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We present a comparison of the Milky Way's star formation rate (SFR) surface density ($\Sigma_{\rm SFR}$) obtained with two independent state-of-the-art observational methods. The first method infers $\Sigma_{\rm SFR}$ from observations of the dust thermal emission from interstellar dust grains in far-infrared wavelengths registered in the Herschel infrared Galactic Plane Survey (Hi-GAL). The second method determines $\Sigma_{\rm SFR}$ by modeling the current population of O-, B-, and A-type stars in a 6 kpc $\times$ 6 kpc area around the Sun. We find an agreement between the two methods within a factor of two for the mean SFRs and the SFR surface density profiles. Given the broad differences between the observational techniques and the independent assumptions in the methods for computing the SFRs, this agreement constitutes a significant advance in our understanding of the star formation of our Galaxy and implies that the local SFR has been roughly constant over the past 10\,Myr., Comment: Accepted for publication at Astronomy & Astrophysics (11SEP2023). 8 pages, 7 figures

[Abridged] We map the stellar age distribution ($\lesssim 1$ Gyr) across a 6kpc$\,\times\,$6kpc area of the Galactic disc to constrain our Galaxy's recent star-formation history. Our modelling draws on the sample of Zari et al. (2021) that encompasses all presumed disc OBA stars ($\sim 500,000$ sources) with $G<16$. To be less sensitive to reddening, we do not forward model the detailed CMD distribution of these stars, but instead the K-band absolute magnitude distribution, $n(M_K)$, among stars with $M_K<0$ and $T_{\mathrm{eff}} > 7000$K at a certain positions $\vec{x}$ in the disc as a step function with five age bins, $b(\tau~|~\vec{x}, \vec{\alpha})$, logarithmically-spaced in age from $\tau = 5$ Myr to $\tau = 1$ Gyr. Given a set of isochrones and a Kroupa (2001) initial mass function, we sample $b(\tau\,|\,\vec{x}, \vec{\alpha})$ to maximize the likelihood of the data, accounting for the selection function. This results in a set of mono-age stellar density maps across a sizeable portion of the Galactic disc. These maps show that some, but not all, spiral arms are reflected in overdensities of stars younger than 500 Myr. The maps of the youngest stars ($<10$ Myr) trace major star forming regions. The maps at all ages exhibit an outward density gradient and distinct spiral-like spatial structure, which is qualitatively similar on large scales among all age intervals. When summing over the maps' area and extrapolating to the whole disc, we find an effective star-formation rate over the last 10 Myr of $\approx 3.3 \mathrm{M_{\odot}/yr}$, higher than previously published estimates. Remarkably, our stellar age distribution implies that the star-formation rate has been three times lower throughout most of the last Gyr, having risen distinctly only in the very recent past., Comment: 11 pages, 9 figures. Submitted to A&A: comments welcome

[Abridged] Luminous hot stars dominate the stellar energy input to the interstellar medium throughout cosmological time, they are laboratories to test theories of stellar evolution and multiplicity, and they serve as luminous tracers of star formation in the Milky Way and other galaxies. Massive stars occupy well-defined loci in colour-colour and colour-magnitude spaces, enabling selection based on the combination of Gaia EDR3 astrometry and photometry and 2MASS photometry, even in the presence of substantive dust extinction. In this paper we devise an all-sky sample of such luminous OBA-type stars, designed to be quite complete rather than very pure, to serve as targets for spectroscopic follow-up with the SDSS-V survey. We estimate "astro-kinematic" distances by combining parallaxes and proper motions with a model for the expected velocity and density distribution of young stars; we show that this adds useful constraints on the stars' distances, and hence luminosities. With these distances we map the spatial distribution of a more stringently selected sub-sample across the Galactic disc, and find it to be highly structured, with distinct over- and under-densities. The most evident over-densities can be associated with the presumed spiral arms of the Milky Way, in particular the Sagittarius-Carina and Scutum-Centaurus arms. Yet, the spatial picture of the Milky Way's young disc structure emerging in this study is complex, and suggests that most young stars in our Galaxy ($t_{age}

Context. Detecting and characterizing substellar companions for which the luminosity, mass, and age can be determined independently is of utter importance to test and calibrate the evolutionary models due to uncertainties in their formation mechanisms. HD 19467 is a bright and nearby star hosting a cool brown dwarf companion detected with RV and imaging, making it a valuable object for such studies. Aims. We aim to further characterize the orbital, spectral, and physical properties of the HD 19467 system. Methods. We present new high-contrast imaging data with the SPHERE and NaCo instruments. We also analyze archival data from HARPS, NaCo, HIRES, UVES, and ASAS. We also use proper motion data of the star from Hipparcos and Gaia. Results. We refine the properties of the host star and derive an age of 8.0$^{+2.0}_{-1.0}$ Gyr based on isochrones, gyrochronology, and chemical and kinematic arguments. This estimate is slightly younger than previous estimates of ~9-11 Gyr. No orbital curvature is seen in the current imaging, RV, and astrometric data. From a joint fit of the data, we refine the orbital parameters for HD 19467B: period 398$^{+95}_{-93}$ yr, inclination 129.8$^{+8.1}_{-5.1}$ deg, eccentricity 0.56$\pm$0.09, longitude of the ascending node 134.8$\pm$4.5 deg, and argument of the periastron 64.2$^{+5.5}_{-6.3}$ deg. We assess a dynamical mass of 74$^{+12}_{-9}$ MJ. The fit with atmospheric models of the spectrophotometric data of HD 19467B indicates an atmosphere without clouds or with very thin clouds, an effective temperature of 1042$^{+77}_{-71}$ K, and a large surface gravity of 5.34$^{+0.08}_{-0.09}$ dex. The comparison to model predictions of the bolometric luminosity and dynamical mass of HD 19467B, assuming our system age estimate, indicates a better agreement with the Burrows et al. models; whereas the other evolutionary models used tend to underestimate its cooling rate., Comment: Accepted for publication in A&A, 25 pages, 23 figures, 9 tables. Abstract slightly abridged to match arXiv requirements. Updated after language editing

The series $$S_k(z)=\sum_{m=1}^{\infty}\frac{m^kz^m}{(\{array}{c} 2m m \{array})}$$ is evaluated in non-recursive closed and analytically continued beyond its domain of convergence $0\le |z|<4$ for $k=0,1,2,\...$. From this we provide a firm basis for Lehmer's observation that $\pi$ emerges from the limiting behavior of $S_k(2)$ as $k\rightarrow\infty$., Comment: 14 pages

A bordering of GUE matrices is considered, in which the bordered row consists of zero mean complex Gaussians N$[0,\sigma/2] + i {\rm N}[0,\sigma/2]$ off the diagonal, and the real Gaussian N$[\mu,\sigma/\sqrt{2}]$ on the diagonal. We compute the explicit form of the eigenvalue probability function for such matrices, as well as that for matrices obtained by repeating the bordering. The correlations are in general determinantal, and in the single bordering case the explicit form of the correlation kernel is computed. In the large $N$ limit it is shown that $\mu$ and/or $\sigma$ can be tuned to induce a separation of the largest eigenvalue. This effect is shown to be controlled by a single parameter, universal correlation kernel., Comment: 16 pages, no figures

A survey of linearized cosmological fluid equations with a number of different matter components is made. To begin with, the one-component case is reconsidered to illustrate some important mathematical and physical points rarely discussed in the literature. The work of some previous studies of two-component systems are examined and re-analyzed to point out some deficiencies of solutions, and further solutions and physical interpretation are then presented. This leads into a general two-component model with variable velocity dispersion parameters and mass density fractions of each component. The equations, applicable to both hot dark matter (HDM) and cold dark matter (CDM) universes are solved in the long wavelength limit. This region is of interest, because some modes in this range of wavenumbers are Jeans unstable. The mixture Jeans wavenumber of the two-component system is introduced and interpreted, and the solutions are discussed, particularly in comparison to analogous solutions previously derived for plasma modes. This work is applicable to that region in the early Universe ($20 < z < 140$), where large scale structure formation is thought to have occurred., Comment: 36 pages, 2 figures

The equations describing a two-component cosmological fluid with linearized density perturbations are investigated in the small wavelength or large $k$ limit. The equations are formulated to include a baryonic component, as well as either a hot dark matter (HDM) or cold dark matter (CDM) component. Previous work done on such a system in static spacetime is extended to reveal some interesting physical properties, such as the Jeans wavenumber of the mixture, and resonant mode amplitudes. A WKB technique is then developed to study the expanding universe equations in detail, and to see whether such physical properties are also of relevance in this more realistic scenario. The Jeans wavenumber of the mixture is re-interpreted for the case of an expanding background spacetime. The various modes are obtained to leading order, and the amplitudes of the modes are examined in detail to compare to the resonances observed in the static spacetime results. It is found that some conclusions made in the literature about static spacetime results cannot be carried over to an expanding cosmology., Comment: 42 pages, 12 figures

An intensive study for both the weak coupling and strong coupling limits of the ground state properties of this classic system is presented. Detailed results for specific values of finite $N$ are given and from them results for general $N$ are determined. We focus on the density matrix and concomitantly its Fourier transform, the occupation numbers, along with the pair correlation function and concomitantly its Fourier transform, the structure factor. These are the signature quantities of the Bose gas. One specific result is that for weak coupling a rational polynomial structure holds despite the transcendental nature of the Bethe equations. All these new results are predicated on the Bethe ansatz and are built upon the seminal works of the past., Comment: 23 pages, 0 figures, uses rotate.sty. A few lines added. Accepted by Phys. Rev. A

In a recent study we have obtained correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N by N matrices, both in the bulk and at the soft edge of the spectrum. In the present study these results are used to similarly analyze the eigenvalue density for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry. As in the case of unitary symmetry, a matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk. In addition, aspects of the asymptotic expansion of the smoothed density, which involves delta functions at the endpoints of the support, are interpreted microscopically., Comment: 28 pages, 1 figure

We obtain correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N by N matrices, both in the bulk of the spectrum and near the spectral edge. This is achieved by using the well known orthogonal polynomial expression for the kernel to construct a double contour integral representation for the density, to which we apply the saddle point method. The main correction to the bulk density is oscillatory in N and depends on the distribution function of the limiting density, while the corrections to the Airy kernel at the soft edge are again expressed in terms of the Airy function and its first derivative. We demonstrate numerically that these expansions are very accurate. A matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk., Comment: 14 pages, 4 figures

Fisher-Hartwig asymptotics refers to the large $n$ form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin-spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher-Hartwig formula to the asymptotic decay of the Ising correlations above $T_c$, while the study of the Bose gas density matrix leads us to generalize the Fisher-Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the Fisher-Hartwig asymptotic form known for Toeplitz determinants., Comment: 25 pages

The density matrix for the impenetrable Bose gas in Dirichlet and Neumann boundary conditions can be written in terms of $<\prod_{l=1}^n| \cos\phi_1-\cos\theta_l| |\cos\phi_2-\cos\theta_l|>$, where the average is with respect to the eigenvalue probability density function for random unitary matrices from the classical groups $Sp(n)$ and $O^+(2n)$ respectively. In the large $n$ limit log-gas considerations imply that the average factorizes into the product of averages of the form $<\prod_{l=1}^n|\cos\phi-\cos\theta_l>$. By changing variables this average in turn is a special case of the function of $t$ obtained by averaging $\prod_{l=1}^n| t-x_l|^{2q}$ over the Jacobi unitary ensemble from random matrix theory. The latter task is accomplished by a duality formula from the theory of Selberg correlation integrals, and the large $n$ asymptotic form is obtained. The corresponding large $n$ asymptotic form of the density matrix is used, via the exact solution of a particular integral equation, to compute the asymptotic form of the low lying effective single particle states and their occupations, which are proportional to $\sqrt{N}$., Comment: typos corrected; ref. 7, Phys Rev A67,043607 (2003); accepted for publication in J Math Phys

We present here an overview of the history, applications and important properties of a function which we refer to as the Levy integral. For certain values of its characteristic parameter the Levy integral defines the symmetric Levy stable probability density function. As we discuss however the Levy integral has applications to a number of other fields besides probability, including random matrix theory, number theory and asymptotics beyond all orders. We exhibit a direct relationship between the Levy integral and a number theoretic series which we refer to as the generalised Euler-Jacobi series. The complete asymptotic expansions for all natural values of its parameter are presented, and in particular it is pointed out that the intricate exponentially small series become dominant for certain parameter values., Comment: 7 pages, no figures

Bosons in the form of ultra cold alkali atoms can be confined to a one dimensional (1d) domain by the use of harmonic traps. This motivates the study of the ground state occupations $\lambda_i$ of effective single particle states $\phi_i$, in the theoretical 1d impenetrable Bose gas. Both the system on a circle and the harmonically trapped system are considered. The $\lambda_i$ and $\phi_i$ are the eigenvalues and eigenfunctions respectively of the one body density matrix. We present a detailed numerical and analytic study of this problem. Our main results are the explicit scaled forms of the density matrices, from which it is deduced that for fixed $i$ the occupations $\lambda_i$ are asymptotically proportional to $\sqrt{N}$ in both the circular and harmonically trapped cases., Comment: 22 pages, 8 figures (.eps), uses REVTeX4

The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlev\'e VI transcendent in $\sigma$-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlev\'e V and VI. We discuss how our results can be used to compute the ground state occupations., Comment: AMSLatex, requires Commun. Math. Phys. style macros

Fractal basin boundaries provide an important means of characterizing chaotic systems. We apply these ideas to general relativity, where other properties such as Lyapunov exponents are difficult to define in an observer independent manner. Here we discuss the difficulties in describing chaotic systems in general relativity and investigate the motion of particles in two- and three-black-hole spacetimes. We show that the dynamics is chaotic by exhibiting the basins of attraction of the black holes which have fractal boundaries. Overcoming problems of principle as well as numerical difficulties, we evaluate Lyapunov exponents numerically and find that some trajectories have a positive exponent., Comment: To appear in "Fractals" March issue (World Scientific), 20 figures available by request, also available from SLAC's gr-qc postscript archive

We investigate the phase-space for trajectories in multi-black hole spacetimes. We find that complete, chaotic geodesics are well described by Lyapunov exponents, and that the attractor basin boundary scales as a fractal in a diffeomorphism invariant manner., Comment: 9 pages, RevTeX, 4 figures (appended as a compressed tar file), UTPT-93-32, UM-P-94/18

We investigate rotation and rotating structures in (2+1)-dimensional Einstein gravity. We show that rotation generally leads to pathological physical situations., Comment: 11 pages, UM-P-93/43 & UTPT-93-10