On conjectures and problems of Ruzsa concerning difference graphs of S-units

Given a finite nonempty set of primes S, we build a graph $\mathcal{G}$ with vertex set $\mathbb{Q}$ by connecting x and y if the prime divisors of both the numerator and denominator of x-y are from S. In this paper we resolve two conjectures posed by Ruzsa concerning the possible sizes of induced nondegenerate cycles of $\mathcal{G}$, and also a problem of Ruzsa concerning the existence of subgraphs of $\mathcal{G}$ which are not induced subgraphs.Comment: 15 page

The VVV Survey reveals classical Cepheids tracing a young and thin stellar disk across the Galaxy's bulge

Solid insight into the physics of the inner Milky Way is key to understanding our Galaxy's evolution, but extreme dust obscuration has historically hindered efforts to map the area along the Galactic mid-plane. New comprehensive near-infrared time-series photometry from the VVV Survey has revealed 35 classical Cepheids, tracing a previously unobserved component of the inner Galaxy, namely a ubiquitous inner thin disk of young stars along the Galactic mid-plane, traversing across the bulge. The discovered period (age) spread of these classical Cepheids implies a continuous supply of newly formed stars in the central region of the Galaxy over the last 100 million years.Comment: Accepted for publication in The Astrophysical Journal Letters. Near-IR finding charts of the Cepheids are available at the following URL: http://www.astro.puc.cl/~idekany/pub/inner_disk_ceph_fc.pn

Effects of Magnetic Field on Josephson Current in SNS System

The effect of a magnetic field on Josephson current has been studied for a superconductor/normal-metal/superconductor (SNS) system, where N is a two-dimensional electron gas in a confining potential. It is found that the dependence of Josephson currents on the magnetic field are sensitive to the width of the normal metal. If the normal metal is wide and contains many channels (subbands), the current on a weak magnetic field shows a dependence similar to a Fraunhofer-pattern in SIS system and, as the field gets strong, it shows another type of oscillatory dependence on the field resulting from the Aharonov-Bohm interference between the edge states. As the number of channels decreases (i.e. normal metal gets narrower), however, the dependence in the region of the weak field deviates from a clear Fraunhofer pattern and the amplitude of the oscillatory dependence in the region of the strong field is reduced.Comment: 14 pages, 9 figure

Orbital Magnetism and Current Distribution of Two-Dimensional Electrons under Confining Potential

The spatial distribution of electric current under magnetic field and the resultant orbital magnetism have been studied for two-dimensional electrons under a harmonic confining potential V(\vecvar{r})=m \omega_0^2 r^2/2 in various regimes of temperature and magnetic field, and the microscopic conditions for the validity of Landau diamagnetism are clarified. Under a weak magnetic field (\omega_c\lsim\omega_0, \omega_c being a cyclotron frequency) and at low temperature (T\lsim\hbar\omega_0), where the orbital magnetic moment fluctuates as a function of the field, the currents are irregularly distributed paramagnetically or diamagnetically inside the bulk region. As the temperature is raised under such a weak field, however, the currents in the bulk region are immediately reduced and finally there only remains the diamagnetic current flowing along the edge. At the same time, the usual Landau diamagnetism results for the total magnetic moment. The origin of this dramatic temperature dependence is seen to be in the multiple reflection of electron waves by the boundary confining potential, which becomes important once the coherence length of electrons gets longer than the system length. Under a stronger field (\omega_c\gsim\omega_0), on the other hand, the currents in the bulk region cause de Haas-van Alphen effect at low temperature as T\lsim\hbar\omega_c. As the temperature gets higher (T\gsim\hbar\omega_c) under such a strong field, the bulk currents are reduced and the Landau diamagnetism by the edge current is recovered.Comment: 15 pages, 11 figure

The Orbit of the New Milky Way Globular Cluster FSR1716 =VVV-GC05

Indexación: Scopus.We use deep, multi-epoch near-IR images of the VISTA Variables in the Via Lictea (VVV) Survey to measure proper motions (PMs) of stars in the Milky Way globular cluster (GC) FSR1716 = VVV-GC05. The colormagnitude diagram of this object, made by using PM-selected members, shows an extended horizontal branch, nine confirmed RR Lyrae (RRL) members in the instability strip, and possibly several hotter stars extending to the blue. Based on the fundamental-mode (ab-type) RRL stars that move coherently with the cluster, we confirmed that FSR1716 is an Oosterhoff I GC with a mean period aPabn = 0.574 days. Intriguingly, we detect tidal extensions to both sides of this cluster in the spatial distribution of PM-selected member stars. Also, one of the confirmed RRabs is located -11 arcmin in projection from the cluster center, suggesting that FSR1716 may be losing stars due to the gravitational interaction with the Galaxy. We also measure radial velocities (RVs) for five cluster red giants selected using the PMs. The combination of RVs and PMs allow us to compute for the first time the orbit of this GC, using an updated Galactic potential. The orbit results to be confined within|Zmax| < 2.0 kpc, and has eccentricity 0.4 < e < 0.6, with perigalactic distance 1.5 < Rperi (kpc) < 2.3, and apogalactic distance 5.3 < Rapo (kpc) < 6.4. We conclude that, in agreement with its relatively low metallicity ([Fe/H] =-1.4 dex), this is an inner-halo GC plunging into the disk of the Galaxy. As such, this is a unique object with which to test the dynamical processes that contribute to the disruption of Galactic GCs. © 2018. The American Astronomical Society. All rights reserved.https://iopscience.iop.org/article/10.3847/1538-4357/aacd0

Discovery of a Pair of Classical Cepheids in an Invisible Cluster Beyond the Galactic Bulge

We report the discovery of a pair of extremely reddened classical Cepheid variable stars located in the Galactic plane behind the bulge, using near-infrared time-series photometry from the VVV Survey. This is the first time that such objects have ever been found in the opposite side of the Galactic plane. The Cepheids have almost identical periods, apparent brightnesses and colors. From the near-infrared Leavitt law, we determine their distances with ~1.5% precision and ~8% accuracy. We find that they have a same total extinction of A(V)~32 mag, and are located at the same heliocentric distance of =11.4+/-0.9 kpc, and less than 1 pc from the true Galactic plane. Their similar periods indicate that the Cepheids are also coeval, with an age of ~48+/-3 Myr, according to theoretical models. They are separated by an angular distance of only 18.3", corresponding to a projected separation of ~1 pc. Their position coincides with the expected location of the Far 3 kpc Arm behind the bulge. Such a tight pair of similar classical Cepheids indicates the presence of an underlying young open cluster, that is both hidden behind heavy extinction and disguised by the dense stellar field of the bulge. All our attempts to directly detect this "invisible cluster" have failed, and deeper observations are needed.Comment: Accepted for publication in ApJL. Photometric data are available online at the CDS and also at http://www.astro.puc.cl/~idekany/pub/twincep.da

The compact multiple system HIP 41431

The nearby (50 pc) K7V dwarf HIP 41431 (EPIC 212096658) is a compact three-tier hierarchy. Three K7V stars with similar masses, from 0.61 to 0.63 solar, make a triple-lined spectroscopic system where the inner binary with a period of 2.9 d is eclipsing, and the outer companion on a 59-d orbit exerts strong dynamical influence revealed by the eclipse time variation in the Kepler photometry. Moreover, the centre of mass of the triple system moves on a 3.9-yr orbit, modulating the proper motion. The mass of the fourth star is 0.35 solar. The Kepler and ground-based photometry and radial velocities from four different spectrographs are used to adjust the spectro-photodynamical model that accounts for dynamical interaction. The mutual inclination between the two inner orbits is 2{^{circ}.}16 ± 0{^{circ}.}11, while the outer orbit is inclined to their common plane by 21°±16°. The inner orbit precesses under the influence of both outer orbits, causing observable variation of the eclipse depth. Moreover, the phase of the inner binary is strongly modulated with a 59-d period and its line of apsides precesses. The middle orbit with eccentricity e = 0.28 also precesses, causing the observed variation of its radial velocity curve. Masses and other parameters of stars in this unique hierarchy are determined. This system is dynamically stable and likely old

FAUSTA: Scaling Dynamic Analysis with Traffic Generation at WhatsApp

We introduce Fausta, an algorithmic traffic gener-ation platform that enables analysis and testing at scale. Fausta has been deployed at Meta to analyze and test the WhatsApp plat-form infrastructure since September 2020, enabling WhatsApp developers to deploy reliable code changes to a code base of millions of lines of code, supporting over 2 billion users who rely on WhatsApp for their daily communications. Fausta covers expected and unexpected program behaviors in a privacy-safe controlled environment to support multiple use cases such as reliability testing, privacy analysis and performance regression detection. It currently supports three different algorithmic input generation strategies, each of which construct realistic backend server traffic that closely simulates production data, without replaying any real user data. Fausta has been deployed and closely integrated into the WhatsApp continuous integration process, catching bugs in development before they hit production. We report on the development and deployment of Fausta's reliability use case between September 2020 and August 2021. During this period it has found 1,876 unique reliability issues, with a fix rate of 74%, indicating a high degree of true positive fault revelation. We also report on the distribution of fault types revealed by Fausta, and the correlation between coverage and faults found. Overall, we do find evidence that higher coverage is correlated with fault revelation