We provide a comprehensive overview of the fundamental structural properties of weighted projective Reed-Muller codes. We give a recursive construction for these codes, under some conditions for the weights, and we use it to derive bounds on the generalized Hamming weights and to obtain a recursive construction for their subfield subcodes and their dual codes. The dual codes are further studied in more generality, where the recursive constructions may not apply, obtaining a description as an evaluation code when the degree is low. We also provide insights into the Schur products of these codes when they are not degenerate.

As artificial intelligence assumes cognitive labor, no quantitative framework predicts when human capability loss becomes catastrophic. We present a two-variable dynamical systems model coupling capability (H) and delegation (D), grounded in three axioms: learning requires capability, practice, and disuse causes forgetting. Calibrated to four domains (education, medicine, navigation, aviation), the model identifies a critical threshold K* approximately 0.85 (scope-dependent; broader AI scope lowers K*) beyond which capability collapses abruptly-the "enrichment paradox." Validated against 15 countries' PISA data (102 points, R^2 = 0.946, 3 parameters, lowest BIC), the model predicts that periodic AI failures improve capability 2.7-fold and that 20% mandatory practice preserves 92% more capability than the simulation baseline (which includes a 5% background AI-failure rate). These findings provide quantitative foundations for AI capability-threshold governance.

This technical note presents a algorithmic approach for generating optimal sets of co-prime moduli within specified integer ranges. The proposed method addresses the challenge of balancing moduli bit-lengths while maximizing the dynamic range in Residue Number System (RNS) implementations. Experimental results demonstrate that the generated moduli sets achieve optimal dynamic range coverage while maintaining balanced bit-length distribution, making them particularly suitable for parallel hardware implementations based on RNS.

Multi-channel speech enhancement aims to recover clean speech from noisy multi-channel recordings. Most deep learning methods employ discriminative training, which can lead to non-linear distortions from regression-based objectives, especially under challenging environmental noise conditions. Inspired by ArrayDPS for unsupervised multi-channel source separation, we introduce ArrayDPS-Refine, a method designed to enhance the outputs of discriminative models using a clean speech diffusion prior. ArrayDPS-Refine is training-free, generative, and array-agnostic. It first estimates the noise spatial covariance matrix (SCM) from the enhanced speech produced by a discriminative model, then uses this estimated noise SCM for diffusion posterior sampling. This approach allows direct refinement of any discriminative model's output without retraining. Our results show that ArrayDPS-Refine consistently improves the performance of various discriminative models, including state-of-the-art waveform and STFT domain models. Audio demos are provided at https://xzwy.github.io/ArrayDPSRefineDemo/.

Leadership in social groups is often a dynamic characteristic that emerges from interactions and opinion exchange. Empirical evidence suggests that individuals with strong opinions tend to gain influence, at the same time maintaining alignment with the social context is crucial for sustained leadership. Motivated by the social psychology literature that supports these empirical observations, we propose a novel dynamical system in which opinions and leadership co-evolve within a social network. Our model extends the Friedkin-Johnsen framework by making susceptibility to peer influence time-dependent, turning it into the leadership variable. Leadership strengthens when an agent holds strong yet socially aligned opinions, and declines when such alignment is lost, capturing the trade-off between conviction and social acceptance. After illustrating the emergent behavior of this complex system, we formally analyze the coupled dynamics, establishing sufficient conditions for convergence to a non-trivial equilibrium, and examining two time-scale separation regimes reflecting scenarios where opinion and leadership evolve at different speeds.

We study higher-order curvature estimates along Kähler-Ricci flows on compact Kähler manifolds of intermediate Kodaira dimension. We prove that away from singular fibers, the Ricci curvature is uniformly bounded in $C^1$, the Laplacian of the Ricci curvature in $C^0$, and the scalar curvature in $C^2$. We identify a geometric obstruction to higher-order curvature bounds, whose non-vanishing causes a specific third-order derivative of the Ricci curvature to blow up at rate $e^{t/2}$. Uniform $C^k$ bounds for every $k$ hold for the Ricci curvature in the isotrivial case, and for the full Riemann curvature in the torus-fibered case.

Hybrid superconductor/semiconductor devices play a crucial role in advancing quantum science and technology by merging the properties of superconductors and semiconductors. To operate these devices at high temperature, Niobium could substitute the widespread aluminum as superconducting element. Niobium devices show the best superconducting properties when shaped by etching, but this technique is often incompatible with semiconductors and two-dimensional materials. Our work investigates the influence of oxygen diffusion on the superconducting transition of Nb nanowires fabricated by lift-off technique. To this scope, we fabricate and measure Nb devices of different width (W) and thickness (t). By using the Berezinskii-Kosterlitz-Thouless (BKT) model for charge transport, we demonstrate that our nanowires behave as two-dimensional superconductors regardless of W and t. While the normal-state transition temperature (TN) remains constant with decreasing W, the temperature of the fully superconducting state (TS) decreases. Thus, the superconducting transition width (δTC) increases as W shrinks, due to oxygen diffusion from the lithography resist occurring during deposition. These insights provide essential knowledge for optimizing Nb-based hybrid quantum devices, paving the way for operating temperatures above 2 K and contributing to the development of next-generation quantum technologies.

Hourly rainfall extremes cause some of the most destructive weather disasters, yet numerical weather prediction models still struggle to forecast them, and a physical basis for their predictability remains unclear. Here, we identify a trivariate clustering of hourly rainfall extremes with surface temperature, phases of the Monsoon Intraseasonal Oscillation (MISO), and precipitable water vapor, establishing a physical foundation for the medium range predictability of these events. This clustering arises from multiscale interactions in which extremes organize into storm systems embedded within mesoscale convective clusters and synoptic low-pressure systems during active MISO phases. We develop an algorithm to identify, track, and monitor these storm systems. Although rapid error growth limits the prediction of isolated hourly extremes, our results provide basis for a physics informed training of deep learning, data driven models to forecast organized clusters of hourly rainfall extremes more than a week in advance, offering substantial potential to reduce losses from extreme rainfall.

We present the first systematic comparison of three TiO2 metamaterial solid immersion lens geometries - sub-hemispherical, super-hemispherical, and full-spherical - for label-free super-resolution imaging. Using SEM, we characterised both the cap profiles and the nanoparticle-fluid immersion at the lens-sample interface, revealing that super-hemispherical lenses achieve the deepest immersion and closest contact with sample features. Imaging experiments under wide-field and laser confocal microscopes show that this enhanced immersion drives superior resolution and contrast. In addition, we introduce a deep learning approach based on a SinCUT image translation model to establish a cross-modal mapping between SEM morphology and optical imaging response, enabling virtual optical predictions and providing a first step toward a digital twin representation of mSIL imaging behaviour. Electromagnetic simulations further confirm a direct correlation between immersion depth and far-field main lobe intensity. Our findings demonstrate that careful control of lens shape and nanoparticle-fluid penetration, together with data-driven modelling, is essential to maximise super-resolution performance in TiO2 mSILs.

This paper introduces soliton_solver, an open-source GPU-accelerated software package for the simulation and real-time visualization of topological solitons in two-dimensional non-linear field theories. The software is structured around a theory-agnostic numerical core implemented using Numba CUDA kernels, while individual physical models are introduced through modular theory components. This separation enables a single computational framework to be applied across a broad class of systems, from nanoscale magnetic spin textures in condensed matter physics to cosmic strings spanning galaxies in high energy physics. The numerical backend provides finite-difference discretization, energy minimization, and GPU-resident evaluation of observables. A CUDA--PyOpenGL rendering pipeline allows direct visualization of evolving field configurations without staging full arrays through host memory. The package is distributed in Python via PyPI and supports both reproducible batch simulations and interactive exploration of metastable configurations, soliton interactions, and model-dependent initial states. We describe the software architecture, numerical workflow, and extensibility model, and we present representative example applications. We also outline how additional theories can be incorporated with minimal modification of the shared numerical infrastructure.

We study a nonlocal SIS epidemic model with free boundaries, advection, and spatial heterogeneity, where the dispersal kernels are not assumed to be symmetric. The model describes the evolution of susceptible and infected populations in a bounded infected habitat whose endpoints move according to nonlocal boundary fluxes. Our goal is to determine the sharp threshold between disease spreading and vanishing, and to characterize the long-time behavior of solutions. The analysis faces several essential difficulties. The linearization around the disease-free equilibrium gives rise to a genuinely coupled nonlocal system with drift, so the relevant spectral quantity cannot be reduced directly to a standard scalar eigenvalue problem. In addition, the presence of advection terms and possibly non-symmetric kernels destroys self-adjointness, so no useful variational characterization is available; in particular, classical Rayleigh quotient and minimax arguments cannot be applied. To overcome these difficulties, we employ the generalized principal eigenvalue theory for nonlocal operators developed by Coville and Hamel, together with the Harnack inequality for non-symmetric nonlocal operators established therein. This non-variational framework is particularly well suited to our setting. Combined with comparison principles, sub- and supersolution constructions, and uniform estimates on time-dependent spatial intervals, it allows us to derive the precise asymptotic behavior of the generalized principal eigenvalue with respect to the spatial domain and the diffusion rate, identify the sharp threshold and the critical habitat size, and determine the long-time dynamics of $S$ and $I$ via an $ω$-limit set approach. To the best of our knowledge, this is the first work on a free-boundary SIS epidemic model with non-symmetric nonlocal dispersal kernels, advection, and spatial periodicity.

The present study examines evaporative cooling and the resulting deposition patterns of a sessile $Al_2O_3$-based nanofluid droplet on a hydrophobic glass substrate at different temperatures. Evaporation predominantly occurs in the pinned contact line mode for both heated and non-heated cases, with only slight recession observed without heating. The droplet height and contact angle decrease linearly with time, and scaling relations are proposed to describe the evolution of droplet geometry and volume. A non-dimensional parameter, $Π_{rel}$, is introduced to characterize transitions in deposition patterns. For $Π_{rel} \leq 1$ ($T_s \leq 26^\circ$C), interconnected irregular polygonal network structures form at the periphery, which are rarely reported in evaporating droplets. With increasing substrate temperature, this structure is suppressed, giving rise to a classical coffee-ring pattern for $1 < Π_{rel} \leq 10$. At higher temperatures ($T_s > 40^\circ$C), dual-ring formation along with central particle deposition is observed for $Π_{rel} > 10$. The interfacial temperature is higher near the contact line and decreases toward the apex, and a universal scaling for the temperature profile is proposed. Internal flow velocity increases with substrate temperature, exhibiting asymmetric multi-vortex structures. Evaporative cooling intensifies with heating, enhancing evaporation flux and capillary flow. Appropriate scaling relations for evaporation flux and capillary velocity are established. Overall, the dynamics are governed by thermocapillary (Marangoni) flow induced by evaporative cooling, which enhances internal circulation and governs nanoparticle deposition morphology.

Left-Right Symmetric theory has proved to be one of the most successful models in explaining the origin of neutrino mass and mixings, and the phenomenological origin of neutrinoless double beta decay, lepton flavor violation as well as leptogenesis within its regime. In the current work, left-right symmetric model has been extended with a sterile neutrino per generation, which acts as a dark matter candidate within the model. The model has been realized using $A_{4}$ modular symmetry and the primary focus of the work rests on investigating the impact of assigning different modular weights to the particle content of the model and hence study its effects on the results pertaining to relic abundance of dark matter and also leptogenesis within its framework.

In the field of computational fluid dynamics, direct numerical simulations generate highly detailed data for the analysis of turbulent flows by resolving all relevant physical scales. Yet their large size, complexity, and heterogeneity make systematic post-processing and data reuse increasingly challenging. Despite the growing availability of high-fidelity simulations through public repositories, extracting meaningful physical insight often requires substantial technical effort, specialized workflows, and access to high-performance computing resources. In this article we introduce \texttt{aPriori}, an open-source Python package developed to address these limitations by providing a dedicated, memory-efficient, and user-oriented framework for the analysis of direct numerical simulation data. The software enables streamlined handling of three-dimensional fields, including filtering, scale separation, gradient evaluation, thermochemical analysis, and visualization, using concise and reproducible scripts. Its pointer-based data management strategy allows very large datasets to be processed on standard workstations without excessive memory usage, significantly lowering the barrier to advanced analysis. Beyond basic post-processing, \texttt{aPriori} supports workflows central to modern turbulence and combustion research, such as a priori model assessment, data-driven closure development, and detailed chemical analyses that include computational singular perturbation. By unifying these capabilities within a coherent and extensible software architecture, \texttt{aPriori} enhances productivity, promotes reproducibility, and facilitates broader and more effective use of high-fidelity simulation data within the computational fluid dynamics community.

We investigate shape optimization for the principal eigenvalue of the Pucci extremal operator \[ \left\{ \begin{aligned} -\mathcal{M}^+_{λ,Λ}(D^{2}u)&=μ^{+}_{1}(Ω)u &&\text{in }Ω,\\ u &=0 &&\text{on }\partialΩ, \end{aligned} \right. \] in dimension three. Since $\mathcal{M}^+_{λ,Λ}$ is fully nonlinear, in non-divergence form, and non-variational, classical symmetrization and rearrangement methods are not available. We introduce a three-dimensional family of double--pyramidal domains $\{Ω^ω_{γ,a}\}$ parametrized by an anisotropy factor $γ\in \left[\frac{1}{\sqrtω},\sqrtω\right]$ and an affine shear parameter $a\in(-π,π)$, under fixed ellipticity ratio $ω=Λ/λ\ge 1$. Within this family and under a fixed-volume constraint, we prove that the volume-normalized principal eigenvalue is uniquely minimized at the symmetric unsheared configuration $(γ,a)=(1,0)$ among domains in the family $\{Ω^ω_{γ,a}\}$. The proof combines an explicit construction of positive eigenfunctions on seven patches with a lower bound under affine shear deformations. Using the homogeneity and orthogonal invariance of the Pucci operator, we identify an involutive symmetry $γ\mapsto γ^{-1}$ in the associated volume functional and establish strict monotonicity away from the self-dual point $γ=1$. In particular, for $ω>1$, any nontrivial anisotropy or shear strictly increases the normalized principal eigenvalue. This reveals a genuinely three-dimensional rigidity mechanism for a fully nonlinear spectral problem and extends to dimension three the symmetry-minimization phenomenon previously known in the planar case.

Solidification governs the microstructure and, therefore, the mechanical response of metal components, yet the atomistic details of nucleation and defect formation are often difficult to determine experimentally. Molecular dynamics can bridge this gap, but only if the interatomic model is both accurate and computationally efficient. Here, we develop a Machine Learning Potential (MLP) for aluminum and demonstrate its near ab initio fidelity when trained with the sequential-refinement workflow that fine-tunes the model on low-energy structures. The favorable scaling of the model enables nanosecond simulations involving millions of atoms, thereby overcoming finite-size effects in simulations of polycrystalline solidification and subsequent mechanical testing. Comparison with classical potentials and recent MLP models, including a general-purpose model, shows that inaccuracies in stacking-fault energetics and diffusion can lead to qualitatively incorrect solidified grain structures and post-solidification mechanical behavior. Since our framework is based on an equivariant graph neural network, it allows for straightforward extensions to multi-component systems, providing valuable guidance for the future design and fine-tuning of both specialized and universal MLPs in computational mechanics simulations.

LLMs are increasingly embedded in programming workflows, from code generation to automated code review. Yet, how gendered communication styles interact with LLM-assisted programming and code review remains underexplored. We present a mixed-methods pilot study examining whether gender-related linguistic differences in prompts influence code generation outcomes and code review decisions. Across three complementary studies, we analyze (i) collected real-world coding prompts, (ii) a controlled user study, in which developers solve identical programming tasks with LLM assistance, and (iii) an LLM-based simulated evaluation framework that systematically varies gender-coded prompt styles and reviewer personas. We find that gender-related differences in prompting style are subtle but measurable, with female-authored prompts exhibiting more indirect and involved language, which does not translate into consistent gaps in functional correctness or static code quality. For LLM code review, in contrast, we observe systematic biases: on average, models approve female-authored code more, despite comparable quality. Controlled experiments show that gender-coded prompt style affect code length and maintainability, while reviewer behavior varies across models. Our findings suggest that fairness risks in LLM-assisted programming arise less from generation accuracy than from LLM evaluation, as LLMs are increasingly deployed as automated code reviewers.

We derive a Riemannian metric on three-dimensional color space from the Fisher information of neural population codes in the visual pathway. Photoreceptor adaptation, retinal opponent channels, and cortical population encoding each map onto a geometric construction, producing a metric tensor whose components correspond to measurable neural quantities. The resulting 17-parameter model is fitted jointly to four independent threshold datasets: MacAdam's (1942) chromaticity ellipses, the Koenderink et al. (2026) three-dimensional ellipsoids, Wright's (1941) wavelength discrimination function, and the Huang et al. (2012) threshold color difference ellipses, covering 96 independently measured discrimination conditions across varied chromaticities and luminances. The joint fit achieves STRESS of 23.9 on MacAdam, 20.8 on Koenderink et al., 30.1 on Wright, and 30.8 on Huang et al.

In this work, we continue the investigation of correlation numbers in $\mathcal{N}=1$ super Minimal Liouville Gravity (SMLG), with physical fields in the Ramond sector. Building upon our previous construction of physical operators and the evaluation of three-point correlation functions involving Ramond and Neveu-Schwarz (NS) insertions, we now turn to the analytic computation of four-point correlation numbers. This development is motivated by the framework established for the bosonic Minimal Liouville Gravity and its supersymmetric NS analog, where the integration over moduli space in correlation functions can be performed explicitly using the higher equations of motion (HEM) in Liouville theory. In particular, if one of the insertions corresponds to a degenerate field, the four-point amplitude can be expressed in terms of boundary contributions obtained from the OPE structure of logarithmic counterparts of ground ring elements. We aim to adapt and generalize this approach to the Ramond sector.Our result is a closed-form analytic expression for four-point correlation numbers involving Ramond fields.

A rigid connection between the optical fiber and the rock makes amplitudes of 'fiber strain' measured with Distributed Acoustic Sensing (DAS) equal to 'rock strain'. We demonstrate this by running four interrogator units (IU) on a DAS testbed with single-fiber patch cables being cemented into a groove in the concrete floor of Black Forest Observatory (BFO). The recorded signals are compared with the recordings of a calibrated Invar wire strain meter array that has been continuously in operation for the last decades. This way we measure 'strain transfer rate' (ratio of 'fiber strain' over 'rock strain') at frequencies below 0.2 Hz. Waveform similarity for strong earthquake signals is high with typical values of the normalized correlation coefficient greater than 0.95. The 'strain transfer rate' is close to 1 for all four IUs, while it was significantly less in a previous study with DAS cables unreeled on the floor and loaded down by sand and sandbags, only. At frequencies up to 14 Hz we make an intercomparison of IUs, showing no significant variation with frequency. The scatter of 'strain transfer rate' in between channels which are spatially near to each other in the same fiber route is about $\pm$10 % in most cases. The variation of median values in between different IUs and earthquakes is less than 5 %. By subtracting the common mode laser noise, which is coherent along the fiber route, we lower the background signal level to an rms-amplitude of 100 pstrain at 0.1 Hz and 5 pstrain at 1 Hz in a bandwidth of 1/6 decade for the best cases. This allows the detection of the marine microseisms during times of moderate amplitude level.

We develop an averaging approach to robust risk measurement under payoff uncertainty. Instead of taking a worst-case value over an uncertainty neighborhood, we weight nearby payoffs more heavily under a chosen metric and average the baseline risk measure. We prove continuity in the neighborhood radius and provide a stable large-radius behavior. In Banach lattices, the approach leads to a convex risk measure and under separability of the space, a dual representation through a penalty term based on an inf-convolution taken over a Gelfand integral constraint. We also relate our veraging to aggregation at the distribution and quantile levels of payoffs, obtaining dominance and coincidence results. Numerical illustrations are conducted to verify calibration and sensitivity.

Our electronic structure theory for crystalline solids is commonly built on the periodic potential assumption $V(\mathbf r)=V(\mathbf r+\mathbf R)$ for every lattice translation $\mathbf R$, enabling Bloch eigenstates, crystal momentum as a good quantum number, and the standard quasiparticle-based description of the behavior of metals. Because the zero-point motion of the ions, however, in correlated electron systems the electronic environment experienced by an itinerant electron is neither static nor self-averaging at the single-particle level, even in perfectly stoichiometric crystals, leading to a distribution of local Kondo scales that spans two orders of magnitude in temperature. We discuss, through a comparison between uniform scenarios and one that breaks with perfect lattice translational symmetry, how incorporating this distribution yields a unified description for all heavy-fermion systems at the quantum critical point.

A recently introduced recurrence-relation ansatz applied to the Bose-Hubbard model is here used in the generalized Aubry-Andre model. The resulting modified Aubry-Andre model allows for a simple parametrization of the solutions in terms of three parameters, viz., the system energy when the quasiperiodicity amplitude Delta = 0, the site mu where the particle is initially localized, and the tuning parameter -1 < alpha < 1 that determines the regions of localized or extended states. The standard Aubry-Andre form corresponds to alpha = 0.

The performance of quantum processors in the noisy intermediate-scale quantum (NISQ) era is severely constrained by environmental noise and other uncertainties. While the recently proposed quantum control robustness landscape (QCRL) offers a powerful framework for generating robust control pulses for parametric gate families, its application has been practically restricted to quasi-static noise. To address the spectrally complex, time-varying noise prevalent in reality, we propose filter function-enhanced QCRL (FF-QCRL), which integrates filter function formalism into the QCRL framework. The resulting FF-QCRL algorithm minimizes a generalized robustness metric that faithfully encodes the impact of stochastic processes, enabling robust pulse-family generation for parametric gates under realistic time-varying noise. Numerical validation in a representative single-qubit setting confirms the effectiveness of the proposed method.

Going from lower to higher multiplicity events in proton-proton and proton-lead collisions, the data show a stronger than linear growth of the D-meson normalized yields. In this contribution we try to understand this behavior using a Monte Carlo event generator which implements the $k_T$-factorization formalism. We use different spatial distribution of matter in the proton and in the lead nucleus at the initial stage of these collisions. We find that, with all the tested spatial distributions, the model reproduces well the behavior seen in the data. We conclude that this observable is not appropriate for a detailed study of the spatial distribution of matter in the proton.

Decoherence in many-body quantum systems can give rise to intrinsically mixed-state phases and phase transitions beyond the pure-state paradigm. Here we study the $(2+1)$D transverse-field Ising model subject to a strongly $\mathbb{Z}_2$-symmetric decoherence channel, with a focus on strong-to-weak spontaneous symmetry breaking (SWSSB). This problem is challenging because the relevant transitions occur in the strong-decoherence regime, beyond the reach of perturbative expansions around the pure-state limit, while conventional quantum Monte Carlo (QMC) methods are hampered by the need to access nonlinear observables and by the sign problem. We overcome these difficulties by developing a QMC algorithm that efficiently evaluates nonlinear Rényi-2 correlators in higher dimensions, complemented by an effective field-theoretic approach. We show that the decohered state realizes a rich mixed-state phase diagram governed by an effective 2D Ashkin-Teller theory. This theory enables analytical predictions for the mixed-state phases and the universality classes of the phase boundaries, all of which are confirmed by large-scale QMC simulations.

In the present study we determine from the available experimental data the cross section of muon transfer to molecular oxygen at low energies with account of the oxygen molecule structure. Building on an earlier work, the results highlight the role of the molecular structure effects and signifcantly improve the agreement with theoretical calculations of the muon transfer rate. An effcient computational model of the kinetics of processes involving muonic hydrogen atoms in gaseous mixture of H2 and O2 is developed and analyzed. The model is applied in the description of the FAMU experiment for the measurement of the hyperfine splitting in muonic hydrogen and the Zemach radius of the proton.

In contemporary biology and medicine, 3D microscopy is one of the most widely-used techniques for imaging and manipulation of various kinds of samples. Navigating such a micrometer-sized, 3-dimensional sample under the microscope -- e.g. to find relevant imaging regions -- can pose a tedious challenge for the experimenter. In this paper, we examine whether 2D desktop, 3D desktop, or Virtual Reality (VR) interfaces provide the best user experience and performance for the exploration of 3D samples. We invited 12 skilled microscope operators to perform two different exploration tasks in 2D, 3D and VR and compared all conditions in terms speed, usability, and completion. Our results show a clear benefit when using VR -- in terms of task efficiency, usability, and user acceptance. Intriguingly, while VR outperformed desktop 2D and 3D in all scenarios, 3D desktop did not outperform 2D desktop.

Given a finite metric space $(X\cup Y, \mathbf{d})$ the $k$-median problem is to find a set of $k$ centers $C\subseteq Y$ that minimizes $\sum_{p\in X} \min_{c\in C} \mathbf{d}(p,c)$. In general metrics, the best polynomial time algorithm computes a $(2+ε)$-approximation for arbitrary $ε>0$ (Cohen-Addad et al. STOC 2025). However, if the metric is doubling, a near linear time $(1+ε)$-approximation algorithm is known (Cohen-Addad et al. J. ACM 2021). We show that the $(1+ε)$-approximation algorithm can be generalized to the case when either $X$ or $Y$ has bounded doubling dimension (but the other set not). The case when $X$ is doubling is motivated by the assumption that even though $X$ is part of a high-dimensional space, it may be that it is close to a low-dimensional structure. The case when $Y$ is doubling is motivated by specific clustering problems where the centers are low-dimensional. Specifically, our work in this setting implies the first near linear time approximation algorithm for the $(k,\ell)$-median problem under discrete Fréchet distance when $\ell$ is constant. We further introduce a novel complexity reduction for time series of real values that leads to a similar result for the case of discrete Fréchet distance. In order to solve the case when $Y$ has a bounded doubling dimension, we introduce a dimension reduction that replaces points from $X$ by sets of points in $Y$. To solve the case when $X$ has a bounded doubling dimension, we generalize Talwar's decomposition (Talwar STOC 2004) to our setting. The running time of our algorithms is $2^{2^t} \tilde O(n+m)$ where $t=O(\mathrm{ddim} \log \frac{\mathrm{ddim}}ε)$ and where $\mathrm{ddim}$ is the doubling dimension of $X$ (resp.\ $Y$). The results also extend to the metric facility location problem.

The knowledge gap hypothesis suggests that the diffusion of information tends to increase rather than reduce social inequalities. Subsequent research on the digital divide has extended this perspective by focusing on unequal access to and use of digital technologies. The emergence of generative artificial intelligence raises the question of whether these frameworks remain sufficient to describe current forms of informational inequality. While access to AI systems is increasingly widespread, differences may arise in how users engage with AI-generated content. This paper proposes a theoretical extension of the knowledge gap perspective by arguing that generative AI shifts the focus from access and usage to the critical evaluation of information. It is assumed that individuals with higher levels of education are more likely to question and contextualize AI-generated outputs, whereas individuals with lower levels of education may rely more directly on them. The contribution is conceptual and does not present empirical findings. It aims to provide a framework for future research on the relationship between education, AI use, and knowledge inequality.