We study a numerical method, built on the tensor neural network (TNN) architecture introduced in \cite{wang2022tensor}, for solving nonlocal diffusion models in high-dimensional spaces. The tensor-product structure of the TNN ansatz, combined with the separability of the Gaussian kernel, reduces the high-dimensional integrals in the nonlocal energy to products of low-dimensional integrals, which are evaluated by Gauss--Legendre quadrature; nonseparable source and boundary data are handled by a TNN-based preconditioning step. For the Dirichlet boundary condition, we establish the asymptotically compatible $L^2$ error estimate \[ \|u_{\mathrm{loc}}-u_{δ,p}\|_{L^2(Ω)} \le C\!\left(\frac{\varepsilon_f}{\sqrtδ} +\frac{\varepsilon_g}δ +\frac{\varepsilon_u}{\sqrtδ} +η_{\mathrm{opt}}\right) +C\sqrtδ, \] where $\varepsilon_f$, $\varepsilon_g$ and $\varepsilon_u$ are the data and trial-class approximation errors and $η_{\mathrm{opt}}$ is the optimization residual. For the Neumann boundary condition, the $L^2$ estimate is improved to $O(\varepsilon_f+\varepsilon_g/\sqrtδ+\varepsilon_u +η_{\mathrm{opt}}+δ)$, and an $H^1$ gradient estimate is further obtained through a smoothing post-processing step. Numerical experiments on tensor-product domains up to $d=20$ support the theoretical results, and additional tests on two- and three-dimensional $L$-shaped domains demonstrate the practical robustness of the method beyond the smooth-domain setting covered by the analysis.

The additive noise mechanism is a foundational tool for differential privacy (DP) of $T$-dimensional real-valued vector queries. The Gaussian mechanism, utilizing Gaussian noise, is the mostly widely used such mechanism, due to its simplicity and strong privacy guarantees. In this work, we provide justification for this choice, showing that as the dimension $T\to\infty$, no additive-noise mechanism can asymptotically improve on the Gaussian mechanism's privacy--utility tradeoff for the strong privacy settings typically used.We also develop a new family of \emph{Spherical Generalized Gamma} DP mechanisms, which contains both the Gaussian mechanism and the recently studied $\ell_2$ mechanism (Joseph \emph{et al.}, ICML 2025). We identify members of this family that outperform both the Gaussian and $\ell_2$ mechanisms in certain low-dimensional settings, and show tight composition of all mechanisms in this family, answering an open question of Joseph \emph{et al.}~regarding the $\ell_2$ mechanism.

Given an algebra B over a field k, we study conditions under which a Lie subalgebra of Der(B) is locally finite as a set of derivations. As an application of our results, we show that if X is a quasi-affine variety over an arbitrary field k, and if L is a finitely generated solvable Lie subalgebra of Der O(X) consisting of locally finite derivations, then L is locally finite. If, moreover, k is algebraically closed and of characteristic zero, and X is irreducible and affine, then L is integrable.

Optimal subsampling efficiently selects the most informative data points, enabling accurate statistical inference while significantly reducing computational burden for massive datasets. However, the existing relevant methods can not directly be applied to pairwise loss problems, particularly for convoluted rank regression (CRR), due to the double summation structure in objective function. To this end, we first propose a new BIweighted Poisson Subsampling (BIPS) framework for such problems through designing a proper weight for a pair of observations instead of for a single observation for objective function. Two concrete inverse probability weighting strategies are considered. Secondly, we focus on the CRR models, under which the BIPS estimator (BIPS-CRR) is formulated. We establish consistency and asymptotic normality for BIPS-CRR, derive its optimal Poisson subsampling probabilities under the L-optimality criterion, and provide a practical algorithm to facilitate implementation. Thirdly, we develop a distributed estimator for CRR that incorporates BIPS as a pilot subsampling strategy. This estimation is globally efficient and is robust to both randomly and non-randomly distributed datasets in distributed computing environments. Extensive simulations and a real-world application demonstrate the excellent finite-sample performance of proposed methodology. Additionally, our BIPS can be readily extended to other U-statistics optimization problems and pairwise learning tasks.

Age verification is rapidly emerging as a central regulatory instrument for protecting minors online, with several jurisdictions mandating its deployment for access to adult and pornographic content. This regulatory direction raises significant privacy concerns, as it risks binding sensitive content access to identity-related attributes. It also introduces security risks, since age-verification mechanisms are often outsourced to third-party providers with limited transparency into the robustness of their verification processes. In this work, we conduct, to the best of our knowledge, the first exploratory security assessment of regulation-mandated age-verification mechanisms deployed by adult websites. Rather than treating age verification as a purely regulatory question, we empirically examine whether current deployments provide security guarantees commensurate with the privacy risks of relying on sensitive identity-related data. Our methodology combines ecosystem mapping, adversary modeling, and empirical testing across four countries, covering document-based verification, biometric age estimation, indirect signals, and website-workflow integration. Our results reveal systemic weaknesses across mechanisms and integrations under realistic threat assumptions, including failures against low-cost, widely accessible attacks. Finally, we derive concrete guidelines and design directions for mitigating the security and privacy risks exposed by current age-verification deployments.

We give a short analytic proof of Raposo's signed refinement of Pohst's inequality.

Muography, commonly known as muon tomography, is a passive, non-destructive imaging technique that utilizes naturally occurring cosmic-ray muons to visualize the internal density structures of large, static, or inaccessible objects. In course of developing a small prototype muography system for material identification that relies upon the multiple Coulomb scattering of muons in matter, we explored the possible use of Thick-GEM detector as muon tracking device. It is a 5-20 fold scaled-up version of traditional GEM technology, that has become increasingly popular in recent years, owing to its mechanical robustness, cost-effective production, and excellent position sensing capabilities. A few prototypes of this detector of dimension $40\,\mathrm{mm} \times 48\,\mathrm{mm}$ with variation in other design parameters, were manufactured from a local industry. Subsequent to conditioning, detailed characterization of the detectors was performed to validate their suitability in muography applications. To identify the optimal operating region, gain variation was studied under various voltage configurations for both single and double-stage configurations. Experimental measurement of muon detection efficiency across the entire operating range yielded a maximum efficiency of 99.5\% in both cases. Using a collimated Fe$^{55}$-source, the best spatial resolution was determined to be 30 $μ$m for both single and double-stage operation.

Motivated by polynomial identity testing with exponentials (Li and Wu, ITCS'26), we study uncertainty principles for the number-theoretic transform (NTT). We show that the NTT satisfies strong sparsity tradeoffs: For every fixed prime $q$ and for all but finitely many primes $p \equiv 1 \pmod q$ every nonzero $f\in \mathbb F_p^{\mathbb Z_q}$ and its number-theoretic transform $\hat f$ satisfy \[ |\mathrm{Supp}(f)| + |\mathrm{Supp}(\hat f)| \ge q+1. \] Thus, a $k$-sparse function has transform support at least $q-k+1$. As our main technical contribution, we prove a probabilistic version of the above uncertainty principle, averaged over primes $p$, in the regime $p=q^{O(1)}$. As an application, we obtain a black-box identity test for $k$-sparse exponential polynomials of degree at most $d$ with vanishing soundness error, for $q$ moderately larger than $k$.

Since its classical origin, Hamilton's principle has been formulated under an exact closure condition: admissible variations vanish at the boundaries of the variational domain. This condition removes the boundary term in the first variation of the action and yields the Euler--Lagrange equation. Although natural for isolated deterministic systems, fixed boundary admissibility is usually treated as a technical condition rather than as a physical closure hypothesis. Here we ask what follows when this hypothesis is made explicit and relaxed. We introduce \emph{variational openness} as the retention of the boundary contribution in the variational balance. The retained term defines a boundary-openness density, which must be projected onto admissible variations before it becomes a dynamical source. In this formulation, the classical Euler--Lagrange equation is recovered as the exact-closure limit of an open variational balance; the source term is therefore identified with incomplete variational closure rather than with an externally imposed force. The framework is illustrated through three elementary examples: an open harmonic oscillator, a finite-compliance boundary, and a delayed oscillator with memory. These examples show how boundary openness can produce forcing, partial closure, history dependence, and non-Markovian structure while preserving standard mechanics in the closed limit. The resulting perspective suggests that Hamiltonian mechanics may be understood as the mechanics of variationally closed systems and motivates an open Hamilton--Jacobi theory in which admissibility itself becomes dynamical.

We establish new $L^2$ restriction estimates for toral eigenfunctions. These estimates are sharp in certain cases, and thus prove a conjecture of Huang-Zhang for smooth submanifolds of large codimension. In particular, they provide new progress toward a conjecture of Bourgain-Rudnick. The proof combines a slicing and packing method with the approximation of the discrete spherical multiplier by Magyar-Stein-Wainger and Magyar.

Let $B_1$ be the unit disk in ${\mathbb R}^2$. We consider the harmonic map equation $$ -Δu=|\nabla u|^2u,$$ subject to the Dirichlet boundary condition $ u(e^{iθ})=(R\cosθ,R\sinθ,\sqrt{1-R^2}):=g_R$, where $0<R<1$ and $u: B_1\to {\mathbb S}^2$ is understood in the weak harmonic-map sense. In 1983, Brezis and Coron proved the existence of two explicit solutions of this nonlinear Dirichlet problem and showed that they are the unique minimizers in their respective relative homotopy classes. In this paper, we resolve a long-standing open question originally posed in their work, later posed as Open Problem 3.1 in Brezis Favorite Open Problems List. Specifically, we prove that these two explicit maps are the only weak harmonic maps with boundary trace $g_{R}$, thereby providing a definitive affirmative answer to Brezis open problem. The proof is based on a boundary rigidity argument. An auxiliary potential $X$ associated with $u$, the Pohozaev identity for the Hopf differential, and the planar isoperimetric inequality imply $$|u_r|\equiv R, \qquad u_r\cdot u_θ\equiv0 \qquad\text{on }\partial B_1. $$ Thus the Hopf differential vanishes on the boundary and hence, by holomorphicity, on the whole disk. The problem is then reduced to the conformal case, where a stereographic-coordinate classification gives exactly the two Brezis--Coron maps.

The ability to predict protein three-dimensional structures from amino acid sequences is a landmark achievement in molecular biology, where recent deep learning approaches such as AlphaFold are the culmination of decades of work. Yet, the quantitative understanding of how protein sequences give rise to dynamic conformational changes and higher-order assemblies remains unsolved. Folding and conformational states are dynamic, stochastic processes, shaped by sequence, energy, co-translational constraints, chaperone machineries, and the physicochemical conditions of the cellular environment. Recent advances now position the field to move beyond static structural endpoints toward a mechanistic understanding of folding dynamics in living systems. Single-molecule techniques enable time-resolved observation of folding trajectories and intermediate states hitherto hidden by traditional structural biology approaches, while computational innovations and data-driven approaches offer new ways to integrate heterogeneous data across scales. In this Roadmap, we review the current conceptual landscape of protein folding, examine the experimental and theoretical gaps that remain, and discuss emerging strategies that integrate high-resolution measurements with multiscale modeling. We outline a roadmap toward a quantitative and predictive science of protein folding dynamics, conformational kinetics, and macromolecular self-assembly. Realizing this vision would transform our understanding of the dynamics of molecular self-organization, from the folding of individual polypeptides to the emergence of dynamic macromolecular complexes. This will enable rational control of folding and misfolding in health and disease, extend protein engineering principles beyond static structural design, and establish a mechanistic foundation for predictive and personalized interventions in proteostasis-related disorders.

A multivariate polynomial on $n$ variables $x_1,\ldots,x_n$ of total degree $n$ over $\mathbf{Z}_2$ containing the multilinear monomial $\prod_{i=1}^n x_i$ is by the combinatorial nullstellensatz [Alon, Comb. Probab. Comput., 1999] known to always have a nonroot. We show that there cannot be a randomised polynomial time algorithm that given an arithmetic circuit of polynomial size formally computing such a polynomial, locates a nonroot with constant nonzero probability unless RP=NP. The result holds even when the individual degree of every variable in the input polynomial is at most two.

We look at Bell inequalities from the lens of information structures in stochastic teams. We consider the usual CHSH game and a dynamic variant of the same to study how various classes of strategies, classical, projective and quantum, behave under team theoretic solution concepts. We find that projective strategies (where each player performs projective measurements) enjoy important properties in the usual CHSH game, but they do not carry over to its dynamic version. These results shed light on the delicate interplay of information structure in quantum strategies and the fragility of some well known ideas under changes of information structure.

The intricate relationship between high temperature superconductivity and antiferromagnetic order in cuprates, and the fundamental origin of electron pairing remain open questions. By utilizing high-resolution laser-based spatially-resolved angle-resolved photoemission spectroscopy, we investigate the seven-layer $Bi_{2}Sr_{2}Ca_{6}Cu_{7}O_{18+δ}$ (Bi2267) and identify a cuprate system that consists of multiple hole Fermi pockets. The observed Fermi pockets exhibit pronounced momentum-, temperature- and Fermi surface-dependent energy gaps. Crucially, high temperature superconductivity with a critical temperature ($T_{\mathrm{c}}$) of $\sim$75 K emerges in a system with multiple Fermi pockets and the presence of strong antiferromagnetic order and correlations. In particular, substantial electron pairing is observed along the Fermi pocket with an energy gap up to $\sim$42 meV in lightly-doped CuO$_{2}$ planes ($p\sim$0.05). These findings challenge the conventional understanding of the roles of the nodal and antinodal electronic states in driving high-temperature superconductivity. They show that superconductivity and antiferromagnetism can coexist in a cuprate with multiple Fermi pockets, offering further insights into the pairing mechanism in cuprate superconductors.

Externally controlled trials (ECTs), including single-arm studies augmented with historical data and hybrid randomized designs with partial external augmentation, are increasingly used when concurrent randomized controls are infeasible or unethical. Regulatory guidance from the FDA, EMA, and NMPA calls for sensitivity analysis of borrowing assumptions, yet provides no structured template for which analyses to run or how to interpret them together. We propose a three-pillar framework organized around three questions: was the borrowing appropriate, did it contribute meaningful value, and are the conclusions robust to perturbation? The framework comprises eight modular analyses covering heterogeneity diagnostics, source influence, no-borrowing references, effective sample size, prior sensitivity, tipping points, alternative borrowing methods, and structural model sensitivity. It is method-agnostic and applies to both Bayesian and frequentist borrowing in patient-level or hybrid settings. We illustrate the framework using simulated data that mimic a hybrid evidence synthesis from a historical approval of ethnic-bridging submission under a real-world-evidence regulatory pathway. That original analysis combined individual patient data from a global pivotal study and a regional real-world study with aggregate data from two published cohorts, fitted via a Bayesian longitudinal model with ethnic-difference parameters. The worked example provides a reproducible template for sensitivity analysis in ECT submissions.

We give some necessary conditions satisfied by special perfect polynomials over $\mathbb{F}_2$.

The interaction of light and matter under strong isotope disorder gives rise to unconventional physics that goes beyond the quantum perturbation theory. In boron arsenide, the large mass difference between the two stable boron isotopes presents a paradigmatic case where perturbation theory fails, yet a unified theoretical framework across the perturbative and nonperturbative regime has remained elusive. Here, we develop a nonperturbative approach to capture isotope-disorder effect in boron arsenide, which fundamentally alters light-matter interactions. We reveal that coherently mixed vibrations between two boron isotopes reshape the dielectric function in the nonperturbative regime. The nonperturbative isotope interactions dictate the properties of coupled surface phonon polaritons and near-field radiative heat transfer. Two-fold tuning of radiative heat flux is achieved by modulating the surface phonon polariton resonance via isotope engineering. This work establishes a unified framework connecting the perturbative and nonperturbative limits, enabling quantitative predictions across weak to strong disorder regime.

Altermagnetism has recently emerged as a highly promising phase for spintronics, offering the combined advantages of both antiferromagnets and ferromagnets. Here, using a first-principles analysis based on density functional theory (DFT), we identify a monolayer V$_2$O crystal in a buckled Lieb lattice as a promising two-dimensional altermagnetic material. The structural and thermal stability of V$_2$O is verified through calculations of the crystal's formation energy, phonon structure, room-temperature ab initio molecular dynamics, and stiffness matrix. The system is found to exhibit auxetic behavior with a negative Poisson's ratio. Our calculations indicate an antiferromagnetic ground state with a local magnetic moment of $2.79\,μ_{\mathrm{B}}$ per V atom and a magnetocrystalline anisotropy that favors an out-of-plane easy axis. The electronic structure exhibits a momentum-dependent spin splitting of 1.2 eV, which is a characteristic of altermagnets. Inclusion of spin-orbit coupling breaks the symmetry of the quadratic band crossing near the Fermi level, resulting in a large Berry curvature and significant intrinsic spin Hall conductivity around $40\,(\hbar/e)\,\mathrm{S\,cm^{-1}}$. The results demonstrate that monolayer V$_2$O serves as a robust room-temperature altermagnetic platform, exhibiting magnetic anisotropy and spin-dependent transport responses.

Protecting an asset against threats is a challenging problem in an era of continuously evolving intelligent attacks. This requires cooperation between the asset and the defender to share information and jointly maneuver. To address this problem, this work proposes a cooperative guidance and control strategy for active asset protection against a maneuvering threat. This work develops a joint maneuver strategy where both the defender and the asset coordinate their time-varying speeds and courses to neutralize/capture the attacker. The control strategy is formulated around three coupled geometric and temporal objectives. The first objective is to set the line-of-sight rate between the asset and the attacker to zero, putting the attacker on a collision course and reducing their maneuvering. The second objective is to maintain the defender on the line-of-sight between the asset and the attacker. This ensures that the attacker faces the defender first before reaching the vicinity of the asset. Lastly, the defender is also guided to pursue the attacker based on the time-to-go estimates between the defender and the attacker. While keeping these objectives in mind, the control actions for the asset and the defender are jointly designed, fostering cooperation between the two. The stability of the proposed strategy is established using a Lyapunov-based approach. Numerical simulations performed show the effectiveness of the proposed cooperative strategy in ensuring the successful capture of a maneuvering threat.

Designing multisensory experiences involves the deliberate combination of sensory elements to shape specific impressions for a given audience. Advances in technologies beyond audiovisual modalities now make it feasible to design across touch, taste, smell, and more. However, HCI still lacks the tools and shared vocabulary needed to systematically create and evaluate such experiences. The xSense Design Cards address this gap with four card types: (1) Experience Cards define purpose, context, and audience; (2) Sensory Cards break down multisensory concepts into elements and events; (3) Technology Cards prompt consideration of relevant technologies; and (4) Exploration Cards guide reflection on the broader context, including responsible innovation. This work introduces the cards and their theoretical grounding, showing how they support structured design, reflection, and evaluation of an experience's multisensory composition. By presenting xSense, we aim to broaden the vocabulary for multisensory design and stimulate discussion within the growing multisensory HCI community.

We study Hawking radiation of the Dymnikova regular black hole. This model replaces the central singularity by a smooth de Sitter core while remaining Schwarzschild-like far from the hole, and its black-hole branch ends in a cold extremal remnant. We compute the greybody factors of the Standard Model test fields and gravitons, compare the precise numerical scattering results with WKB estimates, and use the resulting spectra to estimate an adiabatic evaporation history. The main effect is not a dramatic change in the transmission probabilities: the greybody thresholds move only slightly as the geometry approaches the remnant. Instead, the rapid decrease of the Hawking temperature strongly suppresses the total luminosity. The photon, light-fermion and graviton channels all fade near the endpoint, with the gravitational contribution remaining subdominant. The residual massless flux becomes increasingly fermion dominated because the photon channel is suppressed more efficiently. The black hole approaches the cold remnant only asymptotically, so the quoted lifetime estimates should be interpreted as cutoff times to near-extremal configurations rather than as complete evaporation times.

We investigate the thermal impact of dark matter (DM) capture and annihilation on neutron stars (NSs) in the Galactic Center (GC). Accounting for both kinetic energy deposition and internal annihilation, we systematically evaluate the influence of various DM density profiles, ranging from cored to cuspy distributions, on the late-time thermal evolution of NSs. For NSs older than $\sim 10^7~\mathrm{yr}$, the surface temperature approaches an equilibrium value $T_\mathrm{s}^{\mathrm{eq}} \sim 10^4$--$10^6~\mathrm{K}$, depending on the stellar location and the ambient DM density. In the presence of a density spike, enhanced heating shifts the emission toward ultraviolet (UV) and soft x-ray bands; however, strong interstellar extinction and large hydrogen column densities significantly suppress the observable flux density. We further provide an estimate of the cumulative infrared surface brightness from the NS population in the GC. The predicted flux density from an individual NS remains below $\sim 0.1\,\mathrm{nJy}$, while the integrated emission yields an average surface brightness $I_ν\lesssim 10^{-9}\,\mathrm{Jy\,arcsec^{-2}}$, corresponding to a signal-to-noise ratio well below current detection thresholds. Our results indicate that thermal signatures from DM-heated NSs in the GC remain below the sensitivity limits of current instruments, although nearby systems with lower extinction may provide more promising targets for detection.

In this paper, we develop a bialgebra theory for diassociative algebras. Inspired by the notion of a quadratic diassociative algebra, we introduce the concept of a Manin triple of diassociative algebras. We then define a diassociative bialgebra, which is shown to be equivalent to a Manin triple of diassociative algebras through a specific matched pair of diassociative algebras. We further formulate the diassociative Yang-Baxter equation (DYBE) in a diassociative algebra, and prove that symmetric solutions of the DYBE give rise to diassociative bialgebras. To construct such solutions, we also introduce relative Rota-Baxter operators and pre-diassociative algebras. As a key application, we lift the known relationships between diassociative algebras and other algebraic structures to the bialgebra level. In particular, we show that every diassociative bialgebra naturally induces a Leibniz bialgebra, thereby extending Loday's classical result that a diassociative algebra gives rise to a Leibniz algebra. Moreover, we provide explicit constructions of Lie bialgebras via tensor products of diassociative bialgebras and quadratic dendriform algebras.

We study Minkowski averages on Riemannian manifolds in which the interpolation is by action-minimizing magnetic geodesics with respect to a given magnetic potential. We establish equivalence between Brunn-Minkowski inequalities for this operation and lower bounds on a magnetic Ricci curvature. We then discuss various examples, including natural magnetic fields on Kähler and Sasakian manifolds, and prove a sharp, undistorted Brunn-Minkowski inequality for contact magnetic geodesics on the Heisenberg group. We also observe that closed magnetic potentials from different cohomology classes may give rise to different geodesic Minkowski averages.

In this paper, we establish two families of $q$-supercongruences modulo the third and fourth powers of a cyclotomic polynomial by employing Watson's ${}_{8}ϕ_7$ transformation, the creative microscoping method introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials.

The Atlantic Meridional Overturning Circulation (AMOC) has been identified as a tipping element in the Earth system. Under the current climate change scenarios, it is urgent to develop robust methods for determining the probability of future AMOC transitions. Recent studies using an Earth System Model of Intermediate Complexity (EMIC) have revealed the importance of an AMOC edge state, located on the boundary of the attraction basin of the collapsed state, in AMOC transitions. Here, we provide a characterization of this edge state through climate networks, using instantaneous temporal correlations between geographical locations to define the network links. We apply the climate network analysis to a set of EMIC simulations with CO$_2$ forcing according to an intermediate climate change scenario (SSP2-4.5) that exhibit qualitatively different AMOC responses as a result of interaction with the edge state. We show that network measures, specifically the normalized degree centrality, reveal the presence of teleconnections across the equator as the AMOC approaches the edge state. A similar result is obtained for an Earth System Model (ESM) simulating AMOC collapse or recovery, suggesting that climate networks could be used to detect the onset of an AMOC tipping event in ESMs.

Storage-ring-based fully coherent light sources, including steady-state microbunching (SSMB), as well as compact seeded FELs driven by laser plasma accelerators, typically have relatively large intrinsic energy spreads. Extending the spectral reach of these facilities toward the X-ray regime represents a major challenge, as existing seeded schemes require rather extreme parameters to generate appreciable microbunching at high harmonics. In this Letter, we propose an echo enhanced strong focusing scheme that employs transverse-longitudinal coupling together with the beam echo effect to simultaneously resolve the energy spread bottleneck and enable efficient high-harmonic generation. This approach substantially relaxes the requirements on both the intrinsic energy spread and the transverse emittance, paving the way for soft X-ray production using relatively weak laser modulation. Based on this scheme, we further present an SSMB storage ring capable of generating kW-level average power 6.7 nm soft X-ray radiation.

We study strategyproof mechanisms for eliciting agents' location preferences truthfully in the Euclidean plane $\mathbb R^2$ and locating a facility so as to minimize the $L_p$-norm social cost, defined as the $L_p$-norm of the vector of distances from the facility to the agents' preferred locations, for any $p \ge 1$. While the cases $p=1$ and $p=\infty$ have been well-studied, open questions remain about the optimal approximation ratios achievable by strategyproof mechanisms for general $p$. Our first result resolves an open question of Goel and Hann-Caruthers [Soc. Choice Welf. 2023]. They showed that the coordinate-wise median (CM) mechanism achieves an approximation ratio lying between \(2^{1-\frac{1}{p}}\) and \(2^{\frac{3}{2}-\frac{2}{p}}\) for $p\ge 2$, and they conjectured that it is exactly \(2^{1-\frac{1}{p}}\). We confirm this conjecture, and we further show that CM has a tight $\sqrt 2$-approximation for $1\le p\le 2$. Our second and third results demonstrate that two randomized mechanisms can yield better approximation ratios. In particular, we first consider the uniformly rotated coordinate-wise median (URCM) mechanism, and prove that, for \(1\le p<2\), its approximation ratio strictly improves over the deterministic bound \(\sqrt{2}\), while no such improvement is possible for $p\ge 2$. We then study the centroid random dictatorship mechanism that returns the average location (i.e., centroid) and the random dictatorship each with half probability, and show that its approximation ratio strictly improves over CM and URCM for every finite \(p\gtrsim 1.6\). Moreover, our analysis independently recovers the classical deterministic and randomized results for $p=1$ [Meir, SAGT 2019] [Barak, EC 2026] and $p=\infty$ [Goel and Hann-Caruthers, SCW 2023] [Tang et al., EC 2020] using significantly different techniques.

SQL query rewriting is a well-established technique for improving database performance without schema or index changes, yet finding effective rewrites for modern analytical workloads remains difficult: rule-based methods are limited to predefined transformations, while LLM-based approaches often produce rewrites that are semantically valid but compile to equivalent physical plans or degrade runtime performance. We present SPA, a SQL-Plan-Aware reinforcement learning framework that trains LLMs to rewrite queries using physical execution feedback. SPA formulates rewriting as a policy optimization problem and extends GRPO with rewards spanning semantic equivalence, textual rewrite distance, physical-plan divergence, and runtime speedup. To handle reward sparsity across query difficulty, SPA introduces Probability-Gated Adaptive Reward Shaping, a query-level curriculum that unlocks higher-level rewards only once a rollout group achieves sufficient mastery of lower-level objectives, and further improves sample efficiency through on-policy self-improvement by recycling slowdown rewrites from the current policy as targeted training signals. On both IID and OOD workloads, SPA outperforms rule-based and strong LLM baselines in end-to-end runtime, substantially reduces harmful slowdown rewrites, and yields strong tail-latency gains.