Chains of large gaps between primes

Let $p_n$ denote the $n$-th prime, and for any $k \geq 1$ and sufficiently large $X$, define the quantity $$G_k(X) := \max_{p_{n+k} \leq X} \min( p_{n+1}-p_n, \dots, p_{n+k}-p_{n+k-1} ),$$ which measures the occurrence of chains of $k$ consecutive large gaps of primes. Recently, with Green and Konyagin, the authors showed that $$G_1(X) \gg \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for sufficiently large $X$. In this note, we combine the arguments in that paper with the Maier matrix method to show that $$G_k(X) \gg \frac{1}{k^2} \frac{\log X \log \log X\log\log\log\log X}{\log \log \log X}$$ for any fixed $k$ and sufficiently large $X$. The implied constant is effective and independent of $k$.Comment: 16 pages, no figure

Oxygen Gas Abundances at 0.4<z<1.5: Implications for the Chemical Evolution History of Galaxies

We report VLT-ISAAC and Keck-NIRSPEC near-infrared spectroscopy for a sample of 30 0.47<z<0.92 CFRS galaxies and five [OII]-selected, M_B,AB<-21.5, z~1.4 galaxies. We have measured Halpha and [NII] line fluxes for the CFRS galaxies which have [OII], Hbeta and [OIII] line fluxes available from optical spectroscopy. For the z~1.4 objects we measured Hbeta and [OIII] emission line fluxes from J-band spectra, and Halpha line fluxes plus upper limits for [NII] fluxes from H-band spectra. We derive the extinction and oxygen abundances for the sample using a method based on a set of ionisation parameter and oxygen abundance diagnostics, simultaneously fitting the [OII], Hbeta, [OIII], Halpha and [NII] line fluxes. Our most salient conclusions are: a) the source of gas ionisation in the 30 CFRS and in all z~1.4 galaxies is not due to AGN activity; b) about one third of the 0.47<z<0.92 CFRS galaxies in our sample have substantially lower metallicities than local galaxies with similar luminosities and star formation rates; c) comparison with a chemical evolution model indicates that these low metallicity galaxies are unlikely to be the progenitors of metal-poor dwarf galaxies at z~0, but more likely the progenitors of massive spirals; d) the z~1.4 galaxies are characterized by the high [OIII]/[OII] line ratios, low extinction and low metallicity that are typical of lower luminosity CADIS galaxies at 0.4<z<0.7, and of more luminous Lyman Break Galaxies at z~3.1, but not seen in CFRS galaxies at 0.4<z<1.0; e) the properties of the z~1.4 galaxies suggest that the period of rapid chemical evolution takes place progressively in lower mass systems as the universe ages, and thus provides further support for a downsizing picture of galaxy formation, at least from z~1.4 to today.Comment: Proceedings contribution for "The Fabulous Destiny of Galaxies; Bridging Past and Present", Marseille, 200

Precise Charm- and Bottom-Quark Masses: Theoretical and Experimental Uncertainties

Recent theoretical and experimental improvements in the determination of charm and bottom quark masses are discussed. A new and improved evaluation of the contribution from the gluon condensate $$ to the charm mass determination and a detailed study of potential uncertainties in the continuum cross section for $b\bar b$ production is presented, together with a study of the parametric uncertainty from the $\alpha_s$-dependence of our results. The final results, $m_c(3 \text{GeV})=986(13)$MeV and $m_b(m_b)=4163(16)$MeV, represent, together with a closely related lattice determination $m_c(3\;{\rm GeV})=986(6)$MeV, the presently most precise determinations of these two fundamental Standard Model parameters. A critical analysis of the theoretical and experimental uncertainties is presented.Comment: 12 pages, presented at Quarks~2010, 16th International Seminar of High Energy Physics, Kolomna, Russia, June 6-12, 2010; v2: references adde

Expected exponential loss for gaze-based video and volume ground truth annotation

Many recent machine learning approaches used in medical imaging are highly reliant on large amounts of image and ground truth data. In the context of object segmentation, pixel-wise annotations are extremely expensive to collect, especially in video and 3D volumes. To reduce this annotation burden, we propose a novel framework to allow annotators to simply observe the object to segment and record where they have looked at with a \$200 eye gaze tracker. Our method then estimates pixel-wise probabilities for the presence of the object throughout the sequence from which we train a classifier in semi-supervised setting using a novel Expected Exponential loss function. We show that our framework provides superior performances on a wide range of medical image settings compared to existing strategies and that our method can be combined with current crowd-sourcing paradigms as well.Comment: 9 pages, 5 figues, MICCAI 2017 - LABELS Worksho

On polynomial solutions of Heun equation

By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before identifying the polynomial solutions. The Heun equation extended by the addition of a term, - \s/x, is also amenable for polynomial solutions.Comment: 4 pages, No figur