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1. Global well-posedness for 2D generalized parabolic Anderson model via paracontrolled calculus

   https://doi.org/10.1007/s40072-025-00358-z  

   Shen, Hao; Zhu, Rongchan; Zhu, Xiangchan (May 2025, Stochastics and Partial Differential Equations: Analysis and Computations)

   Abstract This article revisits the problem of global well-posedness for the generalized parabolic Anderson model on$$\mathbb {R}^+\times \mathbb {T}^2$$ $R^{+}\times T^2$within the framework of paracontrolled calculus (Gubinelli et al. in Forum Math, 2015). The model is given by the equation:$$\begin{aligned} (\partial _t-\Delta ) u=F(u)\eta \end{aligned}$$ $\begin{array}{c}({\partial }_t-\Delta )u=F\left(u\right)\eta \end{array}$where$$\eta \in C^{-1-\kappa }$$ $\eta \in C^{-1-\kappa }$with$$1/6>\kappa >0$$ $1/6>\kappa >0$, and$$F\in C_b^2(\mathbb {R})$$ $F\in C_b^2\left(R\right)$. Assume that$$\eta \in C^{-1-\kappa }$$ $\eta \in C^{-1-\kappa }$and can be lifted to enhanced noise, we derive new a priori bounds. The key idea follows from the recent work by Chandra et al. (A priori bounds for 2-d generalised Parabolic Anderson Model,,2024), to represent the leading error term as a transport type term, and our techniques encompass the paracontrolled calculus, the maximum principle, and the localization approach (i.e. high-low frequency argument). 

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2. Computing Euler factors of genus 2 curves at odd primes of almost good reduction

   https://doi.org/10.1007/s40993-024-00605-7  

   Maistret, Céline; Sutherland, Andrew V (March 2025, Research in Number Theory)

   Abstract We present an efficient algorithm to compute the Euler factor of a genus 2 curve$$C/\mathbb {Q}$$ $C/Q$at an odd primepthat is of bad reduction forCbut of good reduction for the Jacobian ofC(a prime of “almost good” reduction). Our approach is based on the theory of cluster pictures introduced by Dokchitser, Dokchitser, Maistret, and Morgan, which allows us to reduce the problem to a short, explicit computation over$$\mathbb {Z}$$ $Z$and$$\mathbb {F}_p$$ $F_p$, followed by a point-counting computation on two elliptic curves over$$\mathbb {F}_p$$ $F_p$, or a single elliptic curve over$$\mathbb {F}_{p^2}$$ $F_{p^2}$. A key feature of our approach is that we avoid the need to compute a regular model forC. This allows us to efficiently compute many examples that are infeasible to handle using the algorithms currently available in computer algebra systems such as Magma and Pari/GP. 

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3. Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

   https://doi.org/10.1007/s00454-022-00426-4  

   Greenfeld, Rachel; Tao, Terence (January 2023, Discrete & Computational Geometry)

   Abstract We construct an example of a group$$G = \mathbb {Z}^2 \times G_0$$ $G=Z^2\times G_0$for a finite abelian group $$G_0$$ $G_0$, a subsetEof $$G_0$$ $G_0$, and two finite subsets$$F_1,F_2$$ $F_1,F_2$of G, such that it is undecidable in ZFC whether$$\mathbb {Z}^2\times E$$ $Z^2\times E$can be tiled by translations of$$F_1,F_2$$ $F_1,F_2$. In particular, this implies that this tiling problem isaperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings ofEby the tiles$$F_1,F_2$$ $F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $$\mathbb {Z}^2$$ $Z^2$). A similar construction also applies for$$G=\mathbb {Z}^d$$ $G=Z^d$for sufficiently large d. If one allows the group$$G_0$$ $G_0$to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles. 

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4. Abelian varieties of prescribed order over finite fields

   https://doi.org/10.1007/s00208-024-03084-4  

   van_Bommel, Raymond; Costa, Edgar; Li, Wanlin; Poonen, Bjorn; Smith, Alexander (March 2025, Mathematische Annalen)

   Abstract Given a prime powerqand$$n \gg 1$$ $n\gg 1$, we prove that every integer in a large subinterval of the Hasse–Weil interval$$[(\sqrt{q}-1)^{2n},(\sqrt{q}+1)^{2n}]$$ $[{(\sqrt{q}-1)}^{2n},{(\sqrt{q}+1)}^{2n}]$is$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$for some ordinary geometrically simple principally polarized abelian varietyAof dimensionnover$${\mathbb {F}}_q$$ $F_q$. As a consequence, we generalize a result of Howe and Kedlaya for$${\mathbb {F}}_2$$ $F_2$to show that for each prime powerq, every sufficiently large positive integer is realizable, i.e.,$$\#A({\mathbb {F}}_q)$$ $\#A\left(F_q\right)$for some abelian varietyAover$${\mathbb {F}}_q$$ $F_q$. Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse–Weil interval. A separate argument determines, for fixedn, the largest subinterval of the Hasse–Weil interval consisting of realizable integers, asymptotically as$$q \rightarrow \infty $$ $q\to \infty$; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if$$q \le 5$$ $q\le 5$, then every positive integer is realizable, and for arbitraryq, every positive integer$$\ge q^{3 \sqrt{q} \log q}$$ $\ge q^{3\sqrt{q}logq}$is realizable. 

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5. Investigating $D^0$ meson production in p-Pb collisions at 5.02 TeV with a multi-phase transport model

   https://doi.org/10.1140/epjc/s10052-024-13265-9  

   Zhang, Chao; Zheng, Liang; Shi, Shusu; Lin, Zi-Wei (September 2024, The European Physical Journal C)

   Abstract We study the production of$$D^0$$ $D^0$meson inp+pandp-Pb collisions using the improved AMPT model considering both coalescence and independent fragmentation of charm quarks after the Cronin broadening is included. After a detailed discussion of the improvements implemented in the AMPT model for heavy quark production, we show that the modified AMPT model can provide a good description of$$D^0$$ $D^0$meson spectra inp-Pb collisions, the$$Q_{\textrm{pPb}}$$ $Q_{\text{pPb}}$data at different centralities and$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$data in both mid- and forward (backward) rapidities. We also studied the effects of nuclear shadowing and parton cascade on the rapidity dependence of$$D^{0}$$ $D^0$meson production and$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$. Our results indicate that using the same strength of the Cronin effect (i.e$$\delta $$ $\delta$value) as that obtained from the mid-rapidity data leads to a considerable overestimation of the$$D^0$$ $D^0$meson spectra and$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$data at high$$p_{\textrm{T}}$$ $p_{\text{T}}$in the backward rapidity. As a result, the$$\delta $$ $\delta$is determined via a$$\chi ^2$$ ${\chi }^2$fitting of the$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$data across various rapidities. This work lays the foundation for a better understanding of cold-nuclear-matter (CNM) effects in relativistic heavy-ion collisions. 

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