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   Asok, Aravind; Deglise, Frederic; Nagel, Johannes (April 2020, Contemporary Mathematics: Motivic Homotopy Theory and Refined Enumerative Geometry)

   In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its E2-page. Our description of the E2-page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost’s cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work. 

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3. The ultraspherical spectral element method

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4. The Inverse Spectral Map for Dimers

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   George, T.; Goncharov, A. B.; Kenyon, R. (September 2023, Mathematical Physics, Analysis and Geometry)

   In 2015, Vladimir Fock proved that the spectral transform, associating to an element of a dimer cluster integrable system its spectral data, is birational by constructing an inverse map using theta functions on Jacobians of spectral curves. We provide an alternate construction of the inverse map that involves only rational functions in the spectral data. 

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5. Spectral calibration of the MethaneAIR instrument

   https://doi.org/10.5194/amt-14-3737-2021  

   Staebell, Carly; Sun, Kang; Samra, Jenna; Franklin, Jonathan; Chan Miller, Christopher; Liu, Xiong; Conway, Eamon; Chance, Kelly; Milligan, Scott; Wofsy, Steven (January 2021, Atmospheric Measurement Techniques)

   Abstract. MethaneAIR is the airborne simulator of MethaneSAT, an area-mapping satellite currently under development with the goal of locating and quantifying large anthropogenic CH4 point sources as well as diffuse emissions at the spatial scale of an oil and gas basin. Built to closely replicate the forthcoming satellite, MethaneAIR consists of two imaging spectrometers. One detects CH4 and CO2 absorption around 1.65 and 1.61 µm, respectively, while the other constrains the optical path in the atmosphere by detecting O2 absorption near 1.27 µm. The high spectral resolution and stringent retrieval accuracy requirements of greenhouse gas remote sensing in this spectral range necessitate a reliable spectral calibration. To this end, on-ground laboratory measurements were used to derive the spectral calibration of MethaneAIR, serving as a pathfinder for the future calibration of MethaneSAT. Stray light was characterized and corrected for through fast-Fourier-transform-based Van Cittert deconvolution. Wavelength registration was examined and found to be best described by a linear relationship for both bands with a precision of ∼ 0.02 spectral pixel. The instrument spectral spread function (ISSF), measured with fine wavelength steps of 0.005 nm near a series of central wavelengths across each band, was oversampled to construct the instrument spectral response function (ISRF) at each central wavelength and spatial pixel. The ISRFs were smoothed with a Savitzky–Golay filter for use in a lookup table in the retrieval algorithm. The MethaneAIR spectral calibration was evaluated through application to radiance spectra from an instrument flight over the Colorado Front Range. 

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