Fractional part sums and divisor functions

AbstractElementary methods are used to study sums of the form Σd≤x{xd}t for integers p and t, t > 0, where {x} denotes the fractional part of x. These sums are then used to study sums of the form Σd≤xdpPt(xd) for integers p and t, t > 0, where Pt(x) = Bt({x}) and Bt(x) are Bernoulli polynomials. some general results on sums of error terms are used to study sums of the form Σn≤xntσa(n) and Σn≤xEt(n) for integers t and a, a ≥ 0, where σa(n) is the sum of the ath powers of the divisors of n and Et(x) is the error term in the sum Σn≤xntσa(n)

Determining trophic niche width: a novel approach using stable isotope analysis

1. Although conceptually robust, it has proven difficult to find practical measures of niche width that are simple to obtain, yet provide an adequate descriptor of the ecological position of the population examined. 2. Trophic niche has proven more tractable than other niche dimensions. However, indices used as a proxy for trophic niche width often suffer from the following difficulties. Such indices rarely lie along a single scale making comparisons between populations or species difficult; have difficulty in combining dietary prey diversity and evenness in an ecologically meaningful way; and fail to integrate diet over ecological time-scales thus usually only comprise single snapshots of niche width. 3. We propose an alternative novel method for the comparison of trophic niche width: the use of variance of tissue stable isotope ratios, especially those of nitrogen and carbon. 4. This approach is a potentially powerful method of measuring trophic niche width, particularly if combined with conventional approaches, because: it provides a single measure on a continuous axis that is common to all species; it integrates information on only assimilated prey over time; the integration period changes with choice of tissue sampled; and data production is theoretically fast and testing among populations simple. 5. Empirical studies are now required to test the benefits of using isotopic variance as a measure of niche width, and in doing so help refine this approach

100 ancient genomes show repeated population turnovers in Neolithic Denmark.

Major migration events in Holocene Eurasia have been characterized genetically at broad regional scales <sup>1-4</sup> . However, insights into the population dynamics in the contact zones are hampered by a lack of ancient genomic data sampled at high spatiotemporal resolution <sup>5-7</sup> . Here, to address this, we analysed shotgun-sequenced genomes from 100 skeletons spanning 7,300 years of the Mesolithic period, Neolithic period and Early Bronze Age in Denmark and integrated these with proxies for diet ( <sup>13</sup> C and <sup>15</sup> N content), mobility ( <sup>87</sup> Sr/ <sup>86</sup> Sr ratio) and vegetation cover (pollen). We observe that Danish Mesolithic individuals of the Maglemose, Kongemose and Ertebølle cultures form a distinct genetic cluster related to other Western European hunter-gatherers. Despite shifts in material culture they displayed genetic homogeneity from around 10,500 to 5,900 calibrated years before present, when Neolithic farmers with Anatolian-derived ancestry arrived. Although the Neolithic transition was delayed by more than a millennium relative to Central Europe, it was very abrupt and resulted in a population turnover with limited genetic contribution from local hunter-gatherers. The succeeding Neolithic population, associated with the Funnel Beaker culture, persisted for only about 1,000 years before immigrants with eastern Steppe-derived ancestry arrived. This second and equally rapid population replacement gave rise to the Single Grave culture with an ancestry profile more similar to present-day Danes. In our multiproxy dataset, these major demographic events are manifested as parallel shifts in genotype, phenotype, diet and land use

Population genomics of post-glacial western Eurasia.

Western Eurasia witnessed several large-scale human migrations during the Holocene <sup>1-5</sup> . Here, to investigate the cross-continental effects of these migrations, we shotgun-sequenced 317 genomes-mainly from the Mesolithic and Neolithic periods-from across northern and western Eurasia. These were imputed alongside published data to obtain diploid genotypes from more than 1,600 ancient humans. Our analyses revealed a 'great divide' genomic boundary extending from the Black Sea to the Baltic. Mesolithic hunter-gatherers were highly genetically differentiated east and west of this zone, and the effect of the neolithization was equally disparate. Large-scale ancestry shifts occurred in the west as farming was introduced, including near-total replacement of hunter-gatherers in many areas, whereas no substantial ancestry shifts happened east of the zone during the same period. Similarly, relatedness decreased in the west from the Neolithic transition onwards, whereas, east of the Urals, relatedness remained high until around 4,000 BP, consistent with the persistence of localized groups of hunter-gatherers. The boundary dissolved when Yamnaya-related ancestry spread across western Eurasia around 5,000 BP, resulting in a second major turnover that reached most parts of Europe within a 1,000-year span. The genetic origin and fate of the Yamnaya have remained elusive, but we show that hunter-gatherers from the Middle Don region contributed ancestry to them. Yamnaya groups later admixed with individuals associated with the Globular Amphora culture before expanding into Europe. Similar turnovers occurred in western Siberia, where we report new genomic data from a 'Neolithic steppe' cline spanning the Siberian forest steppe to Lake Baikal. These prehistoric migrations had profound and lasting effects on the genetic diversity of Eurasian populations

Astronomical Distance Determination in the Space Age: Secondary Distance Indicators

The formal division of the distance indicators into primary and secondary leads to difficulties in description of methods which can actually be used in two ways: with, and without the support of the other methods for scaling. Thus instead of concentrating on the scaling requirement we concentrate on all methods of distance determination to extragalactic sources which are designated, at least formally, to use for individual sources. Among those, the Supernovae Ia is clearly the leader due to its enormous success in determination of the expansion rate of the Universe. However, new methods are rapidly developing, and there is also a progress in more traditional methods. We give a general overview of the methods but we mostly concentrate on the most recent developments in each field, and future expectations. © 2018, The Author(s)

Search for gravitational waves from Scorpius X-1 in the second Advanced LIGO observing run with an improved hidden Markov model

We present results from a semicoherent search for continuous gravitational waves from the low-mass x-ray binary Scorpius X-1, using a hidden Markov model (HMM) to track spin wandering. This search improves on previous HMM-based searches of LIGO data by using an improved frequency domain matched filter, the J-statistic, and by analyzing data from Advanced LIGO's second observing run. In the frequency range searched, from 60 to 650 Hz, we find no evidence of gravitational radiation. At 194.6 Hz, the most sensitive search frequency, we report an upper limit on gravitational wave strain (at 95% confidence) of h095%=3.47×10-25 when marginalizing over source inclination angle. This is the most sensitive search for Scorpius X-1, to date, that is specifically designed to be robust in the presence of spin wandering. © 2019 American Physical Society

Search for Tensor, Vector, and Scalar Polarizations in the Stochastic Gravitational-Wave Background

The detection of gravitational waves with Advanced LIGO and Advanced Virgo has enabled novel tests of general relativity, including direct study of the polarization of gravitational waves. While general relativity allows for only two tensor gravitational-wave polarizations, general metric theories can additionally predict two vector and two scalar polarizations. The polarization of gravitational waves is encoded in the spectral shape of the stochastic gravitational-wave background, formed by the superposition of cosmological and individually unresolved astrophysical sources. Using data recorded by Advanced LIGO during its first observing run, we search for a stochastic background of generically polarized gravitational waves. We find no evidence for a background of any polarization, and place the first direct bounds on the contributions of vector and scalar polarizations to the stochastic background. Under log-uniform priors for the energy in each polarization, we limit the energy densities of tensor, vector, and scalar modes at 95% credibility to Ω0T<5.58×10-8, Ω0V<6.35×10-8, and Ω0S<1.08×10-7 at a reference frequency f0=25 Hz. © 2018 American Physical Society