Revisiting a result of Ko

In this paper we analyze Ko's Theorem 3.4 in \cite{Ko87}. We extend point b) of Ko's Theorem by showing that \poneh(\upcoup)=\upcoup. As a corollary, we get the equality \ph(\upcoup) = \poneh(\upcoup), which is, to our knowledge, a unique result of type \poneh({\cal C})=\ph({\cal C}), for a class $\cal C$ that would not be equal to \DP. With regard to point a) of Ko's Theorem, we observe that it also holds for the classes \upk{k} and for \fewp. In spite of this, we prove that point b) of Theorem 3.4 fails for such classes in a relativized world. This is obtained by showing the relativized separation of \upcoupk{2} from \poneh(\npconp). Finally, we suggest a natural line of research arising from these facts

Revisiting a fundamental test of the disc instability model for X-ray binaries

We revisit a core prediction of the disc instability model (DIM) applied to X-ray binaries. The model predicts the existence of a critical mass transfer rate, which depends on disc size, separating transient and persistent systems. We therefore selected a sample of 52 persistent and transient neutron star and black hole X-ray binaries and verified if observed persistent (transient) systems do lie in the appropriate stable (unstable) region of parameter space predicted by the model. We find that, despite the significant uncertainties inherent to these kinds of studies, the data are in very good agreement with the theoretical expectations. We then discuss some individual cases that do not clearly fit into this main conclusion. Finally, we introduce the transientness parameter as a measure of the activity of a source and show a clear trend of the average outburst recurrence time to decrease with transientness in agreement with the DIM predictions. We therefore conclude that, despite difficulties in reproducing the complex details of the lightcurves, the DIM succeeds to explain the global behaviour of X-ray binaries averaged over a long enough period of time.Comment: 12 pages, 4 figures. Accepted for publication in MNRAS. Version 2: some typos corrected and references adde

Surface energy and boundary layers for a chain of atoms at low temperature

We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard-Jones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature $\beta^{-1}$ goes to zero. Our main results are: (1) As $\beta \to \infty$ at fixed positive pressure $p>0$, the Gibbs measures $\mu_\beta$ and $\nu_\beta$ for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals $\overline{\mathcal{E}}_{\mathrm{bulk}}$ and $\overline{\mathcal{E}}_\mathrm{surf}$. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of $\overline{\mathcal{E}}_\mathrm{surf}$. (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts. (4) Bounds on the decay of correlations are provided, some of them uniform in $\beta$