Ideal properties and integral extension of convolution operators on L∞(G)L^\infty (G)

We investigate operator ideal properties of convolution operators $C_\lambda$ (via measures $\lambda$) acting in ${L^\infty (G)}$, with $G$ a compact abelian group. Of interest is when $C_\lambda$ is compact, as this corresponds to $\lambda$ having an integrable density relative to Haar measure $\mu$, i.e., $\lambda \ll \mu$. Precisely then is there an \textit{optimal} Banach function space $L^1 (m_\lambda)$ available which contains ${L^\infty (G)}$ properly, densely and continuously and such that $C_\lambda$ has a continuous, ${L^\infty (G)}$-valued, linear extension $I_{m_\lambda}$ to $L^1 (m_\lambda)$. A detailed study is made of $L^1 (m_\lambda)$ and $I_{m_\lambda}$. Amongst other things, it is shown that $C_\lambda$ is compact iff the finitely additive, ${L^\infty (G)}$-valued set function $m_\lambda (A) := C_\lambda ({\chi_{_{_{\scriptstyle{A}}}}})$ is norm $\sigma$-additive iff $\lambda \in L^1 (G)$, whereas the corresponding optimal extension $I_{m_\lambda}$ is compact iff $\lambda \in C (G)$ iff $m_\lambda$ has finite variation. We also characterize when $m_\lambda$ admits a Bochner (resp.\ Pettis) $\mu$-integrable, $L^{\infty} (G)$-valued density

Completeness ofL1-Spaces for Measures with Values in Complex Vector Spaces

AbstractWe provide a simple and direct proof of the completeness of the space of C-valued integrable functions with respect to a vector measure taking its values in a complex Fréchet space. A similar result is established (via a quite different argument) for closed measures taking their values in more general sequentially complete, locally convex Hausdorff spaces over C

Asteroseismology of red giants & galactic archaeology

Red-giant stars are low- to intermediate-mass ($M \lesssim 10$~M$_{\odot}$) stars that have exhausted hydrogen in the core. These extended, cool and hence red stars are key targets for stellar evolution studies as well as galactic studies for several reasons: a) many stars go through a red-giant phase; b) red giants are intrinsically bright; c) large stellar internal structure changes as well as changes in surface chemical abundances take place over relatively short time; d) red-giant stars exhibit global intrinsic oscillations. Due to their large number and intrinsic brightness it is possible to observe many of these stars up to large distances. Furthermore, the global intrinsic oscillations provide a means to discern red-giant stars in the pre-helium core burning from the ones in the helium core burning phase and provide an estimate of stellar ages, a key ingredient for galactic studies. In this lecture I will first discuss some physical phenomena that play a role in red-giant stars and several phases of red-giant evolution. Then, I will provide some details about asteroseismology -- the study of the internal structure of stars through their intrinsic oscillations -- of red-giant stars. I will conclude by discussing galactic archaeology -- the study of the formation and evolution of the Milky Way by reconstructing its past from its current constituents -- and the role red-giant stars can play in that.Comment: Lecture presented at the IVth Azores International Advanced School in Space Sciences on "Asteroseismology and Exoplanets: Listening to the Stars and Searching for New Worlds" (arXiv:1709.00645), which took place in Horta, Azores Islands, Portugal in July 201

Particle acceleration mechanisms

We review the possible mechanisms for production of non-thermal electrons which are responsible for non-thermal radiation in clusters of galaxies. Our primary focus is on non-thermal Bremsstrahlung and inverse Compton scattering, that produce hard X-ray emission. We briefly review acceleration mechanisms and point out that in most astrophysical situations, and in particular for the intracluster medium, shocks, turbulence and plasma waves play a crucial role. We consider two scenarios for production of non-thermal radiation. The first is hard X-ray emission due to non-thermal Bremsstrahlung by nonrelativistic particles. Non-thermal tails are produced by accelerating electrons from the background plasma with an initial Maxwellian distribution. However, these tails are accompanied by significant heating and they are present for a short time of <10^6 yr, which is also the time that the tail will be thermalised. Such non-thermal tails, even if possible, can only explain the hard X-ray but not the radio emission which needs GeV or higher energy electrons. For these and for production of hard X-rays by the inverse Compton model, we need the second scenario where there is injection and subsequent acceleration of relativistic electrons. It is shown that a steady state situation, for example arising from secondary electrons produced from cosmic ray proton scattering by background protons, will most likely lead to flatter than required electron spectra or it requires a short escape time of the electrons from the cluster. An episodic injection of relativistic electrons, presumably from galaxies or AGN, and/or episodic generation of turbulence and shocks by mergers can result in an electron spectrum consistent with observations but for only a short period of less than one billion years.Comment: 22 pages, 5 figures, accepted for publication in Space Science Reviews, special issue "Clusters of galaxies: beyond the thermal view", Editor J.S. Kaastra, Chapter 11; work done by an international team at the International Space Science Institute (ISSI), Bern, organised by J.S. Kaastra, A.M. Bykov, S. Schindler & J.A.M. Bleeke

Galaxy Clusters Associated with Short GRBs. II. Predictions for the Rate of Short GRBs in Field and Cluster Early-Type Galaxies

We determine the relative rates of short GRBs in cluster and field early-type galaxies as a function of the age probability distribution of their progenitors, P(\tau) \propto \tau^n. This analysis takes advantage of the difference in the growth of stellar mass in clusters and in the field, which arises from the combined effects of the galaxy stellar mass function, the early-type fraction, and the dependence of star formation history on mass and environment. This approach complements the use of the early- to late-type host galaxy ratio, with the added benefit that the star formation histories of early-type galaxies are simpler than those of late-type galaxies, and any systematic differences between progenitors in early- and late-type galaxies are removed. We find that the ratio varies from R(cluster)/R(field) ~ 0.5 for n = -2 to ~ 3 for n = 2. Current observations indicate a ratio of about 2, corresponding to n ~ 0 - 1. This is similar to the value inferred from the ratio of short GRBs in early- and late-type hosts, but it differs from the value of n ~ -1 for NS binaries in the Milky Way. We stress that this general approach can be easily modified with improved knowledge of the effects of environment and mass on the build-up of stellar mass, as well as the effect of globular clusters on the short GRB rate. It can also be used to assess the age distribution of Type Ia supernova progenitors.Comment: ApJ accepted versio

TOI-257b (HD 19916b): A warm sub-saturn orbiting an evolved F-type star

We report the discovery of a warm sub-Saturn, TOI-257b (HD 19916b), based on data from NASA's Transiting Exoplanet Survey Satellite (TESS). The transit signal was detected by TESS and confirmed to be of planetary origin based on radial velocity observations. An analysis of the TESS photometry, the Minerva-Australis, FEROS, and HARPS radial velocities, and the asteroseismic data of the stellar oscillations reveals that TOI-257b has a mass of MP = 0.138 ± 0.023 M J (43.9 ± 7.3, M⊕), a radius of RP = 0.639 ± 0.013 R J (7.16 ± 0.15, R ⊕), bulk density of 0.65+0.12-0.11 (cgs), and period 18.38818 +0.00085 -0.00084 days. TOI-257b orbits a bright (V = 7.612 mag) somewhat evolved late F-type star with M∗ = 1.390 ± 0.046 rm M sun, R∗ = 1.888 ± 0.033 Rsun, Teff = 6075 ± 90 rm K, and vsin i = 11.3 ± 0.5 km s-1. Additionally, we find hints for a second non-transiting sub-Saturn mass planet on a ∼71 day orbit using the radial velocity data. This system joins the ranks of a small number of exoplanet host stars (∼100) that have been characterized with asteroseismology. Warm sub-Saturns are rare in the known sample of exoplanets, and thus the discovery of TOI-257b is important in the context of future work studying the formation and migration history of similar planetary systems