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1. Coexistence Phenomena in the Hénon Family

   https://doi.org/10.1007/s00574-023-00345-9  

   Benedicks, Michael ; Palmisano, Liviana ( July 2023 , Bulletin of the Brazilian Mathematical Society, New Series)

   We study the classical Hénon family$$f_{a,b}:(x,y)\mapsto (1-ax^2+y,bx)$$$f_{a,b}:(x,y)\mapsto (1-a{x}^2+y,bx)$,$$0$0<a<2$,$$0$0<b<1$, and prove that given an integer$$k\ge 1$$$k\ge 1$, there is a set of parameters$$E_k$$$E_k$of positive two-dimensional Lebesgue measure so that$$f_{a,b}$$$f_{a,b}$, for$$(a,b)\in E_k$$$(a,b)\in E_k$, has at leastkattractive periodic orbits and one strange attractor of the type studied in Benedicks and Carleson (Ann Math (2) 133(1):73–169, 1991). A corresponding statement also holds for the Hénon-like families of Mora and Viana (Acta Math 171:1–71, 1993), and we use the techniques of Mora and Viana (1993) to study homoclinic unfoldings also in the case of the original Hénon maps. The final main result of the paper is the existence, within the classical Hénon family, of a positive Lebesgue measure set of parameters whose corresponding maps have two coexisting strange attractors.

    

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2. Distinct carbon fractions drive a generalisable two‐pool model of fungal necromass decomposition

   https://doi.org/10.1111/1365-2435.13728  

   See, Craig R. ; Fernandez, Chris W. ; Conley, Anna M. ; DeLancey, Lang C. ; Heckman, Katherine A. ; Kennedy, Peter G. ; Hobbie, Sarah E. ; García‐Palacios, ed., Pablo ( December 2020 , Functional Ecology)

   Fungi represent a rapidly cycling pool of carbon (C) and nitrogen (N) in soils. Understanding of how this pool impacts soil nutrient availability and organic matter fluxes is hindered by uncertainty regarding the dynamics and drivers of fungal necromass decomposition.

   Here we assessed the generality of common models for predicting mass loss during fungal necromass decomposition and linked the resulting parameters to necromass substrate chemistry. We decomposed 28 different types of fungal necromass in laboratory microcosms over a 90‐day period, measuring mass loss on all types, and N release on a subset of types. We characterised the initial chemistry of each necromass type using: (a) fibre analysis methods commonly used for plant tissues, (b) initial melanin and nitrogen (N) concentrations and (c) Fourier transform infrared (FTIR) spectroscopy to assess the presence of bonds associated with common biomolecules.

   We found universal support for an asymptotic model of decomposition, which assumes that fungal necromass consists of an exponentially decomposing ‘fast’ pool, and a ‘slow’ pool that decomposes at a rate approaching zero. The strongest predictor of the fast pool decay rate (k) was the proportion of cell soluble components, though initial N concentration also predictedk, albeit more weakly. The size of the slow pool was best predicted by the acid non‐hydrolysable fraction, which was positively correlated with melanin‐associated aromatics. Nitrogen dynamics varied by necromass type, ranging from net N release to net immobilisation. The maximum quantity of N immobilised was inversely related to cell soluble contents andk, as positively related to FTIR spectra associated with cell wall polysaccharides.

   Collectively, our results indicate that the decomposition of fungal necromass in soils can be described as having two distinct stages that are driven by different components of substrate C chemistry, with implications for rates of N availability and organic matter accumulation in soils.

   A freePlain Language Summarycan be found within the Supporting Information of this article.

    

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3. Weyl remainders: an application of geodesic beams

   https://doi.org/10.1007/s00222-023-01178-5  

   Canzani, Yaiza ; Galkowski, Jeffrey ( June 2023 , Inventiones mathematicae)

   Abstract We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold ( M ,  g ) of dimension n , let $$\Pi _\lambda $$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(-\Delta _g)$$ 1 [ 0 , λ 2 ] ( - Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty $$ λ → ∞ $$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$ ∫ W Π λ ( x , x ) d x = ( 2 π ) - n vol R n ( B ) vol g ( W ) λ n + O ( λ n - 1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams. 

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4. 3D Mirror Symmetry for Instanton Moduli Spaces

   https://doi.org/10.1007/s00220-023-04831-5  

   Koroteev, Peter ; Zeitlin, Anton M. ( September 2023 , Communications in Mathematical Physics)

   We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$$C^2$($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$${\text{Hilb}}^k\left[C^2\right]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$$C_{ħ}^{\times }$-action. First, we find a two-parameter family$$X_{k,l}$$$X_{k,l}$of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$${\text{Hilb}}^k\left[C^2\right]$is obtained via direct limit$$l\longrightarrow \infty $$$l⟶\infty$and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$$ħ$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$$P^2$with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

    

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5. Sustainable glucose-based block copolymers exhibit elastomeric and adhesive behavior

   https://doi.org/10.1039/c6py00700g  

   Nasiri, Mohammadreza ; Reineke, Theresa M. ( January 2016 , Polymer Chemistry)

   Herein, we present the direct modification of glucose, an abundant and inexpensive sugar molecule, to produce new sustainable and functional polymers. Glucose-6-acrylate-1,2,3,4-tetraacetate (GATA) has been synthesized and shown to provide a useful glassy component for developing an innovative family of elastomeric and adhesive materials. A series of diblock and triblock copolymers of GATA and n -butyl acrylate (n-BA) were created via Reversible Addition–Fragmentation Chain Transfer (RAFT) polymerization. Initially, poly(GATA)- b -poly(n-BA) copolymers were prepared using 4-cyano-4-[(ethylsulfanylthiocarbonyl)sulfanyl] pentanoic acid (CEP) as a chain transfer agent (CTA). These diblock copolymers demonstrated decomposition temperatures of 275 °C or greater and two glass transition temperatures ( T g ) around −45 °C and 100 °C corresponding to the PnBA and PGATA domains, respectively, as measured by differential scanning calorimetry (DSC). Triblock copolymers of GATA and n-BA, with moderate dispersities ( Đ = 1.15–1.29), were successfully synthesized when S , S -dibenzyl trithiocarbonate (DTC) was employed as the CTA. Poly(GATA)- b -poly(nBA)- b -poly(GATA) copolymers with 14–58 wt% GATA were prepared and demonstrated excellent thermomechanical properties ( T d ≥ 279 °C). Two well-separated glass transitions near the values for homopolymers of n-BA and GATA (∼−45 °C and ∼100 °C, respectively) were measured by DSC. The triblock with 14% GATA exhibited peel adhesion of 2.31 N cm −1 (when mixed with 30 wt% tackifier) that is superior to many commercial pressure sensitive adhesives (PSAs). Use of 3,5-bis(2-dodecylthiocarbonothioylthio-1oxopropoxy)benzoic acid (BTCBA) as the CTA provided a more efficient route to copolymerize GATA and n-BA. Using BTCBA, poly(GATA)- b -poly(nBA)- b -poly(GATA) triblock copolymers containing 12–25 wt% GATA, with very narrow molar mass distributions ( Đ ≤ 1.08), were prepared. The latter series of triblock copolymers showed excellent thermal stability with T d ≥ 275 °C. Only the T g for the PnBA block was observed by DSC (∼−45 °C), however, phase-separation was confirmed by small-angle X-ray scattering (SAXS) for all of these triblock copolymers. The mechanical behavior of the polymers was investigated by tensile experiments and the triblock with 25% GATA content demonstrated moderate elastomeric properties, 573 kPa stress at break and 171% elongation. This study introduces a new family of glucose-based ABA-type copolymers and demonstrates functionality of a glucose-based feedstock for developing green polymeric materials. 

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