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1. The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum

   https://doi.org/10.1090/tran/8822  

   Blumberg, Andrew; Mandell, Michael (January 2022, Transactions of the American Mathematical Society)

   Let p ∈ Z p\in {\mathbb {Z}} be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum S {\mathbb {S}} admits an “eigensplitting” that generalizes known splittings on K K -theory and T C TC . We identify the summands in the fiber as the covers of Z p {\mathbb {Z}}_{p} -Anderson duals of summands in the K ( 1 ) K(1) -localized algebraic K K -theory of Z {\mathbb {Z}} . Analogous results hold for the ring Z {\mathbb {Z}} where we prove that the K ( 1 ) K(1) -localized fiber sequence is self-dual for Z p {\mathbb {Z}}_{p} -Anderson duality, with the duality permuting the summands by i ↦ p − i i\mapsto p-i (indexed mod p − 1 p-1 ). We explain an intrinsic characterization of the summand we call Z Z in the splitting T C ( Z ) p ∧ ≃ j ∨ Σ j ′ ∨ Z TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j’\vee Z in terms of units in the p p -cyclotomic tower of Q p {\mathbb {Q}}_{p} . 

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2. THE BOUNDARY OF THE p -RANK STRATUM OF THE MODULI SPACE OF CYCLIC COVERS OF THE PROJECTIVE LINE

   https://doi.org/10.1017/nmj.2022.12  

   OZMAN, EKIN; PRIES, RACHEL; WEIR, COLIN (December 2022, Nagoya Mathematical Journal)

   Abstract. We study the p-rank stratification of the moduli space of cyclic degree ! covers of the projective line in characteristic p for distinct primes p and !. The main result is about the intersection of the p-rank 0 stratum with the boundary of the moduli space of curves. When ! = 3 and p ≡ 2 mod 3 is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic p, for every genus g, signature type (r,s), and p-rank f satisfying the clear necessary conditions. 

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3. The Importance of Archiving the Seafloor

   https://doi.org/10.1029/2024EO240020  

   DiCenzo, Christina; Kelley, Katherine A; Anest, Nichole; Fritz, Cara; Donnelly, Jeff (January 2024, Eos)

   Marine geological sample repositories are vital for ocean science, climate change studies, and more. The value of their collections is growing amid efforts to meet rising demand for their services. 

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4. Convergence science in the Anthropocene: Navigating the known and unknown

   https://doi.org/10.1002/pan3.10069  

   Angeler, David G.; Allen, Craig R.; Carnaval, Ana; Palmer, ed., Clare (February 2020, People and Nature)

   Abstract Rapidly changing ecological and social systems currently pose significant societal challenges. Navigating the complexity of social‐ecological change requires approaches able to cope with, and potentially solve, both foreseen and unforeseen societal challenges.The emergent field of convergence addresses the intricacies of such challenges, and is thus relevant to a broad range of interdisciplinary issues.This paper suggests a way to conceptualize convergence research. It discusses how it relates to two major societal challenges (adaptation, transformation), and to the generation of policy‐relevant science. It also points out limitations to the further development of convergence research. A freePlain Language Summarycan be found within the Supporting Information of this article. 

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5. On the Determinant Problem for the Relativistic Boltzmann Equation

   https://doi.org/10.1007/s00220-021-04101-2  

   Chapman, James; Jang, Jin Woo; Strain, Robert M. (June 2021, Communications in Mathematical Physics)

   Abstract This article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular we mention the widely used cancellation lemma [1]. In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum p to the post collisional momentum $$p'$$ p ′ ; specifically we calculate the determinant for $$p\mapsto u = \theta p'+\left( 1-\theta \right) p$$ p ↦ u = θ p ′ + 1 - θ p for $$\theta \in [0,1]$$ θ ∈ [ 0 , 1 ] . Afterwards we give an upper-bound for this determinant that has no singularity in both p and q variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (1991) [8] and Guo-Strain (2012) [12]. These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates. 

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