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Abstract In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori ${𝕋}^d${\mathbb{T}^{d}}, where $d\ge 3${d\geq 3}. The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in [J. Bourgain and C. Demeter,The proof of the $l^2$l^{2}decoupling conjecture,Ann. of Math. (2) 182 2015, 1, 351–389]. As a comparison, this result can be regarded as a periodic analogue of [Y. Hong,Strichartz estimates forN-body Schrödinger operators with small potential interactions,Discrete Contin. Dyn. Syst. 37 2017, 10, 5355–5365] though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate. 

Abstract. We apply airborne measurements across three seasons(summer, winter and spring 2017–2018) in a multi-inversion framework toquantify methane emissions from the US Corn Belt and Upper Midwest, a keyagricultural and wetland source region. Combing our seasonal results withprior fall values we find that wetlands are the largest regional methanesource (32 %, 20 [16–23] Gg/d), while livestock (enteric/manure; 25 %,15 [14–17] Gg/d) are the largest anthropogenic source. Naturalgas/petroleum, waste/landfills, and coal mines collectively make up theremainder. Optimized fluxes improve model agreement with independentdatasets within and beyond the study timeframe. Inversions reveal coherentand seasonally dependent spatial errors in the WetCHARTs ensemble meanwetland emissions, with an underestimate for the Prairie Pothole region butan overestimate for Great Lakes coastal wetlands. Wetland extent andemission temperature dependence have the largest influence on predictionaccuracy; better representation of coupled soil temperature–hydrologyeffects is therefore needed. Our optimized regional livestock emissionsagree well with the Gridded EPA estimates during spring (to within 7 %) butare ∼ 25 % higher during summer and winter. Spatial analysisfurther shows good top-down and bottom-up agreement for beef facilities (withmainly enteric emissions) but larger (∼ 30 %) seasonaldiscrepancies for dairies and hog farms (with > 40 % manureemissions). Findings thus support bottom-up enteric emission estimates butsuggest errors for manure; we propose that the latter reflects inadequatetreatment of management factors including field application. Overall, ourresults confirm the importance of intensive animal agriculture for regionalmethane emissions, implying substantial mitigation opportunities throughimproved management.