Low mass variable stars in the globular cluster NGC 6397

We have conducted a photometric survey of the globular cluster NGC 6397 in a search for variable stars. We obtained ~11h of time-resolved photometric images with one ne European Southern Observatory-Very Large Telescope using the FOcal Reducer and low dispersion Spectrograph imager distributed over two consecutive nights. We analyzed 8391 light curves of stars brighter than magnitude 23 with the 465 nm-filter, and we identified 412 variable stars, reaching ~ 4.8 +- 0.2 per cent of variability with timescales between 0.004 and 2d, with amplitudes variation greater than +- 0.2 mag.Comment: 9 figures, complementary dat

Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension

Given a smoothly bounded domain $\Omega\Subset\mathbb{R}^n$ with $n\ge 1$ odd, we study the blow-up of bounded sequences $(u_k)\subset H^\frac{n}{2}_{00}(\Omega)$ of solutions to the non-local equation $$(-\Delta)^\frac n2 u_k=\lambda_k u_ke^{\frac n2 u_k^2}\quad \text{in }\Omega,$$ where $\lambda_k\to\lambda_\infty \in [0,\infty)$, and $H^{\frac n2}_{00}(\Omega)$ denotes the Lions-Magenes spaces of functions $u\in L^2(\mathbb{R}^n)$ which are supported in $\Omega$ and with $(-\Delta)^\frac{n}{4}u\in L^2(\mathbb{R}^n)$. Extending previous works of Druet, Robert-Struwe and the second author, we show that if the sequence $(u_k)$ is not bounded in $L^\infty(\Omega)$, a suitably rescaled subsequence $\eta_k$ converges to the function $\eta_0(x)=\log\left(\frac{2}{1+|x|^2}\right)$, which solves the prescribed non-local $Q$-curvature equation $$(-\Delta)^\frac n2 \eta =(n-1)!e^{n\eta}\quad \text{in }\mathbb{R}^n$$ recently studied by Da Lio-Martinazzi-Rivi\`ere when $n=1$, Jin-Maalaoui-Martinazzi-Xiong when $n=3$, and Hyder when $n\ge 5$ is odd. We infer that blow-up can occur only if $\Lambda:=\limsup_{k\to \infty}\|(-\Delta)^\frac n4 u_k\|_{L^2}^2\ge \Lambda_1:= (n-1)!|S^n|$

Probing Mass Segregation in NGC 6397

In this study, we present a detailed study of mass segregation in the globular clister NGC 6397. First, we carry out a photometric analysis of projected ESO-VLT data (between 1 and 10 arcmin from the cluster centre), presenting the luminosity function corrected by completeness. The luminosity function shows a higher density of bright stars near the central region of the data, with respect to the outer region. We calculate a deprojected model (covering the whole cluster) estimating a total number of stars of 193000 +- 19000. The shapes of the surface brightness and density-number profiles versus the radial coordinate r (instead of the projected coordinate R) lead to a decreasing luminosity for an average star, and thus of mass, up to 1 arcmin, quantifying the mass segregation. The deprojected model does not show evidence of mass segregation outside this region

New SX Phe variables in the globular cluster NGC 288

We report the discovery of two new variable stars in the metal-poor globular cluster NGC 288, found by means of time-series CCD photometry. We classified the new variables as SX Phoenicis due to their characteristic fundamental mode periods (1.02 +- 0.01 and 0.69 +- 0.01 hours), and refine the period estimates for other six known variables. SX Phe stars are known to follow a well-defined Period-Luminosity (P-L) relation and, thus, can be used for determining distances; they are more numerous than RR Lyraes in NGC~288. We obtain the P-L relation for the fundamental mode M_V = (-2.59 +- 0.18) log P_0(d) + (-0.34 +- 0.24) and for the first-overtone mode M_V = (-2.59 +- 0.18) log P_1(d) + (0.50 +- 0.25). Multi-chromatic isochrone fits to our UBV color-magnitude diagrams, based on the Dartmouth Stellar Evolution Database, provide = -1.3 +- 0.1, E(B-V) = 0.02 +- 0.01 and absolute distance modulus (m-M)0 = 14.72 +- 0.01 for NGC 288.Comment: 8 pages, 9 figures, 3 table

A threshold phenomenon for embeddings of H0mH^m_0 into Orlicz spaces

We consider a sequence of positive smooth critical points of the Adams-Moser-Trudinger embedding of $H^m_0$ into Orlicz spaces. We study its concentration-compactness behavior and show that if the sequence is not precompact, then the liminf of the $H^m_0$-norms of the functions is greater than or equal to a positive geometric constant.Comment: 14 Page

A tale of two layers: The mutual relationship between bitcoin and lightning network

A major concern of the adoption and scalability of Blockchain technologies refers to their efficient use for payments. In this work, we analyze how Lightning Network (LN), which represents a relevant infrastructural novelty, is influenced by the market dynamics of its referring cryptocurrency, namely Bitcoin. In so doing, we focus on how the LN is efficient in performing transactions and we relate this feature to the market conditions of Bitcoin. By applying the Toda–Yamamoto variant of Granger-causality, we note that market conditions of Bitcoin do not significantly influence the topological configuration of the LN. Hence, although the LN represents a second layer on the Bitcoin blockchain, our findings suggest that its efficient functioning does not appear to be related to the simple market performance of its underlying cryptocurrency and, in particular, of its volatile market fluctuations. This result may therefore contribute to shed light on the practical usage of the LN as a blockchain technology to favor transactions