It is well known that the eigenvalues of tridiagonal matrices can be identified with the zeros of polynomials satisfying three-term recursion relations and being therefore members of an orthogonal set. A class of such polynomials is identified some of which feature zeros given by simple formulae involving integer numbers. In the process certain neat formulae are also obtained, which perhaps deserve to be included in standard compilations, since they involve classical polynomials such as the Jacobi polynomials and other 'named' polynomials.
Certain Diophantine conjectures are proven, and to do so certain remarkable classes of orthogonal polynomials are identified, yielding additional Diophantine findings.
Monitoring sea surface salinity (SSS) and density variations is crucial to investigate the global water cycle and the ocean dynamics, and to analyse how they are impacted by climate change. Historically, ocean salinity and density have suffered a poor observational coverage, which hindered an accurate assessment of their surface patterns, as well as of associated space and time variability and trends. Different approaches have thus been proposed to extend the information obtained from sparse in situ measurements and provide gap-free fields at regular spatial and temporal resolution, based on the combination of in situ and satellite data. In the framework of the Copernicus Marine Environment Monitoring Service, a daily (weekly sampled) global reprocessed dataset at ¼ • × ¼ • resolution has been produced by modifying a multivariate optimal interpolation (OI) technique originally developed within MyOcean project. The algorithm has been applied to in situ salinity/density measurements covering the period from 1993 to 2016, using satellite sea surface temperature differences to constrain the surface patterns. This improved algorithm and the new dataset are described and validated here with holdout approach and independent data.
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