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1. Interplanetary magnetic correlation and low-frequency spectrum over many solar rotations

   https://doi.org/10.1073/pnas.2519811122  

   Wang, Jiaming; Pecora, Francesco; Chhiber, Rohit; Roy, Sohom; Matthaeus, William H (January 2026, Proceedings of the National Academy of Sciences)

   Fluctuations and structure across a wide range of spatial and temporal scales are frequently studied in the solar wind. The properties of the low-frequency fluctuations are of relevance to turbulent energy injection into the plasma and the transport of high-energy cosmic rays. Correlation analysis of decade-long intervals of interplanetary data permits study of fluctuations at time scales much longer than suitably defined correlation times, and therefore at frequencies well below those associated with the Kolmogorov inertial range of in situ turbulence. At the frequencies of interest, we study the familiar occurrence of the $1/f$ spectral signature. We also study point spectral features due to solar rotation and their relation with the $1/f$ signal. We report properties at timescales ranging from minutes up to years, using data selected by wind speed, phase of solar cycle, and cartesian components of the magnetic field. A surprising finding is that the power in solar rotation harmonics is consistent with an extension of the $1/f$ spectrum, down to frequencies as low as around $5\times {10}^{-7}\mathit{Hz}$ . The presence of a broadband $1/f$ spectrum across different wind types supports the interpretation that $1/f$ signals may be related to or even originate from the solar dynamo. 

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2. Evolution of the Interplanetary Turbulence and the Associated Turbulence Anisotropy in the Outer Heliosphere: VOYAGER 2 Observations

   https://doi.org/10.3847/1538-4357/ad2eae  

   Zhu, Xingyu; He, Jiansen; Zank, Gary P; Verscharen, Daniel; Zhao, Ling-Ling; Duan, Die; Lin, Rong (April 2024, The Astrophysical Journal)

   Abstract We study the radial evolution of the inertial-range solar wind plasma turbulence and its anisotropy in the outer heliosphere. We use magnetic field (B) measurements from the Voyager 2 spacecraft for heliocentric distancesRfrom 1 to 33 au. We find that the perpendicular and trace power spectral densities (PSDs) of the magnetic field ( $E_{{\mathrm{B}}_{\perp }}$ and $E_{{\mathrm{B}}_{\mathrm{Tr}}}$ ) still follow a Kolmogorov-like spectrum until 33 au. The parallel magnetic field PSD, $E_{B_{\parallel }}$ , transits from a power-law index of −2 to −5/3 as the distance crossesR∼ 10 au. The PSD at frequencies 0.01 Hz <f< 0.2 Hz flattens atR> 20 au, gradually approaching anf⁻¹spectrum, probably due to instrument noise. At 0.002 Hz <f< 0.1 Hz, quasi-parallel propagation dominates at 1 au <R< 7 au, with quasi-perpendicular propagation gradually emerging atR> 5 au. ForR> 7 au, oblique propagation becomes the primary mode of propagation. At smaller frequencies off< 0.01 Hz, $E_{{\mathrm{B}}_{\perp }}$ increases with propagation angle at 1 au <R< 5 au, and in contrast decreases with propagation angle atR> 5 au due to the enhanced power level at propagation angles smaller than 20°. Such enhancement may derive from the injection of wave energy from the pickup ion source into the background turbulent cascade, and the injected wave energy is transferred across scales without leaving local enhancements in $E_{{\mathrm{B}}_{\perp }}$ or $E_{{\mathrm{B}}_{\mathrm{Tr}}}$ . 

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3. 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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4. VC-Dimension of Hyperplanes Over Finite Fields

   https://doi.org/10.1007/s00373-025-02909-6  

   Ascoli, Ruben; Betti, Livia; Cheigh, Justin; Iosevich, Alex; Jeong, Ryan; Liu, Xuyan; McDonald, Brian; Milgrim, Wyatt; Miller, Steven J; Romero_Acosta, Francisco; et al (April 2025, Graphs and Combinatorics)

   Abstract Let$$\mathbb {F}_q^d$$ $F_q^d$ be thed-dimensional vector space over the finite field withqelements. For a subset$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$ and a fixed nonzero$$t\in \mathbb {F}_q$$ $t\in F_q$ , let$$\mathcal {H}_t(E)=\{h_y: y\in E\}$$ $H_t\left(E\right)=\{h_y:y\in E\}$ , where$$h_y:E\rightarrow \{0,1\}$$ $h_y:E\to \{0,1\}$ is the indicator function of the set$$\{x\in E: x\cdot y=t\}$$ $\{x\in E:x·y=t\}$ . Two of the authors, with Maxwell Sun, showed in the case$$d=3$$ $d=3$ that if$$|E|\ge Cq^{\frac{11}{4}}$$ $\left|E\right|\ge C{q}^{\frac{11}{4}}$ andqis sufficiently large, then the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$ is 3. In this paper, we generalize the result to arbitrary dimension by showing that the VC-dimension of$$\mathcal {H}_t(E)$$ $H_t\left(E\right)$ isdwhenever$$E\subseteq \mathbb {F}_q^d$$ $E\subseteq F_q^d$ with$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ $\left|E\right|\ge C_d{q}^{d-\frac{1}{d-1}}$ . 

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5. 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 (May 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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