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1. Noncrossing partitions of an annulus (Extended abstract)

   Brestensky, Laura; Reading, Nathan (April 2023, The Séminaire Lotharingien de Combinatoire)

   The noncrossing partition poset associated to a Coxeter group $$W$$ and Coxeter element $$c$$ is the interval $$[1,c]_T$$ in the absolute order on $$W$$. We construct a new model of noncrossing partititions for $$W$$ of classical affine type, using planar diagrams. The model in type $$\afftype{A}$$ consists of noncrossing partitions of an annulus. In type~$$\afftype{C}$$, the model consists of symmetric noncrossing partitions of an annulus or noncrossing partitions of a disk with two orbifold points. Following the lead of McCammond and Sulway, we complete $$[1,c]_T$$ to a lattice by factoring the translations in $$[1,c]_T$$, but the combinatorics of the planar diagrams leads us to make different choices about how to factor. 

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2. The Chow t-structure on the ∞-category of motivic spectra

   https://doi.org/10.4007/annals.2022.195.25  

   Kong, Hana J.; Wang, Guozhen; Xu, Zhouli (February 2022, Annals of mathematics)

   We define the Chow t-structure on the ∞-category of motivic spectra SH(k) over an arbitrary base field k. We identify the heart of this t-structure SH(k)c♡ when the exponential characteristic of k is inverted. Restricting to the cellular subcategory, we identify the Chow heart SH(k)cell,c♡ as the category of even graded MU2∗MU-comodules. Furthermore, we show that the ∞-category of modules over the Chow truncated sphere spectrum 1c=0 is algebraic. Our results generalize the ones in Gheorghe–Wang–Xu in three aspects: to integral results; to all base fields other than just C; to the entire ∞-category of motivic spectra SH(k), rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field k using the Postnikov–Whitehead tower associated to the Chow t-structure and the motivic Adams spectral sequences over k. 

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3. Categories of hypermagmas, hypergroups, and related hyperstructures

   https://doi.org/10.1016/j.jalgebra.2025.03.056  

   Nakamura, So; Reyes, Manuel L (August 2025, Journal of Algebra)

   In order to diagnose the cause of some defects in the category of canonical hypergroups, we investigate several categories of hyperstructures that generalize hypergroups. By allowing hyperoperations with possibly empty products, one obtains categories with desirable features such as completeness and cocompleteness, free functors, regularity, and closed monoidal structures. We show by counterexamples that such constructions cannot be carried out within the category of canonical hypergroups. This suggests that (commutative) unital, reversible hypermagmas—which we call mosaics—form a worthwhile generalization of (canonical) hypergroups from the categorical perspective. Notably, mosaics contain pointed simple matroids as a subcategory, and projective geometries as a full subcategory. 

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4. Rates of convergence in $$W^2_p$$-norm for the Monge-Amp\`ere equation.

   Neilan, Michael; Zhang, Wujun (October 2018, SIAM journal on numerical analysis)

   We develop discrete $$W^2_p$$-norm error estimates for the Oliker--Prussner method applied to the Monge--Ampère equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function contains information on its second-order difference. In addition, we show that the size of the complement of the contact set is controlled by the consistency of the method. Combining both observations, we show that the error estimate $$\| u - u_h \|_{W^2_{f,p}} (\mathcal{N}^I_h)$$ converges in order $$O(h^{1/p})$$ if $p > d$ and converges in order $$O(h^{1/d} \ln (\frac 1 h)^{1/d})$$ if $$p \leq d$$, where $$\|\cdot\|_{W^2_{f,p}(\mathcal{N}^I_h)}$$ is a weighted $$W^2_p$$-type norm, and the constant $C>0$ depends on $$\|{u}\|_{C^{3,1}(\bar\Omega)}$$, the dimension $$d$$, and the constant $$p$$. Numerical examples are given in two space dimensions and confirm that the estimate is sharp in several cases. 

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5. Measurement of the production cross section for a W boson in association with a charm quark in proton–proton collisions at $$\sqrt{s} = 13\,\hbox {TeV}$$

   https://doi.org/10.1140/epjc/s10052-023-12258-4  

   Tumasyan, A; Adam, W; Andrejkovic, J W; Bergauer, T; Chatterjee, S; Damanakis, K; Dragicevic, M; Valle, A_Escalante Del; Hussain, P S; Jeitler, M; et al (January 2024, The European Physical Journal C)

   Abstract The strange quark content of the proton is probed through the measurement of the production cross section for a W boson and a charm (c) quark in proton–proton collisions at a center-of-mass energy of 13$$\,\text {Te}\hspace{-.08em}\text {V}$$ $\text{Te}\text{V}$. The analysis uses a data sample corresponding to a total integrated luminosity of 138$$\,\text {fb}^{-1}$$ ${\text{fb}}^{-1}$collected with the CMS detector at the LHC. The W bosons are identified through their leptonic decays to an electron or a muon, and a neutrino. Charm jets are tagged using the presence of a muon or a secondary vertex inside the jet. The$$\hbox {W}+\hbox {c}$$ $\text{W}+\text{c}$production cross section and the cross section ratio$$R_\textrm{c}^{\pm }= \sigma ({\hbox {W}}^{+}+\bar{\text {c}})/\sigma (\hbox {W}^{-}+{\textrm{c}})$$ $R_{\text{c}}^{\pm }=\sigma ({\text{W}}^{+}+\overline{\text{c}})/\sigma ({\text{W}}^{-}+\text{c})$are measured inclusively and differentially as functions of the transverse momentum and the pseudorapidity of the lepton originating from the W boson decay. The precision of the measurements is improved with respect to previous studies, reaching 1% in$$R_\textrm{c}^{\pm }= 0.950 \pm 0.005\,\text {(stat)} \pm 0.010 \,\text {(syst)} $$ $R_{\text{c}}^{\pm }=0.950\pm 0.005\text{(stat)}\pm 0.010\text{(syst)}$. The measurements are compared with theoretical predictions up to next-to-next-to-leading order in perturbative quantum chromodynamics. 

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