Papers

We consider inverse problems regularized by a trained neural network with convex activations and unrestricted weights. The resulting optimization problem is generally nonconvex, and gradient-based methods may converge to poor local minima. We introduce a penalized epigraph lifting of the network and show that the lifted problem admits a difference-of-convex (DC) decomposition. For Lipschitz continuous activations, we establish finite exactness: the original objective is recovered exactly at a finite and explicitly computable penalty threshold $\bar\rho$. Building on this structure, we propose H-DC (Homotopic DC), a homotopic scheme that solves a sequence of DC subproblems with increasing $\rho$, warm-starting each level from the previous solution. Numerical experiments on synthetic problems, a manifold denoiser regularizer, and image inpainting with a learned Mumford–Shah regularizer show that H-DC consistently escapes poor local minima and reaches solutions with lower objective values than gradient-based methods from the same initialization.

It has recently been demonstrated that the occurrence of extreme solar events is governed by a phase-stable period close to 1.723 years (629.29 days). According to our previously developed tidally-synchronized solar dynamo model, this is one of the periods resulting from two-planet spring tides exerted by the tidally dominant planets Venus, Earth and Jupiter, which trigger magneto-Rossby waves in the solar tachocline. Overall, the squared sum of these three waves incorporates nine periods ranging from 59 to 629 days. Using a standard Rayleigh test, we demonstrate that all nine of these periods can be identified in the data on extreme solar events. For the latter, we use a combination of 72 ground-level enhancement (GLE) events with various X-flare data, the number of which increases when the threshold is successively lowered from X10 to X1. To identify the spectral peaks, it is necessary to analyze the sidebands that emerge due to the modulation with the longer-term periods such as the 11-year Schwabe cycle and the third harmonics of the Hale cycle. In this framework, the seminal Rieger period of 154 days is re-interpreted as the lower sideband of 146 days which is the quarter of the synodic period of Venus and Earth.

Recursive reasoning models such as the Tiny Recursive Model (TRM) iteratively refine two latent states: an answer state y, decoded into the output, and a reasoning state z, which is never decoded. Many of their failures have been attributed to bad basins, regions of latent space that trap the recursion on wrong answers. Recent test-time approaches perturb y, z or both to recover these failures, but the impact of this choice on performance, test-time compute efficiency and scaling properties remains unclear. Through controlled experiments on TRM, we find that a failed trajectory carries its failure in its answer state. At equal compute, in depth and in width, resampling y recovers more failures than any perturbation of z we tested, and the performance gap widens as the test-time compute budget grows. Our approach, Gated Answer Resampling (GAR), pairs this resampling of the answer state with the model's own halting head and restarts only the puzzles that the head rejects. With at most 64 restarts per puzzle and no training or hyperparameter search, GAR takes TRM from 85.9% to 99.4% on Sudoku-Extreme and from 68.8% to 85.6% on Pencil Puzzle Bench validation. On the same checkpoints, GAR also outperforms Probabilistic TRM (PTRM), which perturbs z at each step, even when PTRM uses 42 to 197 times as much compute.

Automated approval systems let coding agents execute high-risk actions with minimal human oversight by delegating the authorization decision to a classifier, typically a large language model (LLM). Such systems, adopted in tools like Claude Code and Codex, remain vulnerable to failure modes such as scope escalation, where the agent takes actions exceeding what the user authorized. Existing benchmarks focus mainly on malicious requests, evaluating refusal under an implicit policy. We introduce AuthGateBench, the first policy-guided benchmark for authorization gating under benign requests. We implement a novel pipeline where an LLM generates contrast pairs with identical action and trajectory but differing request, flipping the label between block and allow. This allows us to assess whether models reason over context rather than the action alone. We further examine whether auxiliary environment signals can help models make better decisions. The resulting benchmark comprises 1,400 instances spanning five failure modes drawn from real-world Claude Code incidents. Evaluations show accuracy degrades unevenly across failure modes as models over-rely on internal safety alignment, and models still block most actions even when explicitly authorized, revealing unstable rather than deterministic decisions; retrospectively retrieved environment signals, however, substantially improve authorization accuracy.

In real-world scenarios, industrial processes can span multiple factories and goods are often produced at multiple sites, leading to a shared yet non-identical distribution of data across locations. To address this, we propose two federated approaches for i) combining existing coresets (Double Subsampling) and ii) creating new coresets (Federated Subsampling) in a distributed anomaly detection setting. In particular, we adapt the memory bank based PatchCore algorithm to a federated setting, resulting in increased cold-start performance, resilience to domain- and sensor-shift, and resilience to unreliable actor participation, a common problem in classical, network based federated learning strategies.

The yielding of amorphous solids has been extensively studied in the athermal limit, but the influence of thermal fluctuations is much less understood, in particular for glasses of different stability. Using molecular dynamics simulations of a model glass where the swap Monte Carlo algorithm can be employed, we study the yield stress at finite temperature and shear rate, both in steady flow and at shear start-up, over a broad range of glass stability. Thermal fluctuations soften both the dynamic yield stress in steady state and the start-up yield stress. In all cases, the temperature dependence is compatible with the $T^{2/3}$ law expected for thermally activated yielding near a saddle-node instability, with no systematic dependence of the exponent on stability or shear rate. Extrapolations to zero temperature recover the athermal yield stresses. The amplitude of the softening at start-up, by contrast, depends strongly on stability: highly stable glasses are much less sensitive to temperature than poorly annealed ones. A description of yielding at finite temperature must therefore account explicitly for the stability of the glass.

Sprite-based animation remains prevalent in video game development, but manual in-betweening—generating intermediate frames between keyframes—is labor-intensive. Existing video frame interpolation methods, optimized for photorealistic content, introduce smoothing artifacts when applied to the sharp edges, limited color palettes, and stylized motion characteristic of hand-drawn sprites. We present a deep learning approach for sprite in-betweening that adapts a U-Net encoder-decoder architecture with domain-specific modifications, including RGBA-aware processing, multi-scale flow refinement, and alpha-gated blending to preserve pixel-perfect sprite aesthetics. Following data-centric AI principles, we demonstrate that a few carefully curated sprite animation triplets—selected for diversity, quality, and representative edge cases—enable effective training without requiring massive generic datasets. Experimental evaluation using the FLIP perceptual metric shows that our method achieves a mean error of 0.258 on 42 test triplets, outperforming state-of-the-art video interpolation RIFE (0.313, 17.5% worse), traditional optical flow Lucas-Kanade (0.449, 42.6% worse), and general frame synthesis CyclicGen (0.603, 57.2% worse). This study shows that quality-over-quantity data curation and domain-specific architectural adaptations yield better interpolation quality than general-purpose methods trained on orders of magnitude more data, validating the data-centric AI paradigm for specialized visual domains. This is particularly important for small-to-medium-sized game development studios, where budget and time constraints are critical.

Ensemble forecasts from the ECMWF's Integrated Forecasting System (IFS) and the Artificial Intelligence Forecasting System (AIFS-ENS) are compared using a multi-dimensional evaluation framework. Within this framework, the uncertainty is interpreted as the linear subspace spanned by the ensemble members in phase space, from which various geometric properties are extracted. As revealed by the analysis, the error does not entirely lie in this subspace, but is reasonably well calibrated in all directions. Moreover, the framework allows to quantify that this subspace critically depends on the number of ensemble members, suggesting that the current setups only partially represent the uncertainty subspace. This phase-space interpretation furthermore allows a direct comparison of the error-representation patterns of AIFS-ENS and IFS by computing the principal angles between the subspaces: IFS and AIFS ensemble subspaces do not entirely overlap in phase space, even though they were shown to have similar abilities to represent the amplitude and the structure of the error.

Angle-resolved photoemission spectroscopy (ARPES) requires a well-ordered surface, whose quality is routinely judged from low-energy electron diffraction (LEED) and x-ray photoelectron spectroscopy (XPS). We compare LEED, XPS, and ARPES of a Cu(001) single crystal measured in situ at four stages of surface preparation: as-prepared, after aging for 24 h at 6 K, after Ar$^+$ sputtering, and after annealing. The sputtered and annealed surfaces show nearly identical $(1\times1)$ LEED patterns and Cu 2$p$ core levels, but the sputtered surface yields a momentum-independent ARPES spectrum, and the band dispersion recovers after annealing. We attribute this contrast to how each probe responds to sputtering-induced defects: LEED reflects the average lattice, which remains periodic, whereas ARPES reflects the electronic states, whose crystal momentum is no longer well defined. Our results indicate that a sharp LEED pattern and clean core-level spectra do not guarantee a surface suitable for ARPES.

We study the inviscid limit of the one-dimensional viscous Burgers equation from smooth initial data through nondegenerate shock formation and subsequent shock development. Chaturvedi and Graham \cite{CG} obtained sharp estimates of order $\nu^{1/4}$ up to the shock-formation time. We extend their analysis beyond this time, comparing the viscous solution with the inviscid entropy solution evolving from the same initial data. With the shock-formation time normalized to $t=0$, we prove a pointwise error bound on a fixed interval $0\leq t \leq \bar{\varepsilon}$, where $\bar{\varepsilon}>0$ is independent of the viscosity $\nu$. The bound consists of a term of order $\nu^{1/4}$ and an exponentially localized contribution originating from the viscous shock layer. The proof is based on matching an outer expansion, an inner expansion near the formation point, and a viscous shock-profile expansion. Combined with the results of Chaturvedi and Graham, this yields a quantitative description of the inviscid limit from the initial time through the development of the young shock.

Classifier-Free Guidance (CFG) with guidance scale $\gamma$ is commonly motivated by the $\omega$-powered conditional distribution obtained by setting $\omega=\gamma$, but CFG does not, in general, sample from this distribution exactly. We study whether the powered target can instead be learned explicitly. We introduce Importance-Weighted Conditional Flow Matching (IW-CFM), which reweights conditional flow-matching examples using density ratios obtained from frozen conditional and unconditional flow models. With exact normalized weights, the resulting objective has the same gradient as flow matching on the powered target; in practice, we use self-normalized estimation and a regularized empirical approximation for high-dimensional data. On synthetic distributions, IW-CFM recovers analytical powered targets for which numerically matched CFG, $\gamma=\omega$, exhibits a visible mismatch. On ImageNet, an amortized lightweight corrector achieves better alignment than fixed- and matched-guidance CFG with the corresponding weighted and filtered reference distributions. We additionally present an exploratory CLIP analysis of powered-density rankings on real ImageNet data. Aggregate shifts across selected prompt-specific contrasts coexist with heterogeneous classwise responses, including nonmonotonic trajectories. These results provide a practical route for studying and approximating model-defined powered conditional distributions.

We demonstrate that, in thermal atomic vapors, the electromagnetically-induced-transparency (EIT) transmission peak exhibits saturation. Investigating the amplitude of the velocity-selected double-resonance transmission peak-a purely kinematic effect arising from finite wavevector mismatch-we identify a threshold atomic column density Ns. For column densities N below Ns, the amplitude of the transmission peak maximum increases with column density. Once it exceeds Ns, the peak amplitude maximum remains constant, while the corresponding detuning shifts away from resonance and scales as the square root of ln(N over Ns). Furthermore, We find that the operating point for optimum photon-shot-noise-limited sensitivity deviates from that of the transmission peak maximum. Our study provides guidance for optimizing EIT readout in vapor cell and for precision measurements built upon this scheme.

Remote identity verification assumes that a camera stream shows a live person, yet public real-time face-swap tools can replace that person before the stream reaches the verifier. Passive deepfake detectors generalize poorly to new generators, while interactive challenges are often selected without evidence of which gestures expose current attacks. We evaluate randomized gesture challenges in 108 live sessions with six subjects, 24 impersonated identities, and two face-swap tools, DeepFaceLive and FaceFusion, injected through a virtual camera. The static calibration segment reaches a session-level AUC of 0.878 with a frozen UCF spatial detector, while the mean across the active-gesture windows reaches 0.988. Pose and combined gestures have the highest point estimates, whereas expressions show no clear improvement. Hand occlusion against FaceFusion, configured with an occlusion mask, scores below the static baseline. The end-to-end time required to satisfy the validator also distinguishes attacks from genuine sessions, achieving a session-level AUC of 0.732, although this signal includes synthesis and transport delays. These results show that challenge sets should be tested against current attack configurations and that response time can provide an additional, deployment-specific signal.

Compressed music restoration aims to recover high-fidelity audio from lossy codecs such as MP3 by restoring missing spectral content and reducing coding artifacts. Recently, generative adversarial networks and band-split models have made progress in compressed music restoration. However, existing methods either operate without explicitly modeling phase or rely on predefined frequency partitions, which may limit restoration performance and the use of input-dependent frequency information. To address these issues, we propose MPASCNet, a Magnitude-Phase-Aware Adaptive Spectral Compression network. Instead of using fixed bands, MPASCNet combines magnitude and phase features to guide cross-attention, adaptively compressing full-resolution spectral features into compact tokens. These tokens are then processed by a spectro-temporal backbone to model spectral-temporal dependencies. Finally, separate decoder branches reconstruct the full-resolution magnitude and wrapped phase spectra. Experiments on MP3 restoration show that MPASCNet outperforms the Apollo and AEROMamba baselines across multiple restoration metrics while using only 6.49 M parameters. The code is available at https://github.com/xinghour/MPASCNet.

Industrial recommendation rankers keep scaling in width and depth to billions of parameters, but this increasingly incurs costs in parameter growth, distributed communication overhead, and training inefficiency. We introduce \underline{C}hain \underline{D}imension \underline{S}caling (CDS), a scaling strategy orthogonal to width and depth. CDS repeatedly applies a single feature-interaction block with parameters shared across iterations, so the number of iterations (the chain length) becomes a new scaling dimension that adds no parameters. A lightweight intra-chain residual connection stabilizes optimization and improves quality. CDS enables two regimes. In the compression regime, shallow blocks with a longer chain match deeper baselines with fewer parameters and faster training; in the quality-scaling regime, increasing chain length under a fixed parameter budget yields consistent offline and online gains, suggesting the chain dimension adds modeling capacity beyond parameter reuse. We evaluate CDS in a large-scale multi-task short-video ranking system in production. In the compression regime, CDS stays within $0.03\%$ of baseline offline quality while reducing trainable parameters by 26.6% and improving training throughput by 6.4%, with neutral-to-positive online participation. In the quality-scaling regime, quality improves monotonically with chain length. Wall-clock analysis shows the efficiency gain comes mainly not from reduced FLOPs but from lower cross-device synchronization overhead and reduced optimizer update cost. These results position parameter-shared iterative computation as a practical, communication-efficient scaling axis for industrial ranking systems.

LLM-agent workflows routinely transform rich observations into plans, summaries, memories, handoffs, and other reusable state. This creates a privacy boundary that task relevance alone does not capture: an observation may be needed as evidence for a task without its full semantics being authorized to persist. We call this mismatch the purpose-to-state semantic gap. The key distinction is that evidence authority is not release authority. We introduce Purpose IR, a bounded typed intermediate representation that compiles natural-language user purpose into an executable semantic boundary for reusable agent state. A Purpose program is an acyclic semantic dataflow graph in which evidence leaves specify task-required semantics, typed dependencies constrain authorized derivations, and declared sinks determine which values or actions may cross the reusable-state boundary. A deterministic compiler constructs and validates the authoritative program, while a typed runtime reconstructs reusable state only from validated sink outputs. Under explicit execution contracts, this design yields release confinement and purpose noninterference, characterizing what factual semantics may be released and which observation differences must become irrelevant to reusable state. Across 1,200 cases spanning LangGraph, CrewAI, and MobileRun, Purpose IR achieves 99.6% Privacy, 83.3% Task, and 83.3% Privacy-Compliant Success (PCS), outperforming the strongest high-privacy baseline by 27.5 PCS points. On 493 PrivacyLens trajectories, it achieves the highest Safe Action Rate at 77.3%, 10.4 points above the next-best method, extending the same boundary from reusable state to downstream actions. These results show that privacy-preserving agent state requires more than filtering sensitive or task-irrelevant content: it requires an explicit, executable representation of what the user's purpose authorizes to survive.

The single production of first-generation $\mathrm{SU}(2)_L$ doublet vector-like leptons (VLLs) at the Compact Linear Collider (CLIC) with $\sqrt{s}=3~\mathrm{TeV}$ is studied. The neutral and charged VLL states are investigated through $e^+e^-\rightarrow\nu_e\bar N+\bar\nu_eN$ and $e^+e^-\rightarrow e^\pm E^\mp$, respectively. The neutral VLL is searched for in the $N\rightarrow W^\mp e^\pm$ channel with $W^\pm\rightarrow jj$, leading to the $e^\pm+2j+\slashed{E}_T$ final state. The charged VLL is explored through $E^\pm\rightarrow e^\pm h$ ($h\rightarrow b\bar b$) and $E^\pm\rightarrow e^\pm Z$ ($Z\rightarrow q\bar q$), yielding the $e^+e^-+2b$ and $e^+e^-+2j$ signatures. Signal and background simulations include initial-state radiation, beamstrahlung, and detector effects. Cut-based analyses are performed for integrated luminosities of $1$, $3$, and $5~\mathrm{ab}^{-1}$ to obtain the exclusion and discovery sensitivities in the VLL mass–mixing parameter space. The neutral-VLL channel gives the best sensitivity. At $1~\mathrm{ab}^{-1}$, the $2\sigma$ exclusion reaches $\varepsilon\simeq0.013$ at $M_N=1.3~\mathrm{TeV}$ and $\varepsilon\simeq0.065$ at $M_N=2.9~\mathrm{TeV}$, while the $5\sigma$ discovery reach corresponds to $\varepsilon\simeq0.02$–$0.10$. At $5~\mathrm{ab}^{-1}$, the charged-VLL channels probe masses up to about $2.9~\mathrm{TeV}$ and $2.8~\mathrm{TeV}$ in the $e^\pm h$ and $e^\pm Z$ modes, respectively. These results demonstrate that a 3 TeV CLIC can provide sensitive probes of first-generation doublet VLLs over a wide mass range, including scenarios with small VLL–electron mixing.

As IT systems grow in complexity, automated agent-driven incident response becomes increasingly critical. A key capability for these agents is summarizing large log streams into human-readable reports. Further progress requires meaningfully evaluating this function, yet existing benchmarks leave end-to-end summarization from large log streams underexplored due to the prohibitive cost and complexity of collecting real-world data. We introduce SLS-Bench, a benchmark for incident log summarization featuring 90 tasks grounded in real-world incident reports, each with a synthetic log stream and a reference summary. To construct SLS-Bench, we developed a data generation pipeline that converts incident reports into structured incident models, which guide the generation of synthetic logs and reference summaries. This pipeline combines narrative grounding, explicit incident modeling, generator-verifier loops, and statistical comparisons with real-world log streams to produce realistic and controllable observability data. Our evaluation of 14 LLMs on SLS-Bench shows that while frontier models can produce useful summaries, they do so at substantial cost and latency, and open-weight models lag significantly. SLS-Bench provides a framework for systematic evaluation of log analysis agents and a scalable blueprint for future synthetic observability benchmarks.

Deploying robots in human-shared environments requires deciding not only how to navigate, but also when dedicated social signals are worth emitting. Classic observation-based inference can be misleading in such settings: confounding factors may bias the estimated relationship between a signal and the task outcome, leading to incorrect decisions. Causal inference provides a framework for disentangling these relationships by modelling the mechanisms linking observations, actions, and outcomes, and by estimating interventional effects after accounting for confounding paths. We study a human-robot interaction scenario in a corridor and use a causal model to estimate whether a social navigation signal causally improves transit success. We represent the interaction as a causal DAG involving robot action, pedestrian occupancy, available space, task outcome, and a controlled confounder (static obstacles). We compare the interventional estimate with the observational estimate obtained by conditioning only on the robot action, which does not explicitly model interventions. Our results show that accounting for confounding variables can reverse the conclusions drawn from naive observational analysis: the observational estimate wrongly suggests that the social signal is detrimental, whereas the causal estimate shows that it is beneficial. This discrepancy highlights how failing to account for confounding can lead to incorrect conclusions and demonstrates the importance of causal models for context-aware signalling decisions in social robot navigation.

With AI automating consequential tasks, human oversight is required for ethical, legal, and practical reasons. Yet these oversight tasks can be demanding and often make for poor work. Empirical evidence on overseers' lived work experiences remains underexplored, yet critical for designing “good” oversight. We provide longitudinal, in-the-wild insights through action research accompanying (N=8) tutors who oversaw an AI-supported learning assistant over three months. Our thematic analysis combines insights from twelve weekly meetings and two workshop iterations and was enriched by four repeated questionnaires. We identified stimulation, mastery, and relatedness as key challenges, stemming from the task's repetitiveness, unclear impact of interventions, and limited connections with students and peers. Informed by our initial findings, we deployed targeted UI modifications with mixed success. This paper documents the long-term evolution of the work characteristics of a real-world oversight task and discusses which oversight work characteristics can and cannot be supported by interface design.

Repetitive structures can give an incorrect pose candidate more inlier support than an accurate one, causing visual localization to select it. History-Guided Pose Reselection (HPR) uses coarse position forecasts from up to five past localization outputs to distinguish well-separated current-query candidates. The best-fitting forecast proposes an alternative; HPR accepts it only if every well-fitting forecast is much closer to it than to the localizer's pose. HPR needs no odometry and returns the selected candidate unchanged, using prediction for selection rather than pose refinement. With one development-set threshold, HPR improves recall within 5 meters and 10 degrees in all 12 main configurations of Hierarchical Localization (HLoc), MASt3R-based pose generation, and JIST-based retrieval, losing successes on fewer than 0.4 percent of queries. On NAVER Gangnam B1, HPR corrects 97 HLoc failures and loses six successes; gains persist within 0.25 meters and 2 degrees on all three HLoc test sets. Median HPR runtime on HLoc candidates is below one millisecond, excluding pose generation. Recovery weakens under less frequent localization.

Extending the methods previously established for plane sextics and quartic surfaces, we develop a unified and systematic framework for the equisingular deformation classification of projective models of $K3$-surfaces of arbitrary polarization. We treat all polarizations of degree $2n$ with $n\le 4$, including the higher-degree cases of sextic surfaces in $\mathbb{P}^{4}$ and octic surfaces in $\mathbb{P}^{5}$. The computational machinery developed here is degree-independent and provides a complete tool for enumerating and distinguishing the deformation classes arising from singular projective models of $K3$-surfaces. This framework lays the foundation for future investigations in higher degrees, particularly the geometry of equisingular strata, extremal configurations, real structures.

We establish lower bounds on the orbit cardinalities of $\text{SL}(2,\mathbb{Z})$ on the $\mathbb{F}_p$-points of the generalized Markoff surfaces given by $x^2 + y^2 + z^2 - xyz = \tau+2$. We show that outside of the reductions of a certain exceptional finite set of $\overline{\mathbb{Q}}$-points found by Dubrovin and Mazzocco, every orbit has cardinality $\ge p$. Together with a recent result of Nathaniel Kingsbury-Neuschotz, this establishes, for all but finitely many primes $p$, the classification conjectures of McCullough-Wanderley (2013), and gives a complete classification of the connected components of the Hurwitz stack of $\text{SL}(2,\mathbb{F}_p)$-covers of elliptic curves, only branched above the origin. This extends earlier work of Bourgain-Gamburd-Sarnak and the author in the Markoff case $\tau = -2$. Our method studies certain noncongruence modular curves attached to a $\text{SL}(2,\mathbb{Z})$-orbit. These modular curves parametrize elliptic curves with a $\text{PSL}(2,\mathbb{F}_p)$-cover. We use this moduli interpretation to show that the noncongruence modular curves involved have bad reduction, as a cover of the $j$-line, at some place above $p$. This implies that the degree of the cover is $\ge p$, which translates into the desired bound on orbits. Key inputs include a result of Bouw-Wewers on bad reduction of three-point $\text{PSL}(2,\mathbb{F}_p)$-covers of the projective line, and the general stable reduction theory of Henrio, Obus, Raynaud, and Wewers.

Opinion dynamics models relate interpersonal influence to consensus or persistent disagreement. We study a discrete-time model combining weighted-average and weighted-median updates on a fixed directed network. Individuals update synchronously with possibly different adjustment coefficients. We characterize equilibria through local balance conditions and opinion level sets, and establish convergence to a Lyapunov-stable equilibrium using nonexpansiveness and Krasnosel'skii–Mann theory. We define type-dependent cohesive sets: each weighted-average member assigns all its weight to the set, whereas each weighted-median member assigns at least one half. Consensus is reached from every initial condition if and only if no two such sets are disjoint. Otherwise, an open set of initial conditions leads to persistent disagreement. Replacing weighted-median individuals by weighted-average individuals preserves consensus whenever it is already guaranteed for all initial conditions. Four numerical studies examine the number and locations of weighted-average individuals. The frequency of reaching a prescribed consensus tolerance is nondecreasing with their fraction in the sampled ensembles, but individual realizations need not follow this trend.

Classical harmonic analysis is built upon Lebesgue (or more general) measure. Such a framework is remarkably successful for sets of full ambient dimension, but becomes intrinsically insensitive to lower-dimensional geometry: curves, surfaces, and more generally fractal subsets of \(\mathbb R^n\) with zero \(n\)-dimensional Lebesgue measure. In contrast, Hausdorff content provides a dimension-sensitive notion of size for lower-dimensional sets or fractal sets, but harmonic analysis in this setting is still largely limited to maximal operators. In this paper, we establish a weighted two-parameter good-\(\lambda\) inequality under Hausdorff content setting, providing a route toward other general operators from maximal operators over Hausdorff content. To do this, we firstly identify an appropriate reverse Hölder class $\mathcal R\mathcal H_{q,\delta}$ associated with Hausdorff content, and then prove the self-improving property of $\mathcal R\mathcal H_{q,\delta}$ by establishing the fundamental equivalence between reverse Hölder weights and capacitary Muckenhoupt weights. As direct applications, we obtain weighted Fefferman–Stein inequality and some boundedness on general operators over Hausdorff content. The methodology developed in this paper, even when reduced to the Lebesgue measure case, also provide novel insights, which shows that the good-\(\lambda\) principle is not intrinsically tied to an underlying measure, but can extended to a non-additive framework.

We introduce a general methodology to construct globally unstable trajectories near the normally hyperbolic invariant manifold (NHIM) of a saddle-center-center Hamiltonian equilibrium. The proposed approach builds upon the classical geometric mechanism of searching for trajectories near heteroclinic connections between invariant tori of the NHIM. The main ingredient is the Birkhoff normal form near the NHIM, which allows us to control the dynamics of the hyperbolic and center components. We construct transition maps on the NHIM that encode the dynamics along heteroclinic connections. By combining the transition maps with the inner dynamics, given by the restriction of the Hamiltonian flow to the NHIM, we show that these connections can be efficiently (in terms of the travel time) shadowed by true trajectories of the full Hamiltonian system. Finally, the efficiency of this methodology is illustrated in the case of the NHIM associated to the libration point $L_1$ of the Spatial Restricted 3-Body Problem (SR3BP) for the Sun-Earth system.

In this paper, we extend the notion of a removable subcategory to $n$-exact categories, and prove that an ideal quotient by a subcategory of projective-injectives inherits an $n$-exact structure from the original category if and only if the subcategory is removable. In the Frobenius case, the quotient remains Frobenius, and the stable $(n+2)$-angulated categories are equivalent. We give a sufficient condition for a subcategory of projective-injectives to be removable. We also show that the ideal quotient construction is compatible with cluster tilting. Finally, we apply the construction to $2$-cluster tilting subcategories of the category of ACM bundles on a Geigle–Lenzing projective plane, obtaining quotients for both natural and relative $2$-exact structures.

Web application development often proceeds through successive requests that extend an existing implementation. Yet web-generation benchmarks primarily evaluate development toward fixed specifications. We introduce EvoWeb, a novel benchmark comprising 320 cumulative requests over 32 web applications, grounded in real-world product features and structured by functional dependency graphs. Agents iteratively modify a persistent codebase, while a multimodal web agent tests each intermediate application in a live browser, assessing newly requested functionality, cross-request integration, and previously requested functionality. To characterize reliability throughout development, we integrate trajectory performance across user-tolerance thresholds rather than impose a fixed stopping criterion. Experiments across open- and closed-source frontier models show that EvoWeb reveals clear performance differences between models, with cross-request integration emerging as a key challenge. Further analysis shows that later requests with deeper or more numerous dependencies are solved less accurately, while runtime errors become more frequent over successive requests. EvoWeb provides a testbed for evaluating and improving coding agents in long-horizon, multi-request development.

As quantum hardware enters the fault-tolerant regime, algorithms are beginning to be executed on encoded qubits, yet scalable certification of how well they run remains an open problem. Here we introduce magic-free certification, a certification protocol for universal fault-tolerant circuits that consumes no magic, the resource that generally places them beyond efficient classical simulation and is by far the most expensive element of a fault-tolerant computation. The protocol is code- and decoder-agnostic, runs interleaved with the target circuit on the same device, and has a sample complexity that grows linearly with the number of logical qubits and non-Clifford gates while remaining decoupled from the logical error rate itself. Given a predefined error tolerance $p_L^*$, the protocol determines at a chosen confidence level whether the device executes the target circuit at a logical error rate below $p_L^*$. The method relies on trap circuits, copies of the target with every magic state replaced by a stabilizer state, whose outcomes are deterministic and which, being indistinguishable from the target to stochastic Pauli noise, fail at the same rate. We illustrate the protocol in Steane-code simulations of a magic state injection gadget and a quantum Fourier transform.

Pedestrians in a crowd largely walk in small groups, dyads above all, and modelling how these groups arrange themselves is a step towards realistic models of the crowd as a whole. A dyad walks side by side in a dilute crowd and one behind the other in a dense one, an abreast to in-file transition recently measured in real life across the full range of naturally occurring crowd conditions (van der Laan and Corbetta, PLOS ONE 2026). The cost-function models of dyad formation by Zanlungo and coworkers (Phys. Rev. E 2014; EPL 2015) hold the pair abreast and have no in-file minimum at any finite density. Building on them, we propose and analyse a qualitative model in which the transition appears as an effective density parameter grows: the two formations become minima that coexist over a window of densities and exchange stability within it. We quantify this transition by computing the Orientation Log-Odds $\Pi_\text{OLO}$ as predicted by the model. This paper is a first step towards quantitative models of dyads and of their formation transitions.