On the Kauffman bracket skein module of the quaternionic manifold

We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over Z[A,A^{-1}] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the Z_2 homology of the manifold, we determine that they are linearly independent.Comment: corrected summation signs in figures 14, 15, 17. Other minor change

Evidence for a constant IMF in early-type galaxies based on their X-ray binary populations

A number of recent studies have proposed that the stellar initial mass function (IMF) of early type galaxies varies systematically as a function of galaxy mass, with higher mass galaxies having bottom heavy IMFs. These bottom heavy IMFs have more low-mass stars relative to the number of high mass stars, and therefore naturally result in proportionally fewer neutron stars and black holes. In this paper, we specifically predict the variation in the number of black holes and neutron stars based on the power-law IMF variation required to reproduce the observed mass-to-light ratio trends with galaxy mass. We then test whether such variations are observed by studying the field low-mass X-ray binary populations (LMXBs) of nearby early-type galaxies. In these binaries, a neutron star or black hole accretes matter from a low-mass donor star. Their number is therefore expected to scale with the number of black holes and neutron stars present in a galaxy. We find that the number of LMXBs per K-band light is similar among the galaxies in our sample. These data therefore demonstrate the uniformity of the slope of the IMF from massive stars down to those now dominating the K-band light, and are consistent with an invariant IMF. Our results are inconsistent with an IMF which varies from a Kroupa/Chabrier like IMF for low mass galaxies to a steep power-law IMF (with slope $x$=2.8) for high mass galaxies. We discuss how these observations constrain the possible forms of the IMF variations and how future Chandra observations can enable sharper tests of the IMF.Comment: 12 pages, 5 figures, 2 tables, submitted to Ap

Generalized Riemann sums

The primary aim of this chapter is, commemorating the 150th anniversary of Riemann's death, to explain how the idea of {\it Riemann sum} is linked to other branches of mathematics. The materials I treat are more or less classical and elementary, thus available to the "common mathematician in the streets." However one may still see here interesting inter-connection and cohesiveness in mathematics

Testing the Universality of the Stellar IMF with Chandra and HST

The stellar initial mass function (IMF), which is often assumed to be universal across unresolved stellar populations, has recently been suggested to be "bottom-heavy" for massive ellipticals. In these galaxies, the prevalence of gravity-sensitive absorption lines (e.g. Na I and Ca II) in their near-IR spectra implies an excess of low-mass ($m <= 0.5$ $M_\odot$) stars over that expected from a canonical IMF observed in low-mass ellipticals. A direct extrapolation of such a bottom-heavy IMF to high stellar masses ($m >= 8$ $M_\odot$) would lead to a corresponding deficit of neutron stars and black holes, and therefore of low-mass X-ray binaries (LMXBs), per unit near-IR luminosity in these galaxies. Peacock et al. (2014) searched for evidence of this trend and found that the observed number of LMXBs per unit $K$-band luminosity ($N/L_K$) was nearly constant. We extend this work using new and archival Chandra X-ray Observatory (Chandra) and Hubble Space Telescope (HST) observations of seven low-mass ellipticals where $N/L_K$ is expected to be the largest and compare these data with a variety of IMF models to test which are consistent with the observed $N/L_K$. We reproduce the result of Peacock et al. (2014), strengthening the constraint that the slope of the IMF at $m >= 8$ $M_\odot$ must be consistent with a Kroupa-like IMF. We construct an IMF model that is a linear combination of a Milky Way-like IMF and a broken power-law IMF, with a steep slope ($\alpha_1=$ $3.84$) for stars < 0.5 $M_\odot$ (as suggested by near-IR indices), and that flattens out ($\alpha_2=$ $2.14$) for stars > 0.5 $M_\odot$, and discuss its wider ramifications and limitations.Comment: Accepted for publication in ApJ; 7 pages, 2 figures, 1 tabl

Deep Chandra observations of NGC 7457, the X-ray point source populations of a low mass early-type galaxy

We present the X-ray point source population of NGC 7457 based on 124 ks of Chandra observations. Previous deep Chandra observations of low mass X-ray binaries (LMXBs) in early-type galaxies have typically targeted the large populations of massive galaxies. NGC 7457 is a nearby, early-type galaxy with a stellar luminosity of $1.7\times10^{10} L_{K\odot}$, allowing us to investigate the populations in a relatively low mass galaxy. We classify the detected X-ray sources into field LMXBs, globular cluster LMXBs, and background AGN based on identifying optical counterparts in new HST/ACS images. We detect 10 field LMXBs within the $r_{ext}$ ellipse of NGC 7457 (with semi-major axis $\sim$ 9.1 kpc, ellipticity = 0.55). The corresponding number of LMXBs with $L_{x}>2\times10^{37}erg/s$ per stellar luminosity is consistent with that observed in more massive galaxies, $\sim 7$ per $10^{10} L_{K\odot}$. We detect a small globular cluster population in these HST data and show that its colour distribution is likely bimodal and that its specific frequency is similar to that of other early type galaxies. However, no X-ray emission is detected from any of these clusters. Using published data for other galaxies, we show that this non-detection is consistent with the small stellar mass of these clusters. We estimate that 0.11 (and 0.03) LMXBs are expected per $10^{6}M_{\odot}$ in metal-rich (and metal-poor) globular clusters. This corresponds to 1100 (and 330) LMXBs per $10^{10} L_{K\odot}$, highlighting the enhanced formation efficiency of LMXBs in globular clusters. A nuclear X-ray source is detected with $L_{x}$ varying from $2.8-6.8\times10^{38}erg/s$. Combining this $L_{x}$ with a published dynamical mass estimate for the central SMBH in NGC 7457, we find that $L_{x}/L_{Edd}$ varies from $0.5-1.3\times10^{-6}$.Comment: 9 pages, 7 figures, 1 table, accepted for publication in MNRA

Lower Bounds for Heights in Relative Galois Extensions

The goal of this paper is to obtain lower bounds on the height of an algebraic number in a relative setting, extending previous work of Amoroso and Masser. Specifically, in our first theorem we obtain an effective bound for the height of an algebraic number $\alpha$ when the base field $\mathbb{K}$ is a number field and $\mathbb{K}(\alpha)/\mathbb{K}$ is Galois. Our second result establishes an explicit height bound for any non-zero element $\alpha$ which is not a root of unity in a Galois extension $\mathbb{F}/\mathbb{K}$, depending on the degree of $\mathbb{K}/\mathbb{Q}$ and the number of conjugates of $\alpha$ which are multiplicatively independent over $\mathbb{K}$. As a consequence, we obtain a height bound for such $\alpha$ that is independent of the multiplicative independence condition

A magnetar-powered X-ray transient as the aftermath of a binary neutron-star merger

Neutron star-neutron star mergers are known to be associated with short gamma-ray bursts. If the neutron star equation of state is sufficiently stiff, at least some of such mergers will leave behind a supramassive or even a stable neutron star that spins rapidly with a strong magnetic field (i.e., a magnetar). Such a magnetar signature may have been observed as the X-ray plateau following a good fraction (up to 50%) of short gamma-ray bursts, and it has been expected that one may observe short gamma-ray burst-less X-ray transients powered by double neutron star mergers. A fast X-ray transient (CDF-S XT1) was recently found to be associated with a faint host galaxy whose redshift is unknown. Its X-ray and host-galaxy properties allow several possibleexplanations including a short gamma-ray burst seen off axis, a low-luminosity gamma-ray burst at high redshift, or a tidal disruption event involving an intermediate mass black hole and a white dwarf. Here we report a second X-ray transient, CDF-S XT2, that is associated with a galaxy at redshift z = 0.738. The light curve is fully consistent with being powered by a millisecond magnetar. More intriguingly, CDF-S XT2 lies in the outskirts of its star-forming host galaxy with a moderate offset from the galaxy center, as short bursts often do. The estimated event rate density of similar X-ray transients, when corrected to the local value, is consistent with the double neutron star merger rate density inferred from the detection of GW170817.Comment: 29 pages, 4 figures, 3 tables, published in Nature on 11 April 201

Sharpenings of Li's criterion for the Riemann Hypothesis

Exact and asymptotic formulae are displayed for the coefficients $\lambda_n$ used in Li's criterion for the Riemann Hypothesis. For $n \to \infty$ we obtain that if (and only if) the Hypothesis is true, $\lambda_n \sim n(A \log n +B)$ (with $A>0$ and $B$ explicitly given, also for the case of more general zeta or $L$-functions); whereas in the opposite case, $\lambda_n$ has a non-tempered oscillatory form.Comment: 10 pages, Math. Phys. Anal. Geom (2006, at press). V2: minor text corrections and updated reference

Preliminary Human-in-the-Loop Assessment of Procedures for Very-Closely-Spaced Parallel Runways

Demand in the future air transportation system concept is expected to double or triple by 2025 [1]. Increasing airport arrival rates will help meet the growing demand that could be met with additional runways but the expansion airports is met with environmental challenges for the surrounding communities when using current standards and procedures. Therefore, changes to airport operations can improve airport capacity without adding runways. Building additional runways between current ones, or moving them closer, is a potential solution to meeting the increasing demand, as addressed by the Terminal Area Capacity Enhancing Concept (TACEC). TACEC requires robust technologies and procedures that need to be tested such that operations are not compromised under instrument meteorological conditions. The reduction of runway spacing for independent simultaneous operations dramatically exacerbates the criticality of wake vortex incursion and the calculation of a safe and proper breakout maneuver. The study presented here developed guidelines for such operations by performing a real-time, human-in-the-loop simulation using precision navigation, autopilot-flown approaches, with the pilot monitoring aircraft spacing and the wake vortex safe zone during the approach