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1. On Waring’s problem for larger powers

   https://doi.org/10.1515/crelle-2023-0072  

   Brüdern, Jörg; Wooley, Trevor D. (November 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let $G\left(k\right)${G(k)}denote the least numbershaving the property that everysufficiently large natural number is the sum of at mostspositive integralk-th powers.Then for all $k\in \mathbb{N}${k\in\mathbb{N}}, one has $G\left(k\right)⩽\lceil k\left(\log k+4.20032\right)\rceil .$G(k)\leqslant\lceil k(\log k+4.20032)\rceil. Our new methods improve on all bounds available hitherto when $k⩾14${k\geqslant 14}. 

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2. Existence of global weak solutions of $ p $-Navier-Stokes equations

   https://doi.org/10.3934/dcdsb.2021051  

   Liu, Jian-Guo; Zhang, Zhaoyun (January 2022, Discrete & Continuous Dynamical Systems - B)

   This paper investigates the global existence of weak solutions for the incompressible \begin{document}$ p $$\end{document}-Navier-Stokes equations in \begin{document}$$ \mathbb{R}^d $$\end{document} \begin{document}$$ (2\leq d\leq p) $$\end{document}. The \begin{document}$$ p $$\end{document}-Navier-Stokes equations are obtained by adding viscosity term to the \begin{document}$$ p $$\end{document}-Euler equations. The diffusion added is represented by the \begin{document}$$ p $$\end{document}-Laplacian of velocity and the \begin{document}$$ p $$\end{document}-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-\begin{document}$$ p $$\end{document}$ distances with constraint density to be characteristic functions. 

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3. Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors

   https://doi.org/10.1515/crelle-2023-0043  

   Argüz, Hülya; Bousseau, Pierrick (July 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Cluster varieties come in pairs: for any $\mathcal{𝒳}${\mathcal{X}}cluster variety there is an associated Fock–Goncharov dual $\mathcal{𝒜}${\mathcal{A}}cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. Particularly, we show that the mirror to the $\mathcal{𝒳}${\mathcal{X}}cluster variety is a degeneration of the Fock–Goncharov dual $\mathcal{𝒜}${\mathcal{A}}cluster varietyand vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi–Yau varieties obtained as blow-ups of toric varieties. 

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4. Realizing arbitrary $$d$$-dimensional dynamics by renormalization of $C^d$-perturbations of identity

   https://doi.org/10.3934/dcds.2021129  

   Fayad, Bassam; Saprykina, Maria (January 2022, Discrete & Continuous Dynamical Systems)

   Any \begin{document}$ C^d $$\end{document} conservative map \begin{document}$$ f $$\end{document} of the \begin{document}$$ d $$\end{document}-dimensional unit ball \begin{document}$$ {\mathbb B}^d $$\end{document}, \begin{document}$$ d\geq 2 $$\end{document}, can be realized by renormalized iteration of a \begin{document}$$ C^d $$\end{document} perturbation of identity: there exists a conservative diffeomorphism of \begin{document}$$ {\mathbb B}^d $$\end{document}, arbitrarily close to identity in the \begin{document}$$ C^d $$\end{document} topology, that has a periodic disc on which the return dynamics after a \begin{document}$$ C^d $$\end{document} change of coordinates is exactly \begin{document}$$ f $$\end{document}$. 

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5. Thermodynamic formalism for dispersing billiards

   https://doi.org/10.3934/jmd.2022013  

   Baladi, Viviane; Demers, Mark F. (January 2022, Journal of Modern Dynamics)

   For any finite horizon Sinai billiard map \begin{document}$ T $$\end{document} on the two-torus, we find \begin{document}$$ t_*>1 $$\end{document} such that for each \begin{document}$$ t\in (0,t_*) $$\end{document} there exists a unique equilibrium state \begin{document}$$ \mu_t $$\end{document} for \begin{document}$$ - t\log J^uT $$\end{document}, and \begin{document}$$ \mu_t $$\end{document} is \begin{document}$$ T $$\end{document}-adapted. (In particular, the SRB measure is the unique equilibrium state for \begin{document}$$ - \log J^uT $$\end{document}.) We show that \begin{document}$$ \mu_t $$\end{document} is exponentially mixing for Hölder observables, and the pressure function \begin{document}$$ P(t) = \sup_\mu \{h_\mu -\int t\log J^uT d \mu\} $$\end{document} is analytic on \begin{document}$$ (0,t_*) $$\end{document}. In addition, \begin{document}$$ P(t) $$\end{document} is strictly convex if and only if \begin{document}$$ \log J^uT $$\end{document} is not \begin{document}$$ \mu_t $$\end{document}-a.e. cohomologous to a constant, while, if there exist \begin{document}$$ t_a\ne t_b $$\end{document} with \begin{document}$$ \mu_{t_a} = \mu_{t_b} $$\end{document}, then \begin{document}$$ P(t) $$\end{document} is affine on \begin{document}$$ (0,t_*) $$\end{document}. An additional sparse recurrence condition gives \begin{document}$$ \lim_{t\downarrow 0} P(t) = P(0) $$\end{document}$. 

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