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Period-luminosity relations of long period variables (LPVs) are a powerful tool to map the distances of stars in our galaxy, and are typically calibrated using stars in the Large Magellanic Cloud (LMC). Recent results demonstrated that these relations show a strong dependence on the amplitude of the variability, which can be used to greatly improve distance estimates. However, one of the only highly sampled catalogs of such variables in the LMC is based on OGLE photometry, which does not provide all-sky coverage. Here, we provide the first measurement of the period-luminosity relation of long-period variables in the LMC using photometry from the Asteroid Terrestrial-impact Last Alert System (ATLAS). We derive conversions between ugriz, Gaia, and ATLAS c and o passbands with a precision of approximately 0.02 mag, which enable the measurement of reliable amplitudes with ATLAS for crowded fields. We successfully reproduce the known PL sequences A through E, and show evidence for sequence F using the ratios of amplitudes observed in both ATLAS pass-bands. Our work demonstrates that the ATLAS survey can recover variability in evolved red giants and lays the foundation for an all-sky distance map of the Milky Way using long-period variables., Comment: 13 pages, 8 figures, 2 tables. Submitted to AAS journals, comments welcome

We present the discovery of TOI-7041 b (TIC 201175570 b), a hot Saturn transiting a red giant star with measurable stellar oscillations. We observe solar-like oscillations in TOI-7041 with a frequency of maximum power of $\nu_{\rm max} = 218.50\pm2.23$ $\mu$Hz and a large frequency separation of $\Delta\nu = 16.5282\pm0.0186$ $\mu$Hz. Our asteroseismic analysis indicates that TOI-7041 has a radius of $4.10 \pm 0.06$(stat) $\pm$ 0.05(sys) $R_\odot$, making it one of the largest stars around which a transiting planet has been discovered with the Transiting Exoplanet Survey Satellite (TESS), and the mission's first oscillating red giant with a transiting planet. TOI-7041 b has an orbital period of $9.691 \pm 0.006$ days and a low eccentricity of $e = 0.04 \pm 0.04$. We measure a planet radius of $1.02 \pm 0.03$ $R_J$ with photometry from TESS, and a planet mass of $0.36 \pm 0.16$ $M_J$ ($114 \pm 51$ $M_\oplus$) with ground-based radial velocity measurements. TOI-7041 b appears less inflated than similar systems receiving equivalent incident flux, and its circular orbit indicates that it is not undergoing tidal heating due to circularization. The asteroseismic analysis of the host star provides some of the tightest constraints on stellar properties for a TESS planet host and enables precise characterization of the hot Saturn. This system joins a small number of TESS-discovered exoplanets orbiting stars that exhibit clear stellar oscillations and indicates that extended TESS observations of evolved stars will similarly provide a path to improved exoplanet characterization., Comment: 16 pages, 9 figures, 3 tables

Gravito-inertial asteroseismology saw its birth from the 4-years long light curves of rotating main-sequence stars assembled by the Kepler space telescope. High-precision measurements of internal rotation and mixing are available for about 600 stars of intermediate mass so far that are used to challenge the state-of-the-art stellar structure and evolution models. Our aim is to prepare for future large ensemble modelling of gravity (g)-mode pulsators by relying on a new sample of such stars recently discovered from the third Data Release of the Gaia space mission and confirmed by space photometry from the TESS mission. This sample of potential asteroseismic targets is about 23 times larger than the Kepler sample. We use the effective temperature and luminosity inferred from Gaia to deduce evolutionary masses, convective core masses, radii, and ages for ~14,000 g-mode pulsators classified as such from their nominal TESS light curves. We do so by constructing two dedicated grids of evolutionary models for rotating stars with input physics from the asteroseismic calibrations of Kepler $\gamma$ Dor pulsators. We find the new g-mode pulsators to cover an extended observational instability region covering masses from about 1.3 to 9Msun. We provide their mass-luminosity and mass-radius relations, as well as convective core masses. Our results suggest that oscillations excited by the opacity mechanism occur uninterruptedly for the mass range above about 2Msun, where stars have a radiative envelope aside from thin convection zones in their excitation layers. Our evolutionary parameters for the sample of Gaia-discovered g-mode pulsators with confirmed modes by TESS offer a fruitful starting point for future TESS ensemble asteroseismology once a sufficient number of modes is identified in terms of the geometrical wave numbers and overtone for each of the pulsators., Comment: Accepted for publication in Astronomy & Astrophysics. 13 pages, 16 figures. Full abstract in PDF

One way to understand planet formation is through studying the correlations between planet occurrence rates and stellar mass. However, measuring stellar mass in the red giant regime is very difficult. In particular, the spectroscopic masses of certain evolved stars, often referred to as "retired A-stars", have been questioned in the literature. Efforts to resolve this mass controversy using spectroscopy, interferometry and asteroseismology have so far been inconclusive. A recent ensemble study found a mass-dependent mass offset, but the result was based on only 16 stars. With NASA's Transiting Exoplanet Survey Satellite (TESS), we expand the investigation of the mass discrepancy to a total of 92 low-luminosity stars, synonymous with the retired A-stars. We measure their characteristic oscillation frequency, $\mathrm{\nu}_{\mathrm{max}}$, and the large frequency separation, $\mathrm{\Delta\nu}$, from their TESS photometric time series. Using these measurements and asteroseismic scaling relations, we derive asteroseismic masses and compare them with spectroscopic masses from five surveys, to comprehensively study the alleged mass-dependent mass offset. We find a mass offset between spectroscopy and seismology that increases with stellar mass. However, we note that adopting the seismic mass scale does not have a significant effect on the planet occurrence-mass-metallicity correlation for the so-called retired A-stars. We also report seismic measurements and masses for 157 higher luminosity giants (mostly helium-core-burning) from the spectroscopic surveys., Comment: 13 pages, 7 figues

The Gaia mission discovered many new candidate \beta Cephei (\beta Cep) pulsators, which are meanwhile confirmed from TESS space photometry. We aim to analyse all currently available TESS data for these \beta Cep pulsators, of which 145 were new discoveries, in order to exploit their asteroseismic potential. \beta Cep stars belong to an under-represented class of pulsators in the current space photometry revolution while being of critical importance to improve evolution models of massive stars. We extracted light curves for 216 star from the TESS full-frame images and performed pre-whitening. Based on Gaia DR3, we deduced stellar properties and compared them to those of known \beta Cep stars. We developed a methodology to identify the dominant pulsation modes of the \beta Cep stars from Gaia and TESS amplitude ratios and from the detection of rotationally-split multiplets. We used grid modelling to gain insights into the population of \beta Cep stars. With the combination of TESS and Gaia, we successfully identified the mode degrees for 176 stars. of which the majority are dipole non-radial modes. Many non-radial modes show splittings in their TESS frequency spectra allowing us to assemble a large set of split multiplets in \beta Cep stars and to calculate their envelope rotation, spin parameter, and the level of differential envelope-to-surface rotation. For the latter, we find an upper limit of 4, with most stars rotating almost rigidly. We also provide the asymmetries of the multiplets. Based on grid modelling, we provide mass, convective core mass, and ages for 164 stars. By combining Gaia and TESS, we enable asteroseismology of \beta Cep stars as a population. Our study prepares for future detailed modelling based on individual frequencies of identified modes leading towards a better understanding of these massive pulsators, as crucial probes of stellar evolution theory. (abridged), Comment: Submitted to A&A on 30 July 2024

The degree of alignment between a star's spin axis and the orbital plane of its planets (the stellar obliquity) is related to interesting and poorly understood processes that occur during planet formation and evolution. Hot Jupiters orbiting hot stars ($\gtrsim$6250 K) display a wide range of obliquities, while similar planets orbiting cool stars are preferentially aligned. Tidal dissipation is expected to be more rapid in stars with thick convective envelopes, potentially explaining this trend. Evolved stars provide an opportunity to test the damping hypothesis, particularly stars that were hot on the main sequence and have since cooled and developed deep convective envelopes. We present the first systematic study of the obliquities of hot Jupiters orbiting subgiants that recently developed convective envelopes using Rossiter-McLaughlin observations. Our sample includes two newly discovered systems in the Giants Transiting Giants Survey (TOI-6029 b, TOI-4379 b). We find that the orbits of hot Jupiters orbiting subgiants that have cooled below $\sim$6250 K are aligned or nearly aligned with the spin-axis of their host stars, indicating rapid tidal realignment after the emergence of a stellar convective envelope. We place an upper limit for the timescale of realignment for hot Jupiters orbiting subgiants at $\sim$500 Myr. Comparison with a simplified tidal evolution model shows that obliquity damping needs to be $\sim$4 orders of magnitude more efficient than orbital period decay to damp the obliquity without destroying the planet, which is consistent with recent predictions for tidal dissipation from inertial waves excited by hot Jupiters on misaligned orbits., Comment: 22 pages, 14 figures, 3 tables

Asteroseismology of dwarf stars cooler than the Sun is very challenging due to the low amplitudes and rapid timescales of oscillations. Here, we present the asteroseismic detection of solar-like oscillations at 4-minute timescales ($\nu_{\mathrm{max}}\sim4300\mu$Hz) in the nearby K-dwarf $\sigma$ Draconis using extreme precision Doppler velocity observations from the Keck Planet Finder and 20-second cadence photometry from NASA's Transiting Exoplanet Survey Satellite. The star is the coolest dwarf star to date with both velocity and luminosity observations of solar-like oscillations, having amplitudes of $5.9\pm0.8\,$cm$\,\text{s}^{-1}$ and $0.8\pm0.2$ ppm, respectively. These measured values are in excellent agreement with established luminosity-velocity amplitude relations for oscillations and provide further evidence that mode amplitudes for stars with $T_{\mathrm{eff}}<\,5500\,$K diminish in scale following a $(L/M)^{1.5}$ relation. By modeling the star's oscillation frequencies from photometric data, we measure an asteroseismic age of $4.5\pm0.9\,\rm{(ran)} \pm 1.2\,\rm{(sys)}$ Gyr. The observations demonstrate the capability of next-generation spectrographs and precise space-based photometry to extend observational asteroseismology to nearby cool dwarfs, which are benchmarks for stellar astrophysics and prime targets for directly imaging planets using future space-based telescopes., Comment: Accepted for publication in The Astrophysical Journal

Based on the rate of change of its orbital period, PSR J2043+1711 has a substantial peculiar acceleration of 3.5 $\pm$ 0.8 mm/s/yr, which deviates from the acceleration predicted by equilibrium Milky Way models at a $4\sigma$ level. The magnitude of the peculiar acceleration is too large to be explained by disequilibrium effects of the Milky Way interacting with orbiting dwarf galaxies ($\sim$1 mm/s/yr), and too small to be caused by period variations due to the pulsar being a redback. We identify and examine two plausible causes for the anomalous acceleration: a stellar flyby, and a long-period orbital companion. We identify a main-sequence star in \textit{Gaia} DR3 and Pan-STARRS DR2 with the correct mass, distance, and on-sky position to potentially explain the observed peculiar acceleration. However, the star and the pulsar system have substantially different proper motions, indicating that they are not gravitationally bound. However, it is possible that this is an unrelated star that just happens to be located near J2043+1711 along our line of sight (chance probability of 1.6\%). Therefore, we also constrain possible orbital parameters for a circumbinary companion in a hierarchical triple system with J2043+1711; the changes in the spindown rate of the pulsar are consistent with an outer object that has an orbital period of 80 kyr, a companion mass of 0.3 $M_\odot$ (indicative of a white dwarf or low-mass star), and a semi-major axis of 2000 AU. Continued timing and/or future faint optical observations of J2043+1711 may eventually allow us to differentiate between these scenarios.

The $\beta$ Pictoris system is the closest known stellar system with directly detected gas giant planets, an edge-on circumstellar disc, and evidence of falling sublimating bodies and transiting exocomets. The inner planet, $\beta$ Pictoris c, has also been indirectly detected with radial velocity (RV) measurements. The star is a known $\delta$ Scuti pulsator, and the long-term stability of these pulsations opens up the possibility of indirectly detecting the gas giant planets through time delays of the pulsations due to a varying light travel time. We search for phase shifts in the $\delta$ Scuti pulsations consistent with the known planets $\beta$ Pictoris b and c and carry out an analysis of the stellar pulsations of $\beta$ Pictoris over a multi-year timescale. We used photometric data collected by the BRITE-Constellation, bRing, ASTEP, and TESS to derive a list of the strongest and most significant $\delta$ Scuti pulsations. We carried out an analysis with the open-source python package maelstrom to study the stability of the pulsation modes of $\beta$ Pictoris in order to determine the long-term trends in the observed pulsations. We did not detect the expected signal for $\beta$ Pictoris b or $\beta$ Pictoris c. The expected time delay is 6 seconds for $\beta$ Pictoris c and 24 seconds for $\beta$ Pictoris b. With simulations, we determined that the photometric noise in all the combined data sets cannot reach the sensitivity needed to detect the expected timing drifts. An analysis of the pulsational modes of $\beta$ Pictoris using maelstrom showed that the modes themselves drift on the timescale of a year, fundamentally limiting our ability to detect exoplanets around $\beta$ Pictoris via pulsation timing., Comment: 16 pages, 16 figures, 4 tables, accepted for publication in A&A

{\delta} Scuti variables are found at the intersection of the classical instability strip and the main sequence on the Hertzsprung-Russell diagram. With space-based photometry providing millions of light-curves of A-F type stars, we can now probe the occurrence rate of {\delta} Scuti pulsations in detail. Using 30-min cadence light-curves from NASA's Transiting Exoplanet Survey Satellite's (TESS) first 26 sectors, we identify variability in 103,810 stars within 5-24 cycles per day down to a magnitude of $T=11.25$. We fit the period-luminosity relation of the fundamental radial mode for {\delta} Scuti stars in the Gaia $G$-band, allowing us to distinguish classical pulsators from contaminants for a subset of 39,367 stars. Out of this subset, over 15,918 are found on or above the expected period-luminosity relation. We derive an empirical red edge to the classical instability strip using Gaia photometry. The center where pulsator fraction peaks at 50-70%, combined with the red edge, agree well with previous work in the Kepler field. While many variable sources are found below the period-luminosity relation, over 85% of sources inside of the classical instability strip derived in this work are consistent with being {\delta} Scuti stars. The remaining 15% of variables within the instability strip are likely hybrid or {\gamma} Doradus pulsators. Finally, we discover strong evidence for a correlation between pulsator fraction and spectral line broadening from the Radial Velocity Spectrometer (RVS) aboard the Gaia spacecraft, confirming that rotation has a role in driving pulsations in {\delta} Scuti stars., Comment: Accepted to Astrophysical Journal, 23 pages, 15 figures

The Gaia mission has delivered hundreds of thousands of variable star light curves in multiple wavelengths. Recent work demonstrates that these light curves can be used to identify (non-)radial pulsations in the OBAF-type stars, despite the irregular cadence and low light curve precision of order a few mmag. With the considerably more precise TESS photometry, we revisit these candidate pulsators to conclusively ascertain the nature of their variability. We seek to re-classify the Gaia light curves with the first two years of TESS photometry for a sample of 58,970 p- and g- mode pulsators, encompassing gamma Dor, delta Scuti, SPB, and beta Cep variables. We also supply four new catalogues containing the confirmed pulsators, along with their dominant and secondary pulsation frequencies, the number of independent mode frequencies, and a ranking according to their usefulness for future asteroseismic ensemble analysis. We find that the Gaia photometry is exceptionally accurate for detecting the dominant and secondary frequencies, reaching approximately 80% accuracy in frequency for p- and g-mode pulsators. The majority of Gaia classifications are consistent with the classifications from the TESS data, illustrating the power of the low-cadence Gaia photometry for pulsation studies. We find that the sample of g-mode pulsators forms a continuous group of variable stars along the main sequence across B, A, and F spectral types, implying that the mode excitation mechanisms for all these pulsators need to be updated with improved physics. Finally, we provide a rank-ordered table of pulsators according to their asteroseismic potential for follow-up studies. Our catalogue offers a major increase in the number of confirmed gravity-mode pulsators with an identified dominant mode suitable for follow-up TESS ensemble asteroseismology of such stars., Comment: 14 pages, 8 figures, accepted to A&A

Gyrochronology, a valuable tool for determining ages of low-mass stars where other techniques fail, relies on accurate calibration. We present a sample of 327 wide ($>$$100$\,au) white dwarf + main sequence (WD + MS) binary systems. Total ages of WDs are computed using all-sky survey photometry, Gaia parallaxes, and current hydrogen atmosphere WD models. Using a magnetic braking law calibrated against open clusters, along with assumptions about initial conditions and angular momentum transport, we construct gyrochrones to predict the rotation periods of the MS stars. Both data and models show that, near the fully convective boundary, MS stars with WD ages up to 7.5\,Gyr experience a rotation period increase by up to a factor of $\approx$$3$ within a $<50\,\mathrm{K}$ effective temperature range. We suggest that rapid braking at this boundary is driven by a sharp rise in the convective overturn timescale ($\tau_{\mathrm{cz}}$) caused by structural changes between partially and fully convective stars and the $^3 \textrm{He}$ instability occurring at this boundary. While the specific location in mass (or temperature) of this feature varies with model physics, we argue that its existence remains consistent. Stars along this feature exhibit rotation periods that can be mapped, within 1$\sigma$, to a range of gyrochrones spanning $\approx 6$\, Gyr. Due to current temperature errors ($\simeq$$50\,\mathrm{K}$), this implies that a measured rotation period cannot be uniquely associated to a single gyrochrone, implying that gyrochronology may not be feasible for M dwarfs very close to the fully convective boundary., Comment: Submitted to ApJ. 31 pages, 17 figures, 2 tables

The TESS mission has provided a wealth of asteroseismic data for solar-like oscillators. However, these data are subject to varying cadences, large gaps, and unequal sampling, which complicates analysis in the frequency domain. One solution is to model the oscillations in the time domain by treating them as stochastically damped simple harmonic oscillators through a linear combination of Gaussian Process kernels. We demonstrate this method on the well-studied subgiant star nu Indi and a sample of Kepler red giant stars observed by TESS, finding that the time domain model achieves an almost two-fold increase in accuracy for measuring {\nu}max compared to typical frequency domain methods. To apply the method to new detections, we use stellar parameters from Gaia DR3 and the TESS input catalog to calculate revised asteroseismic detection probabilities for all TESS input catalog targets with T<12 mag and a predicted {\nu}max>240{\mu}Hz. We also provide a software tool to calculate detection probabilities for any target of interest. Using the updated detection probabilities we show that time-domain asteroseismology is sensitive enough to recover marginal detections, which may explain the current small number of frequency-based detections of TESS oscillations compared to pre-flight expectations., Comment: Submitted with positive referee report. The ATL can be accessed here: https://github.com/danhey/tess-atl

We report the discovery and characterisation of TIC 350842552 ("Zvrk"), an apparently isolated, rapidly-rotating ($P_\text{rot} \sim 99\ \mathrm{d}$) red giant observed by TESS in its Southern Continuous Viewing Zone. The star's fast surface rotation is independently verified by the use of p-mode asteroseismology, strong periodicity in TESS and ASAS-SN photometry, and measurements of spectroscopic rotational broadening. A two-component fit to APOGEE spectra indicates a coverage fraction of its surface features consistent with the amplitude of the photometric rotational signal. Variations in the amplitude of its photometric modulations over time suggest the evolution of its surface morphology, and therefore enhanced magnetic activity. We further develop and deploy new asteroseismic techniques to characterise radial differential rotation, and find weak evidence for rotational shear within Zvrk's convective envelope. This feature, in combination with such a high surface rotation rate, is incompatible with models of angular-momentum transport in single-star evolution. Spectroscopic abundance estimates also indicate a high lithium abundance, among other chemical anomalies. Taken together, all of these suggest a planet-ingestion scenario for the formation of this rotational configuration, various models for which we examine in detail., Comment: 31 pages, 17 figures. Accepted for publication in ApJ

We report the All-Sky Automated Survey for SuperNovae discovery of the tidal disruption event (TDE) ASASSN-23bd (AT 2023clx) in NGC 3799, a LINER galaxy with no evidence of strong AGN activity over the past decade. With a redshift of $z = 0.01107$ and a peak UV/optical luminosity of $(5.4\pm0.4)\times10^{42}$ erg s$^{-1}$, ASASSN-23bd is the lowest-redshift and least-luminous TDE discovered to date. Spectroscopically, ASASSN-23bd shows H$\alpha$ and He I emission throughout its spectral time series, and the UV spectrum shows nitrogen lines without the strong carbon and magnesium lines typically seen for AGN. Fits to the rising ASAS-SN light curve show that ASASSN-23bd started to brighten on MJD 59988$^{+1}_{-1}$, $\sim$9 days before discovery, with a nearly linear rise in flux, peaking in the $g$ band on MJD $60000^{+3}_{-3}$. Scaling relations and TDE light curve modelling find a black hole mass of $\sim$10$^6$ $M_\odot$, which is on the lower end of supermassive black hole masses. ASASSN-23bd is a dim X-ray source, with an upper limit of $L_{0.3-10\,\mathrm{keV}} < 1.0\times10^{40}$ erg s$^{-1}$ from stacking all \emph{Swift} observations prior to MJD 60061, but with soft ($\sim 0.1$ keV) thermal emission with a luminosity of $L_{0.3-2 \,\mathrm{keV}}\sim4\times10^{39}$ erg s$^{-1}$ in \emph{XMM-Newton} observations on MJD 60095. The rapid $(t < 15$ days) light curve rise, low UV/optical luminosity, and a luminosity decline over 40 days of $\Delta L_{40}\approx-0.7$ make ASASSN-23bd one of the dimmest TDEs to date and a member of the growing ``Low Luminosity and Fast'' class of TDEs., Comment: 17 pages, 13 figures, submitted to MNRAS

HIP 65426 hosts a young giant planet that has become the first exoplanet directly imaged with JWST. Using time-series photometry from the Transiting Exoplanet Survey Satellite (TESS), we classify HIP 65426 as a high-frequency $\delta$ Scuti pulsator with a possible large frequency separation of $\Delta \nu =$7.23$\pm$0.02 cycles day$^{-1}$. We check the TESS data for pulsation timing variations and use the nondetection to estimate a 95% dynamical mass upper limit of 12.8 Mjup for HIP 65426 b. We also identify a low-frequency region of signal that we interpret as stellar latitudinal differential rotation with two rapid periods of 7.85$\pm$0.08 hr and 6.67$\pm$0.04 hr. We use our TESS rotation periods together with published values of radius and $v \sin{i}$ to jointly measure the inclination of HIP 65426 to $i_{\star}=107_{-11}^{+12}$$^\circ$. Our stellar inclination is consistent with the orbital inclination of HIP 65426 b ($108_{-3}^{+6}$$^{\circ}$) at the $68\%$ percent level based on our orbit fit using published relative astrometry. The lack of significant evidence for spin-orbit misalignment in the HIP 65426 system supports an emerging trend consistent with preferential alignment between imaged long-period giant planets and their host stars., Comment: 18 pages, 9 figures, 2 tables. Accepted to AJ after minor constructive revisions. Uncertainty on HIP65426 inclination is now more realistic; Figures 5 and 7 are updated accordingly

We present the results of a systematic search of the Transiting Exoplanet Survey Satellite (TESS) 2-min cadence data for new rapidly oscillating Ap (roAp) stars observed during the Cycle 2 phase of its mission. We find seven new roAp stars previously unreported as such and present the analysis of a further 25 roAp stars that are already known. Three of the new stars show multiperiodic pulsations, while all new members are rotationally variable stars, leading to almost 70 per cent (22) of the roAp stars presented being $\alpha^2$ CVn-type variable stars. We show that targeted observations of known chemically peculiar stars are likely to overlook many new roAp stars, and demonstrate that multi-epoch observations are necessary to see pulsational behaviour changes. We find a lack of roAp stars close to the blue edge of the theoretical roAp instability strip, and reaffirm that mode instability is observed more frequently with precise, space-based observations. In addition to the Cycle 2 observations, we analyse TESS data for all known roAp stars. This amounts to 18 further roAp stars observed by TESS. Finally, we list six known roAp stars that TESS is yet to observe. We deduce that the incidence of roAp stars amongst the Ap star population is just 5.5 per cent, raising fundamental questions about the conditions required to excite pulsations in Ap stars. This work, coupled with our previous work on roAp stars in Cycle 1 observations, presents the most comprehensive, homogeneous study of the roAp stars in the TESS nominal mission, with a collection of 112 confirmed roAp stars in total., Comment: Accepted for publication in MNRAS. 32 Pages, 2 Tables, 77 Figures

In Feinstein and Rudloff (2023), it was shown that the set of Nash equilibria for any non-cooperative $N$ player game coincides with the set of Pareto optimal points of a certain vector optimization problem with non-convex ordering cone. To avoid dealing with a non-convex ordering cone, an equivalent characterization of the set of Nash equilibria as the intersection of the Pareto optimal points of $N$ multi-objective problems (i.e.\ with the natural ordering cone) is proven. So far, algorithms to compute the exact set of Pareto optimal points of a multi-objective problem exist only for the class of linear problems, which reduces the possibility of finding the true set of Nash equilibria by those algorithms to linear games only. In this paper, we will consider the larger class of convex games. As, typically, only approximate solutions can be computed for convex vector optimization problems, we first show, in total analogy to the result above, that the set of $\epsilon$-approximate Nash equilibria can be characterized by the intersection of $\epsilon$-approximate Pareto optimal points for $N$ convex multi-objective problems. Then, we propose an algorithm based on results from vector optimization and convex projections that allows for the computation of a set that, on one hand, contains the set of all true Nash equilibria, and is, on the other hand, contained in the set of $\epsilon$-approximate Nash equilibria. In addition to the joint convexity of the cost function for each player, this algorithm works provided the players are restricted by either shared polyhedral constraints or independent convex constraints.

Objectives: An increasing number of CAD/CAM (computer-aided design/computer-aided manufacturing) hybrid materials have been introduced to the dental market in recent years. In addition, CAD/CAM hybrid materials for additive manufacturing (AM) are becoming more attractive in digital dentistry. Studies on material microstructures using micro-computed tomography ($\mu$-CT) combined with scanning electron microscopy (SEM) have only been available to a limited extent so far. Methods: One CAD/CAM three-dimensional- (3D-) printable hybrid material (VarseoSmile Crown plus) and two CAD/CAM millable hybrid materials (Vita Enamic; Voco Grandio), as well as one direct composite material (Ceram.x duo), were included in the present study. Cylindrical samples with a diameter of 2 mm were produced from each material and investigated by means of synchrotron radiation $\mu$-CT at a voxel size of 0.65 $\mu$m. Different samples from the same materials, obtained by cutting and polishing, were investigated by SEM. Results: The 3D-printed hybrid material showed some agglomerations and a more irregular distribution of fillers, as well as a visible layered macrostructure and a few spherical pores due to the printing process. The CAD/CAM millable hybrid materials revealed a more homogenous distribution of ceramic particles. The direct composite material showed multiple air bubbles and microstructural irregularities based on manual processing. Significance: The $\mu$-CT and SEM analysis of the materials revealed different microstructures even though they belong to the same class of materials. It could be shown that $\mu$-CT and SEM imaging are valuable tools to understand microstructure and related mechanical properties of materials., Comment: 24 pages, 4 tables, 12 figures including supplementary material