On Wolstenholme's theorem and its converse

AbstractFor any positive integer n, let wn=(2n−1n−1)=12(2nn). Wolstenholme proved that if p is a prime ⩾5, then wp≡1(modp3). The converse of Wolstenholme's theorem, which has been conjectured to be true, remains an open problem. In this article, we establish several relations and congruences satisfied by the numbers wn, and we deduce that this converse holds for many infinite families of composite integers n. In passing, we obtain a number of congruences satisfied by certain classes of binomial coefficients, and involving the Bernoulli numbers

EVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS

Abstract. Motivated by a recent work about finite sequences where the n-th term is bounded by n

Systematic review of temporal trends of congenital syphilis in Brazil

The present study was aimed to retrieve and analyse the temporal trends of congenital syphilis (CS) in Brazil, as well as to list its main associated factors. Methods: In August 2019 a systematic review was developed in four electronic databases (Lilacs, Pubmed, Scielo and Web of Science) and in manual searches on reference lists. It was established that the synthesis of this review would be composed by ecological studies with CS temporal trends in the Brazilian territory, regardless of the population characteristics and data representativeness. More specifically, it was also established that the prevalence of CS of the first and last year of the time series would be presented in the descriptive synthesis. Results: Of the 2,157 initial studies, 14 adequately met the inclusion criteria and composed the synthesis. Twelve (85.7%) of these studies showed increases in time trends, with particular emphasis on the two nationwide studies, which showed positive trends between 2003–2008 (0.4) and 2010–2015 (3.7). Associations were found between CS and socioeconomic and ethnic factors, especially in the groups of women with low income, low education, brown / black skin colour and who had untreated partners. Conclusion: most of the available research showed an increase in the temporal trends of CS, highlighting that these data were observed at the national, state and municipal levels. Since mothers' socioeconomic and ethnic factors are associated with higher CS frequencies, efforts are needed to increase the coverage of the Unified Health System to vulnerable women

The Infrared Spectra of Very Large Irregular Polycyclic Aromatic Hydrocarbons (PAHs): Observational Probes of Astronomical PAH Geometry, Size and Charge

The mid-IR spectra of six large, irregular PAHs with formulae (C84H24 - C120H36) have been computed using Density Functional Theory (DFT). Trends in the dominant band positions and intensities are compared to those of large, compact PAHs as a function of geometry, size and charge. Irregular edge moieties that are common in terrestrial PAHs, such as bay regions and rings with quartet hydrogens, are shown to be uncommon in astronomical PAHs. As for all PAHs comprised solely of C and H reported to date, mid-IR emission from irregular PAHs fails to produce a strong CCstr band at 6.2 um, the position characteristic of the important, class A astronomical PAH spectra. Earlier studies showed inclusion of nitrogen within a PAH shifts this to 6.2 um for PAH cations. Here we show this band shifts to 6.3 um in nitrogenated PAH anions, close to the position of the CC stretch in class B astronomical PAH spectra. Thus nitrogenated PAHs may be important in all sources and the peak position of the CC stretch near 6.2 um appears to directly reflect the PAH cation to anion ratio. Large irregular PAHs exhibit features at 7.8 um but lack them near 8.6 um. Hence, the 7.7 um astronomical feature is produced by a mixture of small and large PAHs while the 8.6 um band can only be produced by large compact PAHs. As with the CCstr, the position and profile of these bands reflect the PAH cation to anion ratio.Comment: accepted by Ap

The Spectral Energy Distribution of Dust Emission in the Edge-on spiral galaxy NGC 4631 as seen with Spitzer and the James Clerk Maxwell telescope

We explore variations in dust emission within the edge-on Sd spiral galaxy NGC 4631 using 3.6-160 μm Spitzer Space Telescope data and 450-850 μm JCMT data with the goals of understanding the relation between PAHs and dust emission, studying the variations in the colors of the dust emission, and searching for possible excess submillimeter emission compared to what is expected from dust models extrapolated from far-infrared wavelengths. The 8 μm PAH emission correlates best with 24 μm hot dust emission on 1.7 kpc scales, but the relation breaks down on 650 pc scales, possibly because of differences in the mean free paths between photons that excite the PAHs and photons that heat the dust and possibly because the PAHs are destroyed by the hard radiation fields within some star formation regions. The ratio of 8 μm PAH emission to 160 μm cool dust emission appears to vary as a function of radius. The 70 μm/160 μm and 160 μm/450 μm flux density ratios are remarkably constant even though the surface brightnesses vary by factors of 25, which suggests that the emission is from dust heated by a nearly uniform radiation field. Globally, we find an excess of 850-1230 μm emission relative to what would be predicted by dust models. The 850 μm excess is highest in regions with low 160 μm surface brightnesses, although the magnitude depends on the model fit to the data. We rule out variable emissivity functions or ~4 K dust as the possible origins of this 850 μm emission, but we do discuss the other possible mechanisms that could produce the emission

The Radial Distribution of the Interstellar Medium in Disk Galaxies: Evidence for Secular Evolution

One possible way for spiral galaxies to internally evolve would be for gas to flow to the center and form stars in a central disk (pseudo-bulge). If the inflow rate is faster than the rate of star formation, a central concentration of gas will form. In this paper we present radial profiles of stellar and 8 μm emission from polycyclic aromatic hydrocarbons (PAHs) for 11 spiral galaxies to investigate whether the interstellar medium in these galaxies contains a central concentration above that expected from the exponential disk. In general, we find that the two-dimensional CO and PAH emission morphologies are similar, and that they exhibit similar radial profiles. We find that in 6 of the 11 galaxies there is a central excess in the 8 μm and CO emission above the inward extrapolation of an exponential disk. In particular, all four barred galaxies in the sample have strong central excesses in both 8 μm and CO emission. These correlations suggest that the excess seen in the CO profiles is, in general, not simply due to a radial increase in the CO emissivity. In the inner disk, the ratio of the stellar to the 8 μm radial surface brightness is similar for 9 of the 11 galaxies, suggesting a physical connection between the average stellar surface brightness and the average gas surface brightness at a given radius. We also find that the ratio of the CO to 8 μm PAH surface brightness is consistent over the sample, implying that the 8 μm PAH surface brightness can be used as an approximate tracer of the interstellar medium

Warm H2_2 as a probe of massive accretion and feedback through shocks and turbulence across cosmic time

Galaxy formation depends on a complex interplay between gravitational collapse, gas accretion, merging, and feedback processes. Yet, after many decades of investigation, these concepts are poorly understood. This paper presents the argument that warm H$_2$ can be used as a tool to unlock some of these mysteries. Turbulence, shocks and outflows, driven by star formation, AGN activity or inflows, may prevent the rapid buildup of star formation in galaxies. Central to our understanding of how gas is converted into stars is the process by which gas can dissipate its mechanical energy through turbulence and shocks in order to cool. H$_2$ lines provide direct quantitative measurements of kinetic energy dissipation in molecular gas in galaxies throughout the Universe. Based on the detection of very powerful H$_2$ lines from z = 2 galaxies and proto-clusters at the detection limits of {\it Spitzer}, we are confident that future far-IR and UV H$_2$ observations will provide a wealth of new information and insight into galaxy evolution to high-z. Finally, at the very earliest epoch of star and galaxy formation, warm H$_2$ may also provide a unique glimpse of molecular gas collapse at 7 $<$ z $<$ 12 in massive dark matter (DM) halos on their way to forming the very first galaxies. Such measurements are beyond the reach of existing and planned observatories.Comment: Submitted as a science White Paper to the Astronomy and Astrophysics Astro 2020 Decadal Survey call issued by the National Academies of Sciences, Engineering and Medicine (March 11 2019

Warm Dust and Spatially Variable PAH Emission in the Dwarf Starburst Galaxy NGC 1705

We present Spitzer observations of the dwarf starburst galaxy NGC 1705 obtained as part of SINGS. The galaxy morphology is very different shortward and longward of ~5 microns: short-wavelength imaging shows an underlying red stellar population, with the central super star cluster (SSC) dominating the luminosity; longer-wavelength data reveals warm dust emission arising from two off-nuclear regions offset by ~250 pc from the SSC. These regions show little extinction at optical wavelengths. The galaxy has a relatively low global dust mass (~2E5 solar masses, implying a global dust-to-gas mass ratio ~2--4 times lower than the Milky Way average). The off-nuclear dust emission appears to be powered by photons from the same stellar population responsible for the excitation of the observed H Alpha emission; these photons are unassociated with the SSC (though a contribution from embedded sources to the IR luminosity of the off-nuclear regions cannot be ruled out). Low-resolution IRS spectroscopy shows moderate-strength PAH emission in the 11.3 micron band in the eastern peak; no PAH emission is detected in the SSC or the western dust emission complex. There is significant diffuse 8 micron emission after scaling and subtracting shorter wavelength data; the spatially variable PAH emission strengths revealed by the IRS data suggest caution in the interpretation of diffuse 8 micron emission as arising from PAH carriers alone. The metallicity of NGC 1705 falls at the transition level of 35% solar found by Engelbracht and collaborators; the fact that a system at this metallicity shows spatially variable PAH emission demonstrates the complexity of interpreting diffuse 8 micron emission. A radio continuum non-detection, NGC 1705 deviates significantly from the canonical far-IR vs. radio correlation. (Abridged)Comment: ApJ, in press; please retrieve full-resolution version from http://www.astro.wesleyan.edu/~cannon/pubs.htm