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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. From the ashes of VEI: Decoding explosivity, mass eruption rate and hazards in real time

   https://doi.org/10.22541/essoar.176967627.71453176/v1  

   Sahagian, Dork (January 2026, AGU)

   Ash-producing volcanic eruptions are a major natural hazard. The explosivity of these eruptions has been described for decades by a semi-quantitative “Volcanic Explosivity Index (VEI)” based on a number of interdependent factors (eruption duration, volume of ejected material, eruptive column height, tropospheric and stratospheric injection, and many qualitative descriptors), most of which could only be assessed after eruption had ceased. The utility of this semi-quantitative index has been questioned since then, yet its common usage persists. We suggest that if VEI is to be used in the future, it may as well be an index that can be determined in real time during an eruption, and one that strictly reflects eruption intensity. We find that the same value for the “traditional” Volcanic Explosivity Index of past eruptions can be estimated in real time during an eruption, based solely on eruption column height. This approach is not intended to replace the traditional VEI scale or supplant more detailed mechanistic approaches to mass eruption rate (MER); rather it can be used to provide a more immediate way of calculating the VEI and linking to MER, in real time while an eruption is taking place. As such, if the easily observed height can be measured, both MER and VEI can be estimated. This may facilitate emergency response by aviation, hazard management, and others to mitigate societal impact of hazardous eruptions. This scheme can be inverted to use volume-based VEI of ancient eruptions to roughly retrodict unobserved column height for key historic and pre-historic eruptions. Further, ash particle morphology reflects explosivity in terms of decompression rate at the vent of unobserved eruptions. With sufficiently rapid decompression, this could result in an observable bi-modal bubble size distribution. The morphology of ash particles reveals a threshold between VEI 3 and 4, above which there is sufficiently rapid decompression to trigger a second nucleation event near the vent. 

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3. Computing the Caratheodory Number of a Point

   Bereg, Sergey; Haghpanah, Mohammadreza (August 2020, Canadian Conference on Computational Geometry)

   Caratheodory’s theorem says that any point in the convex hull of a set $$P$$ in $R^d$ is in the convex hull of a subset $P'$ of $$P$$ such that $$|P'| \le d + 1$$. For some sets P, the upper bound d + 1 can be improved. The best upper bound for P is known as the Caratheodory number [2, 15, 17]. In this paper, we study a computational problem of finding the smallest set $P'$ for a given set $$P$$ and a point $$p$$. We call the size of this set $P'$, the Caratheodory number of a point p or CNP. We show that the problem of deciding the Caratheodory number of a point is NP-hard. Furthermore, we show that the problem is k-LDT-hard. We present two algorithms for computing a smallest set $P'$, if CNP= 2,3. Bárány [1] generalized Caratheodory’s theorem by using d+1 sets (colored sets) such that their convex hulls intersect. We introduce a Colorful Caratheodory number of a point or CCNP which can be smaller than d+1. Then we extend our results for CNP to CCNP. 

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4. A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces

   https://doi.org/10.54330/afm.127419  

   Alvarado, Ryan; Hajłasz, Piotr; Malý, Lukáš (October 2022, Annales Fennici Mathematici)

   We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space \(X\) supports a \(p\)-Poincaré inequality, then the \(N^{1,p}(X)\) Sobolev space is reflexive and separable whenever \(p\in (1,\infty)\). We also prove separability of the space when \(p=1\). Our proof is based on a straightforward construction of an equivalent norm on \(N^{1,p}(X)\), \(p\in [1,\infty)\), that is uniformly convex when \(p\in (1,\infty)\). Finally, we explicitly construct a functional that is pointwise comparable to the minimal \(p\)-weak upper gradient, when \(p\in (1,\infty)\). 

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5. Updating SWAT+ to Clarify Understanding of In‐Stream Phosphorus Retention and Remobilization: SWAT+P.R&R

   https://doi.org/10.1029/2022WR033283  

   Wallington, Kevin; Cai, Ximing (March 2023, Water Resources Research)

   Abstract Efforts to reduce riverine phosphorus (P) loads have not been as fruitful as expected or hoped. One reason for the failure of these efforts appears to be that models used for watershed P management have understated and misrepresented the role of in‐stream processes in shaping watershed P export. Here, we update the latest release of the Soil and Water Assessment Tool (SWAT+), a widely used watershed management model, to better represent in‐stream P retention and remobilization (SWAT+P.R&R). We add new streambed pools where P is stored and tracked, and we incorporate three new processes driving in‐stream P dynamics: (a) deposition and resuspension of sediment‐associated P, (b) diffusion of dissolved P between the water column and streambed, and (c) adsorption and desorption of mineral P. The objective of this modeling work is to provide a diagnostic tool that enables researchers to challenge existing assumptions regarding how watersheds store, transform, and transport P. Here, in a first diagnostic analysis, SWAT+P.R&R helps reconcile in‐stream P retention theory (that P is retained at low flows and remobilized at high flows) and a discordant data set in our validation watershed. SWAT+P.R&R results (a) clarify that the theorized relationship between P retention and flow is only valid (for this point‐source affected testbed, at least) at the temporal scale of a single rising‐or‐falling hydrograph limb and (b) illustrate that hysteresis obscures the relationship at longer temporal scales. Future work using SWAT+P.R&R could further challenge assumptions regarding timescales of in‐stream P legacies and sources of P load variability. 

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