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arXiv2026

Algorithmic drift in generative adaptive learning

Generative artificial intelligence promises to democratize personalized tutoring, yet the longitudinal stability of adaptive systems remains under-theorized. Current ethical discourse focuses on static bias. This paper argues that a distinct threat lies in algorithmic drift: a feedback loop in which minor, demographically asymmetric observation errors compound over time into systemic inequities. We make three contributions. First, we formalize algorithmic drift as a coupled learner--tutor dynamical system. Second, we corroborate this theory with an agent-based Monte Carlo simulation in which either a transparent online tracker or an LSTM deep-knowledge-tracing network drives an IRT-based curriculum policy. Third, we introduce the temporal fairness stabilizer (TFS), a targeted intervention that periodically re-probes learner capability, and benchmark it against standard exploration baselines with explicit group-fairness metrics and replication-level inference. Within the modeled environment, TFS closes the achievement gap where generic exploration does not, and we characterize the conditions under which it succeeds. We separate what is established analytically, what is demonstrated in simulation, and what remains to be validated with real learners. Our results reframe algorithmic fairness in education as a temporal property of a closed loop rather than a deployment-time checkbox.

cs.AI↗

arXiv2026

Sequential Monte Carlo Sampling of Feasible Regions Represented by R-Functions

Feasible-region sampling and volume estimation become increasingly inefficient when constraints define only a small fraction of a convenient bounding domain. We combine R-functions, which encode compound feasibility as a continuous signed field, with sequential Monte Carlo (SMC) samplers. The proposed construction exploits not only the sign of the R-function, which determines exact feasibility, but also its continuous negative values outside the feasible set, which provide graded information on constraint violation. Logistic intermediate distributions use this information to guide particles progressively towards feasibility, while the final target distribution remains uniform on the feasible region. Experiments on intersections and unions, including disconnected cases of hyperballs, show substantial variance reductions relative to direct rejection sampling, with the statistical advantage generally increasing as the feasible fraction decreases. For sufficiently small feasible regions, these advantages persist even after accounting for computational cost. To isolate the benefit of continuous R-function information, we also compare the proposed method with an SMC sampler with a region-only (RO) construction that uses the same sequential particle machinery but retains only binary feasible/infeasible information. We demonstrate that the R-function-informed SMC method concentrates particles in the feasible region earlier and produces lower-variance volume estimates than RO.

stat.CO↗

arXiv2026

Comparison of the Bergman and Szegő kernels on arbitrary bounded pseudoconvex domains with $C^2$ boundary

On every bounded pseudoconvex domain $Ω\Subset\mathbb C^n$, $n\ge2$, with $C^2$ boundary, we prove $K_Ω/S_Ω\le C_{Ω,\varepsilon} δ_Ω^{-1}|\logδ_Ω|^{1+\varepsilon}$ near the boundary for every $\varepsilon>0$, where $δ_Ω$ denotes the boundary distance. For each positive nondecreasing $H(s)=o(s)$, a smoothly bounded pseudoconvex complete Reinhardt domain in $\mathbb C^2$ shows that the logarithmic factor cannot be replaced by $H(|\logδ_Ω|)$.

math.CV↗

arXiv2026

Assessing the consistency of the CPL dark-energy model with cosmological data: Simulation-calibrated neural posterior disagreement

Recent DESI observations hint at a preference for dynamical dark energy with an equation of state that crosses the phantom divide within the Chevallier-Polarski-Linder (CPL) parameterization. In this work, we assess the consistency of the CPL parameterization with current cosmological observations. We develop a model check based on the disagreement between posteriors inferred by neural posterior estimation and flow-matching posterior estimation. Both estimators are trained on the same CPL simulations, and their posterior disagreement is calibrated using independent simulations to obtain a global $p$-value. Controlled simulations assess the reliability of the calibrated $p$-values and show how detection sensitivity depends on the form of departure from the assumed model. Applying the test to the compressed CMB+DESI+DESY5 data vector yields $p_{\rm global}=0.524$, indicating that the current observations are consistent with the CPL model. The method has a median evaluation time of only $0.53\,{\rm s}$, enabling efficient assessments of cosmological model adequacy. This framework enables efficient tests of dark-energy models, providing a firmer basis for investigating the physical origin of cosmic acceleration.

astro-ph.CO↗

arXiv2026

The Mechanism of Chirality in Plasmonic Structures: Asymmetric Resonance Modes

This study presents a theoretical framework for understanding the mechanism of chirality generation and circular dichroism (CD) in chiral nanostructures, with a particular focus on the role of asymmetric plasmon resonance mode distributions. Although previous studies have linked chiroptical properties to resonance differences, the specific nature and origin of these differences have not been fully elucidated. Here, we provide a more detailed explanation by quantitatively analyzing the asymmetric electric-field components arising from mode asymmetry. Specifically, we examine the near-field symmetry of individual nanorods and corner-stacked coupled structures (CSCS). The results demonstrate that chirality arises from asymmetric electric-field distributions, which give rise to distinct phase and amplitude responses of the electromagnetic fields in the composite structure under right-handed circularly polarized (RCP) and left-handed circularly polarized (LCP) illumination. The absorption and CD spectra, which exhibit distinct features around $910\,\mathrm{nm}$, support this mechanism and provide valuable insights into the design of chiroptical materials. The proposed mechanism is not restricted to a specific chiral structure or resonance and offers a broader explanation for the generation of circular dichroism.

physics.optics↗

arXiv2026

Conflict-Aware Logit Adapters for Utility-Preserving Anti-Distillation

While reasoning traces improve model capability, they also provide supervision signals for downstream distillation. Initial efforts to mitigate this issue rely on decoding-time interventions, but they incur high inference overhead and can harm the teacher model's natural fluency. We propose Conflict-Aware Logit Adapters (CALA), a lightweight training-time defense. CALA attaches a small residual adapter to a frozen teacher model and trains only the adapter. A proxy student identifies positions where imitation is most sensitive, and the adapter perturbs the output distribution at these positions while remaining within high-probability regions. A conflict-aware optimization strategy balances anti-distillation and utility objectives, while regularization preserves the teacher's performance and fluency. At inference time, CALA incurs modest overhead and requires no auxiliary model. Experiments on reasoning benchmarks show that CALA substantially reduces successful student imitation while maintaining near-original teacher performance and generation quality. These results establish CALA as a practical and efficient alternative to decoding-time approaches.

cs.AI↗

arXiv2026

Harm Is Behavioural, Moderation Is Lexical: How Language and Culture Decide Whom Automated Safety Systems Protect and Whom They Silence

Content warning: this paper quotes slurs and hateful statements from published test suites; the racial slur among them is masked. Automated systems decide at scale which messages are harmful, and they decide unevenly. Classifiers label African American English as offensive up to twice as often as other English, the accounts of prominent drag queens can score as more toxic than those of white nationalists, and safety behaviour that holds in English collapses in low-resource languages. We argue that over-flagging and under-protection share one cause: deciding harm from vocabulary and surface form rather than from what a message does. We organise the evidence into seven failure modes and argue, as testable hypotheses, that harmful behaviours (a request for secrecy, a demand for payment, an attempt to isolate someone) travel across languages better than the words that express them. We propose an evaluation protocol that makes the gap between the best- and worst-served language communities a headline result, and show that the naive gap is inflated under equal service and that its usual bootstrap interval is unreliable. Applied to five open systems in eleven languages, from a keyword lexicon to two current multilingual guard models, the protocol finds that none reaches native-speaker accuracy at separating harmful function from harmless form (51 of 55 system-language pairs fall short); that greater capability raises discrimination in nine languages while leaving Arabic and romanised Hindi near chance, so the gap between the best- and worst-served languages grows with capability; that one system both over-flags and under-protects the same language most; and that a language a system reads worst can look the most fairly treated when only false positives are reported.

cs.CL↗

arXiv2026

Directing to Diversify: Multi-Directional Search for Automatic Heuristic Design

Large language models (LLMs) have demonstrated strong potential to automate heuristic design for NP-hard combinatorial optimization (CO) tasks without human intervention. Recent methods predominantly rely on evolutionary computation (EC) frameworks, where heuristic diversification is mainly induced through EC-like operators or downstream selection, while the direction of heuristic transformation remains largely unharnessed. In this paper, we propose a multi-directional Perturbation-Rewrite Search (PRS) framework for automatic heuristic design that factorizes heuristic transformation through directional search primitives at the generation level. Specifically, PRS operates on heuristic ideas through three complementary directions: expansion toward richer and more expressive designs, compression toward simpler and less specialized designs, and steering for fine-grained refinement, together enabling broader and more sustained exploration of diverse regions in the heuristic design space. PRS further organizes candidate exploration within a breadth-first search-based process with adaptive operator control, which ensures thorough candidate utilization and sustained exploration, and maintains strong search dynamics while preventing premature convergence. Extensive empirical results across diverse CO tasks show that PRS consistently discovers high-quality heuristics, demonstrating the effectiveness and generality of explicitly directional exploration design.

cs.AI↗

arXiv2026

The Keller-Segel Model for Chemotaxis on Metapopulation Graphs and Graphons

Many models of spatial populations use directed motion towards environmental attractants as a mechanism for spatial pattern formation. While prior approaches typically focus on partial differential equation models for directed motion in continuous space, it is natural to consider the dynamics of biological and social systems on a metapopulation featuring well-mixed local patches and spatial motion between patches. In this paper, we formulate a version of the Keller-Segel model for chemotaxis on a graph metapopulation, modeling a chemoattractant that can diffuse across patches and a population of individuals who move through a biased random walk through comparing chemoattractant levels along edges of the metapopulation network. We also derive a continuum limit for our model describing chemotaxis in a population on patches connected by a graphon, showing that our resulting integro-differential equation model arises as the limit of our discrete patch model for families of large deterministic or random graphs. Through linear stability analysis, we derive conditions on the chemotactic sensitivity required to produce instability of a spatially uniform equilibrium for our Keller-Segel models in terms of the reaction parameters and the eigenvalues of the Laplacian operator for the metapopulation graph or graphon. Through a mix of weakly nonlinear stability analysis, numerical simulations, and analytical bounds on solutions, we can also characterize the form of pattern-forming instability in our graphon model and show how our choice of demographic dynamics for the population and the graphon adjacency function can impact the long-time total population size and chemoattractant concentration achieved in emergent spatial patterns.

math.DS↗

arXiv2026

Adaptive Uncertainty Visualization With jk-Plots

An important task of uncertainty visualizations is to communicate how the number of available observations influences the precision of estimates, such as a Normal distribution's mean. In this paper, we propose an adaptive approach to visualizing univariate data that produces salient representations while highlighting the uncertainty introduced by the sample size. We introduce jk-plots, inspired by educational material of Bayesian statistics, that combine visualizations of plausible reference models (emulating popular uncertainty visualization idioms) and the observations. We describe a visualization pipeline to create jk-plots, which we implemented in an R package. We demonstrate jk-plots using synthetic datasets of varying size and discuss possible usage scenarios in domain expert interviews.

cs.HC↗

arXiv2026

Tilt-Rotor Quadcopter with Learned Residual Control and Mode-Dependent State Estimation

Tilt-rotor quadcopters redirect thrust for translation without banking, but complicate modeling, state estimation, and disturbance rejection. This paper presents a simulation-based flight stack for a \SI{2.00}{\kilo\gram} tilt-rotor quadcopter. The system uses a six-degree-of-freedom nonlinear model with first-order motor dynamics and two-axis rotor tilt, parameterized using a 3D CAD model. A cascaded PD controller commands forward and lateral tilt angles. A 14-state EKF fuses sensor data across hover, early-transition, and transition modes using a hysteretic (Schmitt-trigger) classifier to prevent covariance discontinuities. Because the proportional position loop leaves a steady-state offset under constant wind, a small bounded neural residual is added on top of the unmodified PD controller. It is trained by behavioral cloning from a teacher with privileged access to the true wind, while the deployed network never observes wind. In simulation, the EKF achieves a post-settling position root mean square error (RMSE) of \SI{0.226}{\metre} (horizontal) and \SI{0.03}{\metre} (vertical) and an attitude RMSE below \SI{0.8}{\degree}, and it estimates a wind velocity of $(\num{2.0}, \num{-1.2})\,\unit{\metre\per\second}$ to within \SI{0.15}{\metre\per\second} per axis. The residual reduces steady-state station-keeping error under wind by approximately \SI{49}{\percent}, from \SI{0.117}{\metre} to \SI{0.060}{\metre} at \SI{1.5}{\metre\per\second} and from \SI{0.319}{\metre} to \SI{0.162}{\metre} at an out-of-distribution \SI{3.9}{\metre\per\second}. The framework provides a basis for hardware-in-the-loop and flight-test studies.

cs.RO↗

arXiv2026

There is no universal $F_σ$ equivalence relation

Rosendal proved that any $K_σ$ equivalence relation is Borel reducible to $\ell_\infty$. We prove that for every $F_σ$ equivalence relation $F$ on a Polish space, there is an $F_σ$ real linear subspace $H$ of $\ell_2$ such that the coset relation $x\mathrel{E_H}y$ if and only if $x-y\in H$ has no Baire-measurable reduction to $F$. In particular, there is no $F_σ$ equivalence relation universal under Borel reducibility for all $F_σ$ equivalence relations. This answers a question of Rosendal.

math.LO↗

arXiv2026

Spatial scaling of network length and shared-node density in bilayer distribution grids

Certain distribution networks are built in layers, each layer collecting a commodity from the layer above it and delivering it to the layer below. We propose a minimal model that gives the length of each layer per unit area from the local density of demand, here the population density. Starting from the Beardwood, Halton and Hammersley law for minimal networks, the length of a layer depends on the number of nodes it reaches and on the area occupied by the users. When two consecutive layers share this area, the gap between their two length exponents is fixed by the exponent of the density of their shared nodes, with no adjusted parameter. We test the model on the complete low and medium-voltage grid of metropolitan France, at the resolution of a population raster. The exponents are fitted by Poisson maximum likelihood on every pixel, with confidence intervals from a spatial block bootstrap. The length exponents imply an effective settled area that grows sublinearly with the density, and the gap between them agrees with the exponent of the substation density, counted independently. The agreement holds over the whole territory, so that the model can serve to estimate the length of such networks where only the density of demand is known.

physics.soc-ph↗

arXiv2026

Cross-Region Cooperation of UAV Swarms Towards Air-Ground ISAC via Hierarchical Multi-Agent Reinforcement Learning

The limited communication resources and sensing perspectives of a single uncrewed aerial vehicle (UAV) motivate UAV swarms for integrated sensing and communication (ISAC) in air-ground networks. However, in multi-region scenarios, regional UAV allocations may not match vehicle demand, while local observations may lack sufficient diversity for accurate sensing. To address these issues, this paper proposes a cross-region cooperation framework for UAV swarms. For communication, cross-region UAV reallocation reduces service disparities by matching resources to vehicle demand. For sensing, combining UAV observations from different regions increases observation diversity, while adaptive handshaking guided by marginal Cramér--Rao bound (CRB) utilities balances phase synchronization gains against handshaking energy consumption. Joint optimization of UAV positions and transmit power within each region then further improves communication service and sensing performance. These coupled decisions are coordinated by a hierarchical multi-agent reinforcement learning (H-MARL) approach through slow-timescale inter-region UAV reallocation and fast-timescale intra-region UAV control. Under the considered simulation settings, reallocation can improve quality of service (QoS) by over 20 percentage points compared with retaining the initial allocation under severe resource-demand mismatches. Cooperative sensing with adaptive handshaking incurs only about a 5.0% increase in mean CRB relative to perfect synchronization.

eess.SY↗

arXiv2026

Wandering exposed Julia components of rational maps

We prove that every wandering exposed Julia component of a rational map is a point. Here a Julia component is exposed if it meets the boundary of a Fatou domain. The theorem extends the corresponding result for polynomials to rational maps, without any assumption on critical recurrence. The proof combines an enhanced nest with a covering estimate for multiply connected pullback domains.

math.DS↗

arXiv2026

Pigeonhole Equal Subset Sum: Subexponential Algorithm, Tight Lower Bounds, and Average-Case Analysis

The Equal Subset Sum problem asks, given n positive integers bounded by M, to find two distinct non-empty subsets with the same sum. We study this problem in the regime $M = 2^{o(n)}$, where the pigeonhole principle guarantees that a solution exists, and the main challenge is to find one efficiently. A straightforward dynamic-programming algorithm runs in time poly(n).$M$. After a sequence of improvements, the current best algorithm, due to Jin, Williams, and Zhang (ESA 2025), achieves running time poly(n).$M^{1/3}$. In this work, we give the first poly(n).$M^{o(1)}$-time algorithm for Equal Subset Sum in the regime $M = 2^{o(n)}$. We complement this algorithmic result with a tight conditional lower bound showing that any substantial improvement would require a major breakthrough in approximation algorithms for lattice problems, building on recent techniques of Vafa and Vaikuntanathan (STOC 2025). Finally, we show that our techniques yield an asymptotically faster algorithm for the regime when $M = Θ(2^{n})$: a poly(n)$\cdot 2^{n/4}$-time algorithm in the average case.

cs.DS↗

arXiv2026

Initial-boundary value problems for linear lattice differential equations

We study initial-boundary value problems for linear lattice differential equations on the half-line. We first give a systematic account of fundamental solutions for equations with general finite-range interactions. We then develop complementary approaches based on reflection, the Laplace transform, and the unified transform method, obtaining contour integral and Bessel-function representations of solutions. For the semidiscrete diffusion equation, we justify these representations for continuous boundary data and initial data of polynomial growth, establish uniqueness in the corresponding growth class, and derive spatial estimates and long-time properties, including asymptotic periodicity. We also treat nonhomogeneous equations, Robin boundary conditions, nonsymmetric coefficients, and a class of time-dependent coefficients. Finally, higher-order examples illustrate how the unified transform approach extends beyond second-order equations.

math.AP↗

arXiv2026

Complete conformal metrics of constant positive scalar curvature with prescribed isolated singularities

Let $(M^n,g)$ be a smooth closed connected Riemannian manifold of dimension $n\ge3$ with positive Yamabe constant. For any prescribed finite set $Λ\subset M$ containing at least two points, we construct a complete metric on $M\setminusΛ$, conformal to $g$, with scalar curvature $n(n-1)$. At each singularity, the local geometric term in the reduced action comes from the first nonzero homogeneous metric term of degree at most $(n-2)/2$ in finite conformal normal coordinates, or from the mass of the Green function metric when all such terms vanish. We completely solve Schoen's conjecture about singular Yamabe problem.

math.DG↗