On sums of three squares

Let $r_3(n)$ be the number of representations of a positive integer $n$ as a sum of three squares of integers. We give two distinct proofs of a conjecture of Wagon concerning the asymptotic value of the mean square of $r_3(n)$.Comment: 11 pages, minor revisions made; to appear in Internat. J. Number Theor

On two 10th order mock theta identities

We give short proofs of conjectural identities due to Gordon and McIntosh involving two 10th order mock theta functions.Comment: 5 pages, to appear in the Ramanujan Journa

Fractionation of Hydrogen Isotopes by Sulfate- and Nitrate-Reducing Bacteria.

Hydrogen atoms from water and food are incorporated into biomass during cellular metabolism and biosynthesis, fractionating the isotopes of hydrogen-protium and deuterium-that are recorded in biomolecules. While these fractionations are often relatively constant in plants, large variations in the magnitude of fractionation are observed for many heterotrophic microbes utilizing different central metabolic pathways. The correlation between metabolism and lipid δ(2)H provides a potential basis for reconstructing environmental and ecological parameters, but the calibration dataset has thus far been limited mainly to aerobes. Here we report on the hydrogen isotopic fractionations of lipids produced by nitrate-respiring and sulfate-reducing bacteria. We observe only small differences in fractionation between oxygen- and nitrate-respiring growth conditions, with a typical pattern of variation between substrates that is broadly consistent with previously described trends. In contrast, fractionation by sulfate-reducing bacteria does not vary significantly between different substrates, even when autotrophic and heterotrophic growth conditions are compared. This result is in marked contrast to previously published observations and has significant implications for the interpretation of environmental hydrogen isotope data. We evaluate these trends in light of metabolic gene content of each strain, growth rate, and potential flux and reservoir-size effects of cellular hydrogen, but find no single variable that can account for the differences between nitrate- and sulfate-respiring bacteria. The emerging picture of bacterial hydrogen isotope fractionation is therefore more complex than the simple correspondence between δ(2)H and metabolic pathway previously understood from aerobes. Despite the complexity, the large signals and rich variability of observed lipid δ(2)H suggest much potential as an environmental recorder of metabolism

Lorenz gauge gravitational self-force calculations of eccentric binaries using a frequency domain procedure

We present an algorithm for calculating the metric perturbations and gravitational self-force for extreme-mass-ratio inspirals (EMRIs) with eccentric orbits. The massive black hole is taken to be Schwarzschild and metric perturbations are computed in Lorenz gauge. The perturbation equations are solved as coupled systems of ordinary differential equations in the frequency domain. Accurate local behavior of the metric is attained through use of the method of extended homogeneous solutions and mode-sum regularization is used to find the self-force. We focus on calculating the self-force with sufficient accuracy to ensure its error contributions to the phase in a long term orbital evolution will be $\delta\Phi \lesssim 10^{-2}$ radians. This requires the orbit-averaged force to have fractional errors $\lesssim 10^{-8}$ and the oscillatory part of the self-force to have errors $\lesssim 10^{-3}$ (a level frequently easily exceeded). Our code meets this error requirement in the oscillatory part, extending the reach to EMRIs with eccentricities of $e \lesssim 0.8$, if augmented by use of fluxes for the orbit-averaged force, or to eccentricities of $e \lesssim 0.5$ when used as a stand-alone code. Further, we demonstrate accurate calculations up to orbital separations of $a \simeq 100 M$, beyond that required for EMRI models and useful for comparison with post-Newtonian theory. Our principal developments include (1) use of fully constrained field equations, (2) discovery of analytic solutions for even-parity static modes, (3) finding a pre-conditioning technique for outer homogeneous solutions, (4) adaptive use of quad-precision and (5) jump conditions to handle near-static modes, and (6) a hybrid scheme for high eccentricities

Wage differentials and their determinants in US tourism and tourism-associated industries

This paper examines variations in wages for tourism related industries in the US for the period 2004-2009. It critically assesses the extent to which tourism related activities conform to their low wage stereotype and finds this to be true in general but not universally. It then considers the possibility that wages in US tourism related industries can be explained by observable characteristics of these industries. Recent research has suggested that the use of wage data at the level of highly detailed occupations is an effective alternative to other ways of capturing underlying skill differences. In line with this literature data from the US Occupational Employment Statistics (OES) were used to provide this detail. The results strongly support the importance of difference between occupations in wages in understanding differences between industries. They also support the importance of a number of industry characteristics including profitability, multi-factor productivity and demand growth. The paper also considers the relevance of an industry wage premium or discount for tourism related activities in the US over the same period. To assess this it estimates an industry wage model separately for five individual occupations across all industries which employ the occupation concerned. Within the small number of occupations covered the analysis find that workers in the two more highly paid occupations exhibit evidence of a tourism related discount but that workers in the three more lowly paid occupations exhibit a tourism related wage premium

Principal forms X^2 + nY^2 representing many integers

In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X^2+nY^2. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n=2. In this paper, we prove that in fact this constant is unbounded as n runs through positive integers with a fixed number of prime divisors.Comment: 10 pages, title has been changed, Sections 2 and 3 are new, to appear in Abh. Math. Sem. Univ. Hambur

Controls on the composition and lability of dissolved organic matter in Siberia's Kolyma River basin

High-latitude northern rivers export globally significant quantities of dissolved organic carbon (DOC) to the Arctic Ocean. Climate change, and its associated impacts on hydrology and potential mobilization of ancient organic matter from permafrost, is likely to modify the flux, composition, and thus biogeochemical cycling and fate of exported DOC in the Arctic. This study examined DOC concentration and the composition of dissolved organic matter (DOM) across the hydrograph in Siberia's Kolyma River, with a particular focus on the spring freshet period when the majority of the annual DOC load is exported. The composition of DOM within the Kolyma basin was characterized using absorbance-derived measurements (absorbance coefficienta330, specific UV absorbance (SUVA254), and spectral slope ratio SR) and fluorescence spectroscopy (fluorescence index and excitation-emission matrices (EEMs)), including parallel factor analyses of EEMs. Increased surface runoff during the spring freshet led to DOM optical properties indicative of terrestrial soil inputs with high humic-like fluorescence, SUVA254, and low SRand fluorescence index (FI). Under-ice waters, in contrast, displayed opposing trends in optical properties representing less aromatic, lower molecular weight DOM. We demonstrate that substantial losses of DOC can occur via biological (∼30% over 28 days) and photochemical pathways (>29% over 14 days), particularly in samples collected during the spring freshet. The emerging view is therefore that of a more dynamic and labile carbon pool than previously thought, where DOM composition plays a fundamental role in controlling the fate and removal of DOC at a pan-Arctic scale

Interpolated sequences and critical LL-values of modular forms

Recently, Zagier expressed an interpolated version of the Ap\'ery numbers for $\zeta(3)$ in terms of a critical $L$-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier's six sporadic sequences are essentially critical $L$-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown's cellular integrals and critical $L$-values of modular forms of odd weight.Comment: 23 pages, to appear in Proceedings for the KMPB conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theor

Perspectives on Proterozoic surface ocean redox from iodine contents in ancient and recent carbonate

© The Author(s), 2017. This is the author's version of the work. It is posted here under a nonexclusive, irrevocable, paid-up, worldwide license granted to WHOI. It is made available for personal use, not for redistribution. The definitive version was published in Earth and Planetary Science Letters 463 (2017): 159-170, doi:10.1016/j.epsl.2017.01.032.The Proterozoic Eon hosted the emergence and initial recorded diversification of eukaryotes. Oxygen levels in the shallow marine settings critical to these events were lower than today’s, although how much lower is debated. Here, we use concentrations of iodate (the oxidized iodine species) in shallow-marine limestones and dolostones to generate the first comprehensive record of Proterozoic near-surface marine redox conditions. The iodine proxy is sensitive to both local oxygen availability and the relative proximity to anoxic waters. To assess the validity of our approach, Neogene-Quaternary carbonates are used to demonstrate that diagenesis most often decreases and is unlikely to increase carbonate-iodine contents. Despite the potential for diagenetic loss, maximum Proterozoic carbonate iodine levels are elevated relative to those of the Archean, particularly during the Lomagundi and Shuram carbon isotope excursions of the Paleo- and Neoproterozoic, respectively. For the Shuram anomaly, comparisons to Neogene-Quaternary carbonates suggest that diagenesis is not responsible for the observed iodine trends. The baseline low iodine levels in Proterozoic carbonates, relative to the Phanerozoic, are linked to a shallow oxic-anoxic interface. Oxygen concentrations in surface waters would have at least intermittently been above the threshold required to support eukaryotes. However, the diagnostically low iodine data from mid-Proterozoic shallow-water carbonates, relative to those of the bracketing time intervals, are consistent with a dynamic chemocline and anoxic waters that would have episodically mixed upward and laterally into the shallow oceans. This redox instability may have challenged early eukaryotic diversification and expansion, creating an evolutionary landscape unfavorable for the emergence of animals.TL, ZL, and DH thank NSF EAR-1349252. ZL further thanks OCE-1232620. DH, ZL, and TL acknowledge further funding from a NASA Early Career Collaboration Award. TL, AB, NP, DH, and AK thank the NASA Astrobiology Institute. TL and NP received support from the Earth-Life Transitions Program of the NSF. AB acknowledges support from NSF grant EAR-05-45484 and an NSERC Discovery and Accelerator Grants. CW acknowledges support from NSFC grant 40972021