Shifted distinct-part partition identities in arithmetic progressions

The partition function $p(n)$, which counts the number of partitions of a positive integer $n$, is widely studied. Here, we study partition functions $p_S(n)$ that count partitions of $n$ into distinct parts satisfying certain congruence conditions. A shifted partition identity is an identity of the form $p_{S_1}(n-H) = p_{S_2}(n)$ for all $n$ in some arithmetic progression. Several identities of this type have been discovered, including two infinite families found by Alladi. In this paper, we use the theory of modular functions to determine the necessary and sufficient conditions for such an identity to exist. In addition, for two specific cases, we extend Alladi's theorem to other arithmetic progressions

GWAS of thyroid stimulating hormone highlights pleiotropic effects and inverse association with thyroid cancer

Correction: Volume12, Issue1 Article Number7354 DOI10.1038/s41467-021-27675-w PublishedDEC 16 2021Thyroid stimulating hormone (TSH) is critical for normal development and metabolism. To better understand the genetic contribution to TSH levels, we conduct a GWAS meta-analysis at 22.4 million genetic markers in up to 119,715 individuals and identify 74 genome-wide significant loci for TSH, of which 28 are previously unreported. Functional experiments show that the thyroglobulin protein-altering variants P118L and G67S impact thyroglobulin secretion. Phenome-wide association analysis in the UK Biobank demonstrates the pleiotropic effects of TSH-associated variants and a polygenic score for higher TSH levels is associated with a reduced risk of thyroid cancer in the UK Biobank and three other independent studies. Two-sample Mendelian randomization using TSH index variants as instrumental variables suggests a protective effect of higher TSH levels (indicating lower thyroid function) on risk of thyroid cancer and goiter. Our findings highlight the pleiotropic effects of TSH-associated variants on thyroid function and growth of malignant and benign thyroid tumors. Thyroid stimulating hormone (TSH) is critical for normal development and metabolism. Here, the authors conduct a GWAS and suggest protective effect of higher TSH on risk of thyroid cancer and goitre.Peer reviewe

Author Correction:GWAS of thyroid stimulating hormone highlights the pleiotropic effects and inverse association with thyroid cancer

The original version of this article contained an error in the results, in the second paragraph of the subsection entitled “Fine-mapping for potentially causal variants among TSH loci”, in which effect sizes for two variants were incorrectly reported

The Surface Array planned for IceCube-Gen2

IceCube-Gen2, the extension of the IceCube Neutrino Observatory, will feature three main components: an optical array in the deep ice, a large-scale radio array in the shallow ice and firn, and a surface detector above the optical array. Thus, IceCube-Gen2 will not only be an excellent detector for PeV neutrinos, but also constitutes a unique setup for the measurement of cosmic-ray air showers, where the electromagnetic component and low-energy muons are measured at the surface and high-energy muons are measured in the ice. As for ongoing enhancement of IceCube’s current surface array, IceTop, we foresee a combination of elevated scintillation and radio detectors for the Gen2 surface array, aiming at high measurement accuracy for air showers. The science goals are manifold: The in-situ measurement of the cosmic-ray flux and mass composition, as well as more thorough tests of hadronic interaction models, will improve the understanding of muons and atmospheric neutrinos detected in the ice, in particular, regarding prompt muons. Moreover, the surface array provides a cosmic-ray veto for the in-ice detector and contributes to the calibration of the optical and radio arrays. Last but not least, the surface array will make major contributions to cosmic-ray science in the energy range of the transition from Galactic to extragalactic sources. The increased sensitivities for photons and for cosmic-ray anisotropies at multi-PeV energies provide a chance to solve the puzzle of the origin of the most energetic Galactic cosmic rays and will serve IceCube’s multimessenger mission