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1. 2-chains and square roots of Thompson’s group

   https://doi.org/10.1017/etds.2019.14  

   KOBERDA, THOMAS; LODHA, YASH (September 2020, Ergodic Theory and Dynamical Systems)

   We study 2-generated subgroups $$\langle f,g\rangle <\operatorname{Homeo}^{+}(I)$$ such that $$\langle f^{2},g^{2}\rangle$$ is isomorphic to Thompson’s group $$F$$ , and such that the supports of $$f$$ and $$g$$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with non-abelian free subgroups, examples which do not admit faithful actions by $$C^{2}$$ diffeomorphisms on 1-manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to Brin. We also show that many relatively uncomplicated groups of homeomorphisms can have very complicated square roots, thus establishing the behavior of square roots of $$F$$ as part of a general phenomenon among subgroups of $$\operatorname{Homeo}^{+}(I)$$ . 

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2. Contrasting adaptation and optimization of stomatal traits across communities at continental scale

   https://doi.org/10.1093/jxb/erac266  

   Liu, Congcong; Sack, Lawren; Li, Ying; He, Nianpeng (June 2022, Journal of Experimental Botany)

   Lawson, Tracy (Ed.)

   Abstract Shifts in stomatal trait distributions across contrasting environments and their linkage with ecosystem productivity at large spatial scales have been unclear. Here, we measured the maximum stomatal conductance (g), stomatal area fraction (f), and stomatal space-use efficiency (e, the ratio of g to f) of 800 plant species ranging from tropical to cold-temperate forests, and determined their values for community-weighted mean, variance, skewness, and kurtosis. We found that the community-weighted means of g and f were higher in drier sites, and thus, that drought ‘avoidance’ by water availability-driven growth pulses was the dominant mode of adaptation for communities at sites with low water availability. Additionally, the variance of g and f was also higher at arid sites, indicating greater functional niche differentiation, whereas that for e was lower, indicating the convergence in efficiency. When all other stomatal trait distributions were held constant, increasing kurtosis or decreasing skewness of g would improve ecosystem productivity, whereas f showed the opposite patterns, suggesting that the distributions of inter-related traits can play contrasting roles in regulating ecosystem productivity. These findings demonstrate the climatic trends of stomatal trait distributions and their significance in the prediction of ecosystem productivity. 

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3. On the Degree of Polynomials Computing Square Roots Mod p

   https://doi.org/10.4230/LIPIcs.CCC.2024.25  

   Kedlaya, Kiran S; Kopparty, Swastik (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Santhanam, Rahul (Ed.)

   For an odd prime p, we say f(X) ∈ F_p[X] computes square roots in F_p if, for all nonzero perfect squares a ∈ F_p, we have f(a)² = a. When p ≡ 3 mod 4, it is well known that f(X) = X^{(p+1)/4} computes square roots. This degree is surprisingly low (and in fact lowest possible), since we have specified (p-1)/2 evaluations (up to sign) of the polynomial f(X). On the other hand, for p ≡ 1 mod 4 there was previously no nontrivial bound known on the lowest degree of a polynomial computing square roots in F_p. We show that for all p ≡ 1 mod 4, the degree of a polynomial computing square roots has degree at least p/3. Our main new ingredient is a general lemma which may be of independent interest: powers of a low degree polynomial cannot have too many consecutive zero coefficients. The proof method also yields a robust version: any polynomial that computes square roots for 99% of the squares also has degree almost p/3. In the other direction, Agou, Deliglése, and Nicolas [Agou et al., 2003] showed that for infinitely many p ≡ 1 mod 4, the degree of a polynomial computing square roots can be as small as 3p/8. 

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4. Square-planar Co( iii ) in {O ₄ } coordination: large ZFS and reactivity with ROS

   https://doi.org/10.1039/c8cc04464c  

   Steele, Jennifer L.; Tahsini, Laleh; Sun, Chen; Elinburg, Jessica K.; Kotyk, Christopher M.; McNeely, James; Stoian, Sebastian A.; Dragulescu-Andrasi, Alina; Ozarowski, Andrew; Ozerov, M.; et al (October 2018, Chemical Communications)

   Oxidation of distorted square-planar perfluoropinacolate Co compound [Co II (pin F ) 2 ] 2− , 1 , to [Co III (pin F ) 2 ] 1− , 2 , is reported. Rigidly square-planar 2 has an intermediate-spin, S = 1, ground state and very large zero-field splitting (ZFS) with D = 67.2 cm −1 ; | E | = 18.0 cm −1 , ( E / D = 0.27), g ⊥ = 2.10, g ‖ = 2.25 and χ TIP = 1950 × 10 −6 cm 3 mol −1 . This Co( iii ) species, 2 , reacts with ROS to oxidise two (pin F ) 2− ligands to form tetrahedral [Co II (Hpfa) 4 ] 2− , 3 . 

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5. Face flips in origami tessellations

   Akitaya, Hugo A; Dujmović, Vida; Eppstein, David; Hull, Thomas C; Jain, Kshitij; Lubiw, Anna (January 2020, Journal of computational geometry)

   null (Ed.)

   Given a locally flat-foldable origami crease pattern $G=(V,E)$ (a straight-line drawing of a planar graph on the plane) with a mountain-valley (MV) assignment $$\mu:E\to\{-1,1\}$$ indicating which creases in $$E$$ bend convexly (mountain) or concavely (valley), we may \emph{flip} a face $$F$$ of $$G$$ to create a new MV assignment $$\mu_F$$ which equals $$\mu$$ except for all creases $$e$$ bordering $$F$$, where we have $$\mu_F(e)=-\mu(e)$$. In this paper we explore the configuration space of face flips that preserve local flat-foldability of the MV assignment for a variety of crease patterns $$G$$ that are tilings of the plane. We prove examples where $$\mu_F$$ results in a MV assignment that is either never, sometimes, or always locally flat-foldable, for various choices of $$F$$. We also consider the problem of finding, given two locally flat-foldable MV assignments $$\mu_1$$ and $$\mu_2$$ of a given crease pattern $$G$$, a minimal sequence of face flips to turn $$\mu_1$$ into $$\mu_2$$. We find polynomial-time algorithms for this in the cases where $$G$$ is either a square grid or the Miura-ori, and show that this problem is NP-complete in the case where $$G$$ is the triangle lattice. 

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