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1. A Weighted Formulation of Refined Decoupling and Inequalities of Mizohata–Takeuchi-Type for the Moment Curve

   https://doi.org/10.1007/s12220-025-02262-3  

   Carbery, Anthony; Li, Zane Kun; Pang, Yixuan; Yung, Po-Lam (February 2026, The Journal of Geometric Analysis)

   Abstract Let$$\Gamma $$ $\Gamma$ be a compact patch of a well-curved$$C^{n+1}$$ $C^{n+1}$ curve in$$\mathbb {R}^n$$ $R^n$ with induced Lebesgue measure$$\textrm{d} \lambda $$ $\text{d}\lambda$ , and let$$g \mapsto \widehat{g \,\textrm{d}\lambda }$$ $g\mapsto \widehat{g\text{d}\lambda }$ be the Fourier extension operator for$$\Gamma $$ $\Gamma$ . Then we have, for arbitrary non-negative weightsw,$$\begin{aligned} \int _{B_R} |\widehat{g \,\textrm{d}\lambda }|^2w \le C_{n,a} R^{a} \sup _S \left( \int _S w\right) \int _\Gamma |g|^2 \, \textrm{d} \lambda \end{aligned}$$ $\begin{array}{c}{\int }_{B_R}|\widehat{g\text{d}\lambda }{|}^{2}w\le C_{n,a}{R}^a\underset{S}{sup}\left({\int }_S,w\right){\int }_{\Gamma }{\left|g\right|}^2\text{d}\lambda \end{array}$ for any$$a> \frac{n-3}{2} + \frac{2}{n} - \frac{2}{n^2(n+1)}$$ $a>\frac{n-3}{2}+\frac{2}{n}-\frac{2}{n^2(n+1)}$ , where the$$\sup $$ $sup$ is over all 1-neighbourhoodsSof hyperplanes whose normals are parallel to the tangent at some point of$$\Gamma $$ $\Gamma$ . This represents partial progress on the Mizohata–Takeuchi conjecture for curves in dimensions$$n \ge 3$$ $n\ge 3$ , improving upon the exponent$$a=n-1$$ $a=n-1$ which can be obtained as a consequence of the Agmon–Hörmander trace inequality. Our main tool in establishing this inequality will be a weighted formulation of refined decoupling for well-curved curves. We also discuss the sharpness of the exponents we obtain in this and in auxiliary results, and further explore this in the context of axiomatic decoupling for curves. 

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2. Don’t Follow Reinforcement Learning Blindly: Lower Sample Complexity of Learning Optimal Inventory Control Policies with Fixed Ordering Cost

   https://doi.org/10.1177/10591478251378851  

   Fan, Xiaoyu; Chen, Boxiao; Olsen, Tava; Qin, Hanzhang; Zhou, Zhengyuan (April 2026, Production and Operations Management)

   We study the sample complexity of offline learning for a class of structured Markov decision processes (MDPs) describing the inventory control system with fixed ordering cost/setup cost, a fundamental problem in supply chains. We find that a naive plug-in sampling-based approach applied to the inventory MDPs leads to strictly lower sample complexity bounds compared to the optimal bounds recently obtained for the general MDPs. More specifically, in the infinite-horizon discounted cost setting, we obtain an $\tilde{O}(min\{\frac{(\overline{S}-\underset{\_}{s}{)}^2}{(1-\gamma {)}^2{ϵ}^2},\frac{1}{(1-\gamma {)}^4{ϵ}^2}\})$ sample complexity bound, where ${(\overline{S}-\underset{\_}{s})}^2$ corresponds to the number of state-action pairs in a generic MDP with state space $\mathcal{S}$ and action space $\mathcal{A}$ . As such, $\tilde{O}\left(\frac{(\overline{S}-\underset{\_}{s}{)}^2}{(1-\gamma {)}^2{ϵ}^2}\right)$ improves on the optimal generic reinforcement learning (RL) bound $\tilde{\mathrm{\Theta }}\left(\frac{(\overline{S}-\underset{\_}{s}{)}^2}{(1-\gamma {)}^3{ϵ}^2}\right)$ (when directly applying $\tilde{\mathrm{\Theta }}\left(\frac{|\mathcal{S}||\mathcal{A}|}{(1-\gamma {)}^3{ϵ}^2}\right)$ here) by a factor of $(1-\gamma {)}^{-1}$ , and $\tilde{O}\left(\frac{1}{(1-\gamma {)}^4{ϵ}^2}\right)$ is able to completely remove the dependence on state and action cardinality. In the infinite-horizon average cost setting, we obtain an $\tilde{O}\left(\frac{(\overline{S}-\underset{\_}{s}{)}^2}{{ϵ}^2}\right)$ bound, improving on the generic optimal RL bound $\tilde{\mathrm{\Theta }}\left(\frac{(\overline{S}-\underset{\_}{s}{)}^2{t}_{mix}}{{ϵ}^2}\right)$ (when directly applying $\tilde{\mathrm{\Theta }}\left(\frac{|\mathcal{S}||\mathcal{A}|t_{mix}}{{ϵ}^2}\right)$ here) by a factor of $t_{mix}$ , and hence removing the mixing time dependence. By carefully leveraging the structural properties of the inventory dynamics in various settings, we are able to improve on those “best-possible” bounds developed in the RL literature. Our results demonstrate the drawbacks one could face by blindly following RL algorithms and the necessity of designing sample efficient algorithms that properly incorporate the special structures of the inventory systems. 

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3. On illposedness of the Hall and electron magnetohydrodynamic equations without resistivity on the whole space

   https://doi.org/10.1088/1361-6544/ae6c8a  

   Jeong, In-Jee; Oh, Sung-Jin (June 2026, Nonlinearity)

   Abstract It has been shown in Jeong and Oh (2019 On the Cauchy problem for the Hall and electron magnetohy-drodynamic equations without resistivity I: illposedness near degenerate stationary solutionsAnn. PDE815) that the incompressible and irresistive Hall- and electron-magneto-hydrodynamic (MHD) equations are illposed on flat domains $M={\mathbb{R}}^k\times {\mathbb{T}}^{3-k}$ for $0\text{⩽}k\text{⩽}2$ . The data and solutions therein were assumed to be independent of one coordinate, which not only significantly simplifies the systems but also allows for a large class of steady states. In this work, we remove the assumption of independence and conclude strong illposedness for compactly supported data in ${\mathbb{R}}^3$ . This is achieved by constructing degenerating wave packets for linearized systems around time-dependent axisymmetric magnetic fields. A few main additional ingredients are: a more systematic application of the generalized energy estimate, use of the Bogovskiǐ operator, anda prioriestimates for axisymmetric solutions to the Hall- and electron-MHD systems. 

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4. Comments on entire functions of the derivative operator

   https://doi.org/10.1088/1361-6382/ae9110  

   Woodard, R P (August 2026, Classical and Quantum Gravity)

   Abstract Many attempts to introduce fundamental nonlocality into quantum (or classical) field theory are based on the assumption that exponentials of the d’Alembertian are positive-definite, so that these operators can be employed without engendering the Ostrogradskian instability associated with higher derivative Lagrangians.This assumption is false.Working in the simple context of a 1-dimensional, point particle $q\left(t\right)$ , I demonstrate that the equation $\exp \left[T^2\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}\right]q\left(t\right)=0$ has an infinite number of rapidly oscillating, exponentially rising and falling solutions. This infinite kernel is in one-to-one correspondence with the ability to specify ‘initial value data’arbitrarilyoveranyfinite interval $t_1<t<t_2$ . 

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5. Magnetotransport properties of layered Fe ₃ GeTe ₂ crystals

   https://doi.org/10.1088/2515-7639/ae5345  

   Weerahennedige, Hiruni; Karki, Yog; Aljeailan, Huda; Weeraratna, Hansaka; Ronau, Zane; Zeltmann, Steven E; Muller, David A; Jasinski, Jacek B; Sumanasekera, Gamini (April 2026, Journal of Physics: Materials)

   Abstract Van der Waals layered magnetic materials have recently received significant attention for their ability to exhibit antiferromagnetic or ferromagnetic (FM) properties, even at the few-layer or monolayer scale. Among them, Fe₃GeTe₂is one of the most extensively studied systems, crystallizing in a hexagonal structure as an itinerant FM with a Curie temperature (T_C) of ∼220 K in bulk form and strong magnetic anisotropy. In this study, temperature and magnetic field dependence of the four-probe resistance ( $R_{xx}),$ thermopower (TEP) (S), and Hall resistance ( $R_{xy}$ ) were investigated in thick Fe₃GeTe₂flakes with different thicknesses to understand electron and spin transport, as well as spin and magnetic states. $R_{xx}$ decreased with decreasing temperature, confirming metallic behavior, consistent with the observed reduction in the magnitude of the negative TEP. Negative magnetoresistance (MR) with the magnetic field normal to the sample plane exhibited a quadratic field dependence belowT_C. An anomalous Hall effect was observed belowT_C, where $R_{xy}\left(B\right)$ showed a linear field dependence at low fields and saturation at higher fields. The anomalous Hall resistance ( $R_{xy}^A$ ) followed a dependence of $\alpha R_{xx}+\beta R_{xx}^2$ . A positive in-plane MR was observed when the current was perpendicular to the magnetic field, attributed to increased scattering from the enhanced Lorentz force and related orbital effects. Additionally, a hysteresis behavior was observed when cycling the in-plane magnetic field, likely due to the delay in domain alignment in response to the changing field. 

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