Supercongruences of Atkin and Swinnerton-Dyer type for Legendre polynomials

AbstractIn this paper we prove some generalisations of congruences of Atkin and Swinnerton-Dyer type. This is done in the form of congruences for numbers Pn(A√Δ), where Pn(t) are the orthogonal polynomials of Legendre. The proofs are based on complex multiplication of elliptic functions

An operational overview of the EXport processes in the ocean from RemoTe sensing (EXPORTS) northeast pacific field deployment

The goal of the EXport Processes in the Ocean from RemoTe Sensing (EXPORTS) field campaign is to develop a predictive understanding of the export, fate, and carbon cycle impacts of global ocean net primary production. To accomplish this goal, observations of export flux pathways, plankton community composition, food web processes, and optical, physical, and biogeochemical (BGC) properties are needed over a range of ecosystem states. Here we introduce the first EXPORTS field deployment to Ocean Station Papa in the Northeast Pacific Ocean during summer of 2018, providing context for other papers in this special collection. The experiment was conducted with two ships: a Process Ship, focused on ecological rates, BGC fluxes, temporal changes in food web, and BGC and optical properties, that followed an instrumented Lagrangian float; and a Survey Ship that sampled BGC and optical properties in spatial patterns around the Process Ship. An array of autonomous underwater assets provided measurements over a range of spatial and temporal scales, and partnering programs and remote sensing observations provided additional observational context. The oceanographic setting was typical of late-summer conditions at Ocean Station Papa: a shallow mixed layer, strong vertical and weak horizontal gradients in hydrographic properties, sluggish sub-inertial currents, elevated macronutrient concentrations and low phytoplankton abundances. Although nutrient concentrations were consistent with previous observations, mixed layer chlorophyll was lower than typically observed, resulting in a deeper euphotic zone. Analyses of surface layer temperature and salinity found three distinct surface water types, allowing for diagnosis of whether observed changes were spatial or temporal. The 2018 EXPORTS field deployment is among the most comprehensive biological pump studies ever conducted. A second deployment to the North Atlantic Ocean occurred in spring 2021, which will be followed by focused work on data synthesis and modeling using the entire EXPORTS data set

Impact of Distance Determinations on Galactic Structure. I. Young and Intermediate-Age Tracers

Here we discuss impacts of distance determinations on the Galactic disk traced by relatively young objects. The Galactic disk, ∼40kpc in diameter, is a cross-road of studies on the methods of measuring distances, interstellar extinction, evolution of galaxies, and other subjects of interest in astronomy. A proper treatment of interstellar extinction is, for example, crucial for estimating distances to stars in the disk outside the small range of the solar neighborhood. We’ll review the current status of relevant studies and discuss some new approaches to the extinction law. When the extinction law is reasonably constrained, distance indicators found in today and future surveys are telling us stellar distribution and more throughout the Galactic disk. Among several useful distance indicators, the focus of this review is Cepheids and open clusters (especially contact binaries in clusters). These tracers are particularly useful for addressing the metallicity gradient of the Galactic disk, an important feature for which comparison between observations and theoretical models can reveal the evolution of the disk