For an $\ell$-adic sheaf on a variety of arbitrary dimension over a perfect field, we define the Swan class measuring the wild ramification as a 0-cycle class supported on the ramification locus. We prove a Lefschetz trace formula for open varieties and a generalization of the Grothendieck-Ogg-Shararevich formula using the Swan class.

In this study, arsenate, As(V), and fluoride (F) were simultaneously removed from contaminated water using MgO, Mg(OH)₂, and MgCO₃ as Mg-based adsorbents, as existing studies only focus on their individual removal. The removal performance of As(V) and F followed the order MgCO₃ < Mg(OH)₂ < MgO. Under the test conditions, MgO and Mg(OH)₂ met the environmental standards for As and F (0.01 and 0.8 mg/L, respectively), but MgCO₃ did not. The As(V) removal performance was not significantly affected by an increase in the initial F concentration. It was concluded that As(V) was adsorbed and removed more preferentially than F by Mg-based adsorbents because a considerable amount of F remained even when the majority of As(V) was removed. Most arsenic (As)-adsorption data for MgO fit the Langmuir and Freundlich models, whereas those for Mg(OH)₂ did not fit either model well. Additionally, the As-adsorption data for MgCO₃ fit the Freundlich model but not the Langmuir model. Most of the F-adsorption data for the Mg-based adsorbents fit the Langmuir and Freundlich models. The removal mechanisms of As(V) and F using Mg-based adsorbents were assumed to be predominantly caused by ion-exchange and chemical-adsorption reactions on the adsorbent surface because no magnesium arsenate, magnesium fluoride, or magnesium hydroxide fluoride species were observed in the X-ray diffraction analysis. This research advances the sustainable As–F simultaneous treatment method using inexpensive adsorbents.

Period integrals arise from the comparison between Betti cohomologies and de Rham cohomologies. In this paper, we give a formula for the determinant of period integrals in terms of the canonical pairing between Picard group of rank 1 realizations and relative Chow group. It is compatible with the similar formula for Galois modules proved by the first author, and as a consequence, under the assumption of Gersten resolution, the determinant of the cohomology of a determinantally motivic object is motivic.