because stellar metallicity in unbarred galaxies depends on age as well as radial and vertical positions, the initial actions are particularly well suited for assigning metallicities. We argue that assigning metallicities based on single actions, or on positions, results in metallicity distributions inconsistent with those observed in real galaxies. We therefore use all three actions to assign metallicity to an N-body model by comparing with the actions of a star-forming, unbarred simulation. The resulting metallicity distribution is pinched on the vertical axis, has a realistic vertical gradient and has a stronger X-shape in metal-rich populations, as found in real galaxies., Comment: Submitted to MNRAS. Comments welcome. The appendix of figures for the simulation suite will be available, until journal publication, at http://www.star.uclan.ac.uk/~vpd/appendix.pdf (Caution: 57Mb)

Smart metering is an essential feature of smart grids, allowing residential customers to monitor and reduce electricity costs. Devices called smart meters allows residential customers to monitor and reduce electricity costs, promoting energy saving, demand management, and energy efficiency. However, monitoring a households' energy consumption through smart meters poses serious privacy threats, and have thus become a major privacy issue. Hence, a significant amount of research has appeared recently with the purpose of providing methods and mechanisms to reconcile smart metering technologies and privacy requirements. However, most current approaches fall short in meeting one of several of the requirements for privacy preserving smart metering systems. In this paper we show how Intel SGX technology can be used to provide a simple and general solution for the smart metering privacy problem that meets all these requirements in a satisfactory way. Moreover, we present also an implementation of the proposed architecture as well as a series of experiments that have been carried out in order to assess how the proposed solution performs in comparison to a second implementation of the architecture that completely disregards privacy issues.

The time irreversibility and fast relaxation of collapsing $N$-body gravitating systems (as opposed to the time reversibility of the equations of motion for individual stars or particles) are traditionally attributed to information loss due to coarse-graining in the observation. We show that this subjective element is not necessary once one takes into consideration the fundamental fact that these systems are discrete, i.e. composed of a finite number $N$ of stars or particles. We show that a connection can be made between entropy estimates for discrete systems and the Nyquist-Shannon sampling criterion. Specifically, given a sample with $N$ points in a space of $d$ dimensions, the Nyquist-Shannon criterion constrains the size of the smallest structures defined by a function in the continuum that can be uniquely associated with the discrete sample. When applied to an $N$-body system, this theorem sets a lower limit to the size of phase-space structures (in the continuum) that can be resolved in the discrete data. As a consequence, the finite $N$ system tends to a uniform distribution after a relaxation time that typically scales as $N^{1/d}$. This provides an explanation for the fast achievement of a stationary state in collapsing $N$-body gravitating systems such as galaxies and star clusters, without the need to advocate for the subjective effect of coarse-graining., Comment: Accepted for publication in ApJ

We investigate the old problem of the fast relaxation of collisionless $N$-body systems which are collapsing or perturbed, emphasizing the importance of (non-collisional) discreteness effects. We integrate orbit ensembles in fixed external potentials, estimating the entropy of the ensemble to analyze the time evolution of the distribution function. We show that these estimates capture the correct physical behavior expected from the 2nd Law of Thermodynamics, without any spurious entropy production. For self-consistent (i.e. stationary) samples, the entropy is conserved, while for non-self-consistent samples, it increases within a few dynamical times up to a maximum where it stabilizes (even in integrable potentials). Our results shed light on the main ingredients for this fast collisionless relaxation. The fundamental ingredient is the discreteness (finite $N$) of gravitational systems in any potential. Additionally, in non-integrable potentials, the presence of chaotic orbits accelerates the entropy production. Contrary to the traditional violent relaxation scenario, our results indicate that a time-dependent potential is not necessary for this fast relaxation. For the first time, in connection with the Nyquist-Shannon theorem we determine the $N$-dependence of this discreteness-driven relaxation, deriving a typical relaxation time $T/\tau_{cr}\approx 0.1 N^{1/6}$, with slightly weaker $N$-dependencies for non-integrable potentials with substantial fractions of chaotic orbits. This timescale is much smaller than the collisional timescale even for small-$N$ systems such as open clusters and represents an upper limit for real collisionless $N$-body systems. Additionally, our results reinforce the conclusion draw in Beraldo e Silva et al. (2017) that the Vlasov equation does not provide an adequate kinetic description of the fast relaxation of collapsing collisionless $N$-body systems., Comment: Accepted for publication in ApJ