Search

For many equation-theoretical questions about modular lattices, Hall and Dilworth give a useful construction: Let $L_0$ be a lattice with largest element $u_0$, $L_1$ be a lattice disjoint from $L_0$ with smallest element $v_1$, and $a \in L_0$, $b \in L_1$ such that the intervals $[a, u_0]$ and $[v_1, b]$ are isomorphic. Then, after identifying those intervals you obtain $L_0 \cup L_1$, a lattice structure whose partial order is the transitive relation generated by the partial orders of $L_0$ and $L_1$. It is modular if $L_0$ and $L_1$ are modular. Since in this construction the index set $\{0, 1\}$ is essentially a chain, this work presents a method -- termed S-glued -- whereby a general family $L_x\ (x \in S)$ of lattices can specify a lattice with the small-scale lattice structure determined by the $L_x$ and the large-scale structure determined by $S$. A crucial application is representing finite-length modular lattices using projective geometries., Comment: 16 pages, 3 figures. German original by Christian Herrmann, English translation by Dale R. Worley. Added a footnote 1 to give more detail about Dilworth's work on the subject

Logics closed under classes of substitutions broader than class of uniform substitutions are known as hyperformal logics. This paper extends known results about hyperformal logics in two ways. First: we examine a very powerful form of hyperformalism that tracks, for bunched natural deduction systems, essentially all the intensional content that can possibly be tracked. We demonstrate that, after a few tweaks, the well-known relevant logic $\mathbf{B}$ exhibits this form of hyperformalism. Second: we demonstrate that not only can hyperformalism be extended along these lines, it can also be extended to accommodate not just what is proved in a given logic but the proofs themselves. Altogether, the paper demonstrates that the space of possibilities for the study of hyperformalism is much larger than might have been expected.

The Milky Way is a mosaic of stars from different origins. In particular, metal-poor accreted star candidates offer a unique opportunity to better understand the accretion history of the Milky Way. In this work, we aim to explore the assembly history of the Milky Way by investigating accreted stars in terms of their ages, dynamical properties, and chemical abundances. We also aim to better characterize the impact of incorporating asteroseismic information on age and chemical abundance calculations of metal-poor accreted stars for which TESS data is available. In this study, we conducted an in-depth examination of 30 metal-poor accreted star candidates, using TESS and Gaia data, as well as MIKE spectra. We find satisfactory agreement between seismic and predicted/spectroscopic surface gravity (log g) values, demonstrating the reliability of spectroscopic data from our methodology. We found that while age determination is highly dependent on the log g and asteroseismic information used, the overall chemical abundance distributions are similar for different log g. However, we found that calcium (Ca) abundances are more sensitive to the adopted log g. Our study reveals that the majority of our stars have properties compatible to those reported for the Gaia-Sausage-Enceladus, with a minority of stars that might be associated to Splash. We found an age distribution with a median of 11.3 Gyr with lower and upper uncertainties of 4.1 and 1.3 Gyr respectively when including asteroseismic information. As regarding some key chemical signatures we note that these stars are metal-poor ([Fe/H]) < -0.8), alpha-rich ([alpha]/Fe] > 0.2), copper-poor ([Cu/Fe] < 0 ) and with chemical abundances typical of accreted stars. These findings illustrate the importance of multi-dimensional analyses in unraveling the complex accretion history of the Milky Way., Comment: Accepted for publication in A&A

(abridged) The third data release of Gaia, has provided stellar parameters, metallicity [M/H], [{\alpha}/Fe], individual abundances, broadening parameter from its RVS spectra for about 5.6 million objects thanks to the GSP-Spec module. The catalogue publishes the radial velocity of 33 million sources. We took advantage of the intersections between Gaia RVS and Gaia-ESO to compare their stellar parameters, abundances and radial and rotational velocities. We aimed at verifying the overall agreement between the two datasets, considering the various calibrations and the quality-control flag system suggested for the Gaia GSP-Spec parameters. For the targets in common between Gaia RVS and Gaia-ESO, we performed several statistical checks on the distributions of their stellar parameters, abundances and velocities of targets in common. For the Gaia surface gravity and metallicity we considered both the uncalibrated and calibrated values. We find an excellent agreement between the Gaia and Gaia-ESO radial velocities given the uncertainties affecting each dataset. Less than 25 of ~2100 Gaia-ESO spectroscopic binaries are flagged as non-single stars by Gaia. The temperature scales are in good agreement. The calibrated GSP-Spec gravity should be preferred. We note that the quality (accuracy, precision) of the GSP-Spec parameters degrades quickly for objects fainter than G~11. We find that the somewhat imprecise GSP-Spec abundances due to its medium-resolution spectroscopy over a short wavelength window and the faint G regime of the sample under study can be counterbalanced by working with averaged quantities. We studied some properties of the open-cluster population: our combined sample traces very well the radial [Fe/H] and [Ca/Fe] gradients, the age-metallicity relations in different radial regions, and it places the clusters in the thin disc., Comment: Accepted for publication in Astronomy & Astrophysics

The BlueWalker 3 satellite is now fainter than during the first months after deployment. The greatest improvement is that the average maximum luminosity near zenith has been reduced from magnitude 1.0 to 2.2. However, the spacecraft is still usually bright enough to interfere with astronomical research.

Observations of Starlink V2 Mini satellites during orbit-raising suggest that SpaceX applies brightness mitigation when they reach a height of 357 km. The mean apparent magnitudes for objects below that height threshold is 2.68 while the mean for those above is 6.46. When magnitudes are adjusted to a uniform distance of 1000 km the means are 4.58 and 7.52, respectively. The difference of 2.94 between distance-adjusted magnitudes above and below threshold implies that mitigation is 93% effective in reducing the brightness of orbit-raising spacecraft. Orbit-raising Mini spacecraft have a smaller impact on astronomical observations than higher altitude on-station spacecraft because they are relatively few in number. They also spend less time traversing the sky and spend longer in the Earth's shadow. These low-altitude objects will be more out-of-focus in large telescopes such as the LSST which reduces their impact, too. However, they attract considerable public attention and airline pilots have reported them as Unidentified Aerial Phenomena.

The Li abundance observed in pre-main sequence and main sequence late-type stars is strongly age-dependent, but also shows a complex pattern depending on several parameters, such as rotation, chromospheric activity and metallicity. The best way to calibrate these effects, with the aim of studying Li as an age indicator for FGK stars, is to calibrate coeval groups of stars, such as open clusters (OCs) and associations. We present a considerable target sample of 42 OCs and associations, ranging from 1 Myr to 5 Gyr, observed within the Gaia-ESO survey (GES), and using the latest data provided by GES iDR6 and the most recent release of Gaia that was then available, EDR3. As part of this study, we update and improve the membership analysis for all 20 OCs presented in our previous article. We perform detailed membership analyses for all target clusters to identify likely candidates, using all available parameters provided by GES and based on numerous criteria: from radial velocity distributions, to the astrometry and photometry provided by Gaia, to gravity indicators, [Fe/H] metallicity, and Li content. We obtain updated lists of cluster members for the whole target sample, as well as a selection of Li-rich giant contaminants obtained as an additional result of the membership process. Each selection of cluster candidates was thoroughly contrasted with numerous existing membership studies using data from Gaia to ensure the most robust results. These final cluster selections will be used in the third and last paper of this series, which reports the results of a comparative study characterising the observable Li dispersion in each cluster and analysing its dependence on several parameters, allowing us to calibrate a Li-age relation and obtain a series of empirical Li envelopes for key ages in our sample., Comment: 83 pages, 246 figures. To be published in A&A, accepted January 27, 2024

Motivated by recent developments regarding the product structure of planar graphs, we study relationships between treewidth, grid minors, and graph products. We show that the Cartesian product of any two connected $n$-vertex graphs contains an $\Omega(\sqrt{n})\times\Omega(\sqrt{n})$ grid minor. This result is tight: The lexicographic product (which includes the Cartesian product as a subgraph) of a star and any $n$-vertex tree has no $\omega(\sqrt{n})\times\omega(\sqrt{n})$ grid minor.

This is a survey of characterizations and relationships between some properties of lattices, particularly the modular, Arguesian, linear, and distributive properties, but also some other related properties. The survey emphasizes finite and finitary lattices and deemphasizes complemented lattices., Comment: 31 pages

The Gaia-ESO Survey is an European Southern Observatory (ESO) public spectroscopic survey that targeted $10^5$ stars in the Milky Way covering the major populations of the disk, bulge and halo. The observations were made using FLAMES on the VLT obtaining both UVES high ($R\sim47,000$) and GIRAFFE medium ($R\sim20,000$) resolution spectra. The analysis of the Gaia-ESO spectra was the work of multiple analysis teams (nodes) within five working groups (WG). The homogenisation of the stellar parameters within WG11 (high resolution observations of FGK stars) and the homogenisation of the stellar parameters within WG10 (medium resolution observations of FGK stars) is described here. In both cases, the homogenisation was carried out using a bayesian Inference method developed specifically for the Gaia-ESO Survey by WG11. The WG10 homogenisation primarily used the cross-match of stars with WG11 as the reference set in both the stellar parameter and chemical abundance homogenisation. In this way the WG10 homogenised results have been placed directly onto the WG11 stellar parameter and chemical abundance scales. The reference set for the metal-poor end was sparse which limited the effectiveness of the homogenisation in that regime. For WG11, the total number of stars for which stellar parameters were derived was 6,231 with typical uncertainties for Teff, log g and [Fe/H] of 32~K, 0.05 and 0.05 respectively. One or more chemical abundances out of a possible 39 elements were derived for 6,188 of the stars. For WG10, the total number of stars for which stellar parameters were derived was 76,675 with typical uncertainties for Teff, log g and [Fe/H] of 64~K, 0.15 and 0.07 respectively. One or more chemical abundances out of a possible 30 elements were derived for 64,177 of the stars., Comment: 31 pages, 19 figures

The illumination phase functions for Starlink Mini satellites are determined for times of twilight and darkness. Those functions are then evaluated to give apparent magnitudes over a grid of points across the sky and over a range of solar angles below the horizon. Sky maps and a table of satellite magnitude distributions are presented. The largest areas of sky with satellites brighter than magnitudes 6 and 7 both occur during twilight. Brightness surges, known as flares, are also characterized.

The High-Resolution Multi-Object Spectrograph (HRMOS) is a facility instrument that we plan to propose for the Very Large Telescope (VLT) of the European Southern Observatory (ESO), following the initial presentation at the VLT 2030 workshop held at ESO in June 2019. HRMOS provides a combination of capabilities that are essential to carry out breakthrough science across a broad range of active research areas from stellar astrophysics and exoplanet studies to Galactic and Local Group archaeology. HRMOS fills a gap in capabilities amongst the landscape of future instrumentation planned for the next decade. The key characteristics of HRMOS will be high spectral resolution (R = 60000 - 80000) combined with multi-object (20-100) capabilities and long term stability that will provide excellent radial velocity precision and accuracy (10m/s). Initial designs predict that a SNR~100 will be achievable in about one hour for a star with mag(AB) = 15, while with the same exposure time a SNR~ 30 will be reached for a star with mag(AB) = 17. The combination of high resolution and multiplexing with wavelength coverage extending to relatively blue wavelengths (down to 380\,nm), makes HRMOS a spectrograph that will push the boundaries of our knowledge and that is envisioned as a workhorse instrument in the future. The science cases presented in this White Paper include topics and ideas developed by the Core Science Team with the contributions from the astronomical community, also through the wide participation in the first HRMOS Workshop (https://indico.ict.inaf.it/event/1547/) that took place in Firenze (Italy) in October 2021., Comment: 88 pages, 39 figures. Comments and expressions of interest are welcome by contacting members of the Core Science Team

The Gaia-ESO Survey (GES) is a large public spectroscopic survey which acquired spectra for more than 100000 stars across all major components of the Milky Way. In addition to atmospheric parameters and stellar abundances that have been derived in previous papers of this series, the GES spectra allow us to detect spectroscopic binaries with one (SB1), two (SB2) or more (SBn $\ge$ 3) components. Cross-correlation functions (CCFs) have been re-computed thanks to a dozen spectral masks probing a range of effective temperatures, surface gravities and metallicities. By optimising the mask choice for a given spectrum, the new computed so-called Nacre (Narrow cross-correlation experiment) CCFs are narrower and allow to unblend more stellar components than standard masks. The Doe (Detection of Extrema) extremum-finding code then selects the individual components and provides their radial velocities. From the sample of HR10 and HR21 spectra corresponding to 37565 objects, the present study leads to the detection of 322 SB2, ten (three of them being tentative) SB3, and two tentative SB4. In particular, compared to our previous study, the Nacre CCFs allow us to multiply the number of SB2 candidates by $\approx$ 1.5. The colour-magnitude diagram reveals, as expected, the shifted location of the SB2 main sequence. A comparison between the SB identified in Gaia DR3 and the ones detected in the present work is performed and the complementarity of the two censuses is discussed. An application to mass-ratio determination is presented, and the mass-ratio distribution of the GES SB2 is discussed. When accounting for the SB2 detection rate, an SB2 frequency of $\approx$ 1.4% is derived within the present stellar sample of mainly FGK-type stars. As primary outliers identified within the GES data, SBn spectra produce a wealth of information and useful constraints for the binary population synthesis studies.

We present the first large-scale 3D kinematic study of ~2000 spectroscopically-confirmed young stars (<20 Myr) in 18 star clusters and OB associations (hereafter groups) from the combination of Gaia astrometry and Gaia-ESO Survey spectroscopy. We measure 3D velocity dispersions for all groups, which range from 0.61 to 7.4 km/s (1D velocity dispersions of 0.35 to 4.3 km/s). We find the majority of groups have anisotropic velocity dispersions, suggesting they are not dynamically relaxed. From the 3D velocity dispersions, measured radii and estimates of total mass we estimate the virial state and find that all systems are super-virial when only the stellar mass is considered, but that some systems are sub-virial when the mass of the molecular cloud is taken into account. We observe an approximately linear correlation between the 3D velocity dispersion and the group mass, which would imply that the virial state of groups scales as the square root of the group mass. However, we do not observe a strong correlation between virial state and group mass. In agreement with their virial state we find that nearly all of the groups studied are in the process of expanding and that the expansion is anisotropic, implying that groups were not spherical prior to expansion. One group, Rho Oph, is found to be contracting and in a sub-virial state (when the mass of the surrounding molecular cloud is considered). This work provides a glimpse of the potential of the combination of Gaia and data from the next generation of spectroscopic surveys., Comment: 26 pages, 15 figures, accepted for publication in MNRAS

Context. Large spectroscopic surveys devoted to the study of the Milky Way, including Gaia, use automated pipelines to massively determine the atmospheric parameters of millions of stars. The Gaia FGK Benchmark Stars are reference stars with Teff and log g derived through fundamental relations, independently of spectroscopy, to be used as anchors for the parameter scale. The first and second versions of the sample have been extensively used for that purpose, and more generally to help constrain stellar models. Aims. We provide the third version of the Gaia FGK Benchmark Stars, an extended set intended to improve the calibration of spectroscopic surveys, and their interconnection. Methods. We have compiled about 200 candidates which have precise measurements of angular diameters and parallaxes. We determined their bolometric fluxes by fitting their spectral energy distribution. Masses were determined using two sets of stellar evolution models. In a companion paper we describe the determination of metallicities and detailed abundances. Results. We provide a new set of 192 Gaia FGK Benchmark Stars with their fundamental Teff and logg, and with uncertainties lower than 2% for most stars. Compared to the previous versions, the homogeneity and accuracy of the fundamental parameters are significantly improved thanks to the high quality of the Gaia data reflecting on distances and bolometric fluxes., Comment: accepted in A&A

Photometric characteristics for all models of Starlink satellites launched to date are reviewed. The Original design that lacked brightness mitigation is the most luminous. SpaceX installed a sunshade on the VisorSat model which reduced its luminosity by a factor of 3. The visor was omitted on Post-VisorSat spacecraft with laser communication which followed, but the company added a reflective layer which resulted in an intermediate brightness between Original and VisorSat. SpaceX is applying advanced brightness mitigation techniques to their Generation 2 Starlink satellites which are larger. The first of these, called Minis, are dimmer than Gen 1 Starlinks despite their greater size. Photometric observations verify that brightness mitigation efforts employed by SpaceX reduce spacecraft luminosity substantially. However, the satellites still have some negative impact on astronomical observations and the very large satellites planned for later in Gen 2 may interfere more seriously.

Many algorithms for inserting elements into tableaux are known, starting with the Robinson-Schensted algorithm. Much of those processes can be incorporated into the general framework of Fomin's "growth diagrams". Even for single types of tableaux, there are various alternative insertion algorithms and, due to the varying ways they are described, the relationships between the algorithms can be obscure. The distinguishing features of many algorithms can be codified into graphic "insertion diagrams" which make important aspects of the algorithms immediately apparent. We use insertion diagrams to build a graphic catalog or picture book of many of the tableau insertion algorithms in the literature., Comment: 31 pages, 45 figures. 2nd version includes supporting Python code and corrects minor errors

Starlink Generation 2 Mini satellites are fainter than Gen 1 spacecraft despite their larger size. The mean of apparent magnitudes for satellites in brightness mitigation mode is 7.06 +/- 0.10. When these magnitudes are adjusted to a uniform distance of 1,000 km that mean is 7.87 +/- 0.09. The brightness mitigation mode reduces distance-adjusted satellite luminosity by a factor of 12 relative to spacecraft that are not mitigated.

We present an empirical model of age-dependent photospheric lithium depletion, calibrated using a large, homogeneously-analysed sample of 6200 stars in 52 open clusters, with ages from 2--6000 Myr and $-0.3<{\rm [Fe/H}]<0.2$, observed in the Gaia-ESO spectroscopic survey. The model is used to obtain age estimates and posterior age probability distributions from measurements of the Li I 6708A equivalent width for individual (pre) main sequence stars with $3000 < T_{\rm eff}/{\rm K} <6500$, a domain where age determination from the HR diagram is either insensitive or highly model-dependent. In the best cases, precisions of 0.1 dex in log age are achievable; even higher precision can be obtained for coeval groups and associations where the individual age probabilities of their members can be combined. The method is validated on a sample of exoplanet-hosting young stars, finding agreement with claimed young ages for some, but not others. We obtain better than 10 per cent precision in age, and excellent agreement with published ages, for seven well-studied young moving groups. The derived ages for young clusters ($<1$ Gyr) in our sample are also in good agreement with their training ages, and consistent with several published, model-insensitive lithium depletion boundary ages. For older clusters there remain systematic age errors that could be as large as a factor of two. There is no evidence to link these errors to any strong systematic metallicity dependence of (pre) main sequence lithium depletion, at least in the range $-0.29 < {\rm [Fe/H]} < 0.18$. Our methods and model are provided as software -- "Empirical AGes from Lithium Equivalent widthS" (EAGLES)., Comment: Accepted for publication in Monthly Notices of the Royal Astronomical Society. Code available at https://github.com/robdjeff/eagles

The Gaia-ESO Survey is a public spectroscopic survey that has targeted $\gtrsim10^5$ stars covering all major components of the Milky Way from the end of 2011 to 2018, delivering its public final release in May 2022. Unlike other spectroscopic surveys, Gaia-ESO is the only survey that observed stars across all spectral types with dedicated, specialised analyses: from O ($T_\mathrm{eff} \sim 30,000-52,000$~K) all the way to K-M ($\gtrsim$3,500~K). The physics throughout these stellar regimes varies significantly, which has previously prohibited any detailed comparisons between stars of significantly different type. In the final data release (internal data release 6) of the Gaia-ESO Survey, we provide the final database containing a large number of products such as radial velocities, stellar parameters and elemental abundances, rotational velocity, and also, e.g., activity and accretion indicators in young stars and membership probability in star clusters for more than 114,000 stars. The spectral analysis is coordinated by a number of Working Groups (WGs) within the Survey, which specialise in the various stellar samples. Common targets are analysed across WGs to allow for comparisons (and calibrations) amongst instrumental setups and spectral types. Here we describe the procedures employed to ensure all Survey results are placed on a common scale to arrive at a single set of recommended results for all Survey collaborators to use. We also present some general quality and consistency checks performed over all Survey results., Comment: A&A accepted, minor revision, 36 pages, 38 figures

Birkhoff's representation theorem for finite distributive lattices states that any finite distributive lattice is isomorphic to the lattice of order ideals (lower sets) of the partial order of the join-irreducible elements of the lattice. This theorem can be extended as follows: A non-finite distributive lattice that is locally finite and has a $\hat{0}$ is isomorphic to the lattice of finite order ideals of the partial order of the join-irreducible elements of the lattice. In addition, certain ``well ordering'' properties are shown to be equivalent to the premises of the extended theorem., Comment: The main result of this paper has been previously published; this version adds a reference to that previous publication