Papers

We prove there are $1442$ four-bar coupler curves through nine generic points in the plane, and thus resolve Alt's problem. We obtain this proof in three steps. First, we identify the space of coupler curves with a Zariski open subset of $\textrm{Gr}(3,6)$. Next, we formulate the polynomial system representing the nine-point path synthesis problem in these coordinates and modify it to obtain the mixed volume $5538$. Finally, we prove that $4096$ of the branches of the generic sparse polynomial system with that support escape the torus in the sparse limit. Thus, we obtain an upper bound of $5538-4096=1442$ for the generic solution count. A lower bound of $1442$ is achieved via numerical certification on one instance.

Efficient bidirectional microwave-optical photon conversion is a key capability for scaling superconducting quantum processors into distributed networks. However, achieving the necessary conversion efficiency requires filling a critical knowledge gap in understanding the loss mechanisms of electro-optic materials. Here, we characterize the microwave properties of single-crystal bulk LiNbO3 and AlN over a broad range of powers, down to single-photon levels, and spanning from millikelvin temperatures to above 1 K. We demonstrate that both materials exhibit two-level systems (TLS) behavior, while piezoelectric-related losses are excluded. We show that TLS-induced dissipation is predominantly localized on the surface rather than being an intrinsic bulk property, a result further corroborated by room-temperature 3D XPS and time-of-flight SIMS analyses. These findings provide useful insights to engineer hybrid architectures that integrate bulk electro-optic crystals within superconducting cavities, proving that microwave quality factors compatible with high-efficiency microwave-optical transduction are within reach.

We present a theoretical study of photocurrents generated by direct optical transitions induced by linearly polarized radiation in monolayers of transition metal dichalcogenides. A microscopic theory of both the ballistic and shift contributions to the photocurrent is developed that accounts for Coulomb attraction between photoexcited electrons and holes. It is demonstrated that, in typical 2D semiconductors, the Coulomb correlation causes the ballistic contribution to dominate over the shift photocurrent.

Computer-use agents (CUAs) have advanced along two separate lines: graphical interaction and software development through code and the command line. Real digital work requires both, interleaved rather than stacked end to end. We study hybrid CUAs that autonomously decide when to explore an interface, implement software, and run and visually verify their artifacts. We introduce RecreationWorld, a five-platform framework built around recreation: given a running reference, an agent must discover its behavior and build a faithful implementation with no prescribed workflow. RecreationWorld provides reproducible environments on Ubuntu, macOS, Windows, Android, and Web, plus a unified harness with native GUI control and coding tools. The running reference serves as an oracle for hidden behavioral tests, providing execution-grounded rewards. We scale trajectory generation with high-quality open-source applications. Models trained on these trajectories improve across five out-of-distribution coding and hybrid computer-use benchmarks and more frequently verify their rendered outputs, providing evidence of transfer beyond recreation. For held-out evaluation, we introduce RecreationBench, comprising 250 diverse tasks across domains and platforms. Reference-grounded programmatic and visual assertions cover action-conditioned outcomes at multiple interaction depths; each is validated on the reference and by human reviewers before the suite is frozen for automatic scoring. GPT-6 Astra leads at 58.1% overall, but passes all programmatic tests on just 2.8% of tasks. Agents reproduce static interface structure more reliably than interactions and computed outputs, while generated applications remain smaller and more monolithic than their references. We release the benchmark, environments, and test suites.

Bent-core liquid crystals are a canonical example of a soft matter system whose behavior is controlled by a local preference for order that cannot be universally achieved. This geometric frustration, arising from the rod's curved shape, has been shown to stabilize a variety of equilibrium phases, such as the helically ordered nematic twist-bend phase ($N_{\rm TB}$). Unlike a traditional nematic, the twist-bend state has 1D translational order arising from a periodic rotation of bend orientation along the helical axis. Here, we use molecular dynamics simulations to study the effect of shearing the $N_{\rm TB}$ phase along directions parallel and perpendicular to the helical axis. In the case of shear perpendicular to the helical axis, the nematic twist-bend phase is stable and flows without disordering. Conversely, shear along the helical axis disrupts order and leads to the emergence of fractionally charged Skyrmion defects, i.e. merons. These defects act as topological machines, locally rotating rods into a re-ordered and stable orientation of the $N_{\rm TB}$ phase. These findings reveal a new mechanism to create and control merons and highlight the potential application of bent-core liquid crystals in designing functional material with specific optical and computational properties.

The One Big, Beautiful Bill Act reinstates immediate expensing of domestic research expenses, reducing the tax burden and incentivizing reinvestment in U.S.-based research and development including software development across a wide range of industries. In this context, accurate and defensible estimation of qualified research expenses (QREs) is of high importance. Statistical sampling provides a practical framework for estimating QREs for a well-defined population of business components (sampling frame) documented in accordance with Internal Revenue Code Section 41 (Form 6765). IRS Revenue Procedure 2011-42 permits both attribute and variable statistical methods, although the latter (stratified mean and difference estimators) are often regarded as the standard approach to QRE estimation. In this paper, we compare attribute and variable statistical methods within a simulation study and demonstrate that attribute methods offer several theoretical and practical advantages. A key concern with the simple attribute estimator is the possibility of upward bias in QRE determination arising from a preponderance of high potential QRE (pQRE), non-qualified projects in the sampling frame. We address this concern by introducing a stratified sampling design that ensures adequate representation of high pQRE projects in the sample. We demonstrate that a convolution-based stratified attribute estimator produces a valid one-sided 95% lower confidence bound on total QREs across a range of sampling frame structures.

LLM agents in social simulation revise their opinions implicitly, in context: how open an agent is to persuasion can neither be specified nor verified, and collective outcomes inherit the model's training prior. We introduce Bayesian Chronicle Agents (BCA), a minimal belief layer separating what an agent believes from how it speaks. Each stance is a probability, updated by one Bayesian step per utterance heard. A single prior-strength parameter $\kappa$ encodes stubbornness, modeled after its role in Friedkin–Johnsen (FJ) opinion dynamics. We then sweep this parameter to yield three canonical regimes of opinion dynamics on demand (consensus, persistent disagreement, committed-minority influence), with persistent disagreement matching the FJ closed-form fixed points at $R^2\!=\!0.93$–$0.99$. We further show that prescribed $\kappa$ remains recoverable after the language round-trip, with perfect rank-order recovery across all four models. Explicit belief also makes simulation auditable: the layer surfaces systematic per-model stance biases that end-to-end simulation would silently absorb.

Large language models can hold knowledge they do not report. A model may sandbag on a capability evaluation, or answer against what it internally knows, and its outputs alone cannot tell whether it is hiding an answer or simply does not have one. We borrow the Concealed Information Test, a forensic method that identifies guilty knowledge by presenting a suspect with the true detail among plausible decoys and measuring a stronger response to the item they recognize. Our method, Probe of Internal Recognition (PIR), does the same inside a model. It presents a question with its candidate answers and reads, from the model's internal states, which candidate the model recognizes as correct. PIR is reference-free, needing no honest reference model and no labeled truth corpus. Across eight models from five families (Gemma, Qwen, Llama, Mistral, and Phi), PIR recovers the recognized answer at 0.70 to 0.87 balanced accuracy, well above the 0.28 to 0.40 unknown-item baseline and the 0.25 chance rate. It stays readable across every form of concealment we test, from prompted deception and trained sandbagging to external password-locked and circuit-broken checkpoints, with recognition between 0.85 and 0.93. When the model hides a known answer, recognition stays high. When unlearning removes the knowledge, recognition drops to the level of a question the model never knew. PIR therefore separates a model that will not answer from one that cannot, which supports sandbagging audits and unlearning verification. The signal is causal, adds information beyond black-box behavioral cues, and extends from multiple-choice questions to free-form generation.

The prediction of critical heat flux (CHF), a key safety-related quantity in nuclear thermal hydraulics, remains an important challenge due to its direct relationship with fuel performance and reactor safety. Recent studies have demonstrated that relative to traditional empirical correlations and lookup tables (LUTs), machine learning (ML) methods can substantially improve CHF prediction accuracy. Most ML-based CHF models, however, have been developed and evaluated using tube databases, leaving their applicability to reactor-relevant rod bundle geometries largely unexplored. This study evaluates ML-based CHF models deployed within the CTF subchannel code using the Electric Power Research Institute (EPRI) rod bundle CHF database. Both pure and hybrid residual correction models are considered in local and semilocal formulations. The tube-trained ML CHF models generally transferred favorably to rod bundle applications and outperformed traditional CHF methods across most geometries and operating conditions. The local hybrid LUT model produced the strongest overall performance, and the semilocal pure ML model remained highly competitive. Comparison against the Bowring correlation, W-3 correlation, and 2006 Groeneveld LUT demonstrated that substantial improvements in rod bundle CHF prediction are possible even when models are trained exclusively on tube data. These findings provide one of the first large-scale assessments of ML-based CHF models in square rod bundles within a production-level subchannel analysis environment and support their broader application in reactor thermal hydraulic analysis.

Determining the exact minimum distances of BCH codes remains a open problem. We establish the minimum distances of several families of primitive narrow-sense BCH codes, showing that they attain their designed distances. Our approach centers on $\mathbb{F}_q$-good zero-sets, which we introduce through a derivative condition on their vanishing polynomials. We show that a $q$-ary primitive narrow-sense BCH code of length $q^m-1$ and designed distance $2\leq\delta\leq q^m-1$ has minimum distance $\delta$ if and only if there exists an $\mathbb{F}_q$-good zero-set of cardinality $\delta+1$ in the finite field $\mathbb{F}_{q^m}$ with $q^m$ elements. To construct $\mathbb{F}_q$-good zero-sets, we develop several methods based on polynomial substitutions, power maps, and shifted inverses, as well as direct constructions using polynomials of special forms. Together with suitable initial $\mathbb{F}_q$-good zero-sets, including those arising from known minimum-distance results, these methods yield new good zero-sets of various cardinalities and hence families of primitive narrow-sense BCH codes whose minimum distances equal their designed distances. These families cover a broad range of designed distances, with several known minimum-distance results recovered as special cases.

We introduce a low-recourse rounding paradigm for packing problems in fully dynamic settings, which we name Dynamic Contention Resolution Schemes (DCRSs). These are dynamic analogs of (Online) Contention Resolution Schemes (or (O)CRSs) for low-recourse dynamic optimization and offer a variety of benefits. Similarly to their offline and online counterparts, DCRSs for different constraints can be combined to obtain DCRSs for the constraints' intersection. Furthermore, together with the Positive Body Chasing framework of Bhattacharya, Buchbinder, Levin, and Saranurak [FOCS 2023], DCRSs imply competitive recourse algorithms for fully dynamic packing problems with submodular objectives: these are algorithms that, for any input sequence, incur recourse that is itself competitive with the best possible recourse for that sequence. We show the existence of $\Omega(1)$-balanced and $O(\log \mathrm{rank})$-recourse DCRSs for matroid constraints, and $\Omega(1)$-balanced/$O(1)$-recourse DCRSs for matching and knapsack constraints. In particular, these yield the first non-trivial recourse bound for fully dynamic knapsack, as well as the first competitive-recourse algorithm for non-bipartite matching, and both of these apply even to monotone submodular objectives. Beyond our particular results, we view the DCRS framework as a principled step towards mechanizing the relax-and-round paradigm of approximation algorithms in the context of dynamic optimization.

Most work in computational ethics treats annotator disagreement on moral content as noise to be voted away, collapsed into majority vote or the more permissive any-annotator rule the moment a single annotator flags an item. We argue this uncertainty should instead be modeled and learned from. We introduce Moral Entropy, a Bayesian framework that keeps a full posterior over the true label and decomposes its entropy into aleatoric uncertainty (irreducible disagreement about the moral content) and epistemic uncertainty (from insufficient or noisy annotation) – and lets any heuristic consensus rule be audited against a calibrated ground truth via entropy methods such as cross-entropy/KL, Brier score, and expected calibration error. Across three corpora and fifteen discourse domains, auditing the standard aggregation rules against this posterior reveals bias that no current pipeline reports: the any-annotator rule disagrees with the calibrated posterior on roughly 30% of items – pooled, almost entirely false positives, though the errors invert at the foundation level (19.9%/38.9% mean FPR/FNR on MFTC) – while the stricter majority and two-vote rules miss 63-83% of true positives.

Cosmic-ray interactions in the Earth's atmosphere produce copious secondary mesons, providing a natural laboratory for searches for new physics. We investigate the atmospheric production of axions through the rare kaon decay $K^{+}\rightarrow\pi^{+}a$. If sufficiently long-lived, these axions can reach large underground detectors such as Super-Kamiokande and IceCube and decay through the diphoton channel, $a\rightarrow\gamma\gamma$. Using data from Super-Kamiokande and IceCube, we derive constraints on the axion decay constant $f_a$ for masses up to $m_a\sim 350~\mathrm{MeV}$. We demonstrate that atmospheric kaon decays provide a novel probe of axion parameter space, yielding constraints complementary to existing bounds.

We investigate the weak and strong gravitational lensing of a dyonic black hole in Kalb-Ramond gravity. After fixing the asymptotic normalization, we distinguish the effective charge governing the local null trajectories from the global angular identification associated with Lorentz-symmetry breaking. In the weak-deflection regime, we evaluate the Gaussian curvature explicitly and apply the Gauss-Bonnet theorem with a perturbed ray boundary, recovering the complete second order mass contribution. Independent calculations based on the orbit equation, the turning point integral, and Fermat's principle reproduce the same bending angle. In the strong-deflection regime, Tsukamoto's method yields closed expressions for both strong-deflection coefficients, with exact charge dependence. We derive finite-distance lens equations and the associated image positions, magnifications, flux ratios, and differential arrival times, retaining the physical winding condition. Shadow sizes inferred from observations of Sgr~A$^*$ and M87$^*$ yield conditional charge bounds, complemented by an estimate from S2 precession. We further construct combinations of strong-lensing observables that separate the effective charge from the conical parameter within the model: angular separation and relative brightness remove the explicit winding dependence, while a timing ratio incorporating an independent distance estimate isolates the conical deformation.

Latent flow models have proven to be a reliable and cost-effective method for time series generation. However, the latent compression induces unwanted artefacts, such as a spectral mismatch with respect to the underlying dataset, thus hindering their use as training surrogates. In this article, we propose a spectrally-aligned latent-flow time series generator, where the latent space for flow matching is trained to preserve dynamical properties that are relevant for the suitability of synthetic samples. We find that incorporating fine-tuning losses based on canonical signal representations such as the Fourier, wavelet and signature transforms helps overcome these issues. The interpretability of these transformations allows us to ensure that the synthetic signals are aligned with the true ones in terms of relevant features, such as smoothness or targeted spectral content, as opposed to relying on pointwise reconstruction losses only. We compare the proposed aligned models against a base latent-flow model and the state of the art over real-world long-range univariate and multivariate benchmark datasets. Our quantitative results validate the superiority of the proposed method in terms of its performance on metrics reflecting signal realness and computational efficiency, while being aligned to the training set with respect to its local structure.

We study the quantitative codimension-two estimate for the singular set in the boundary neighborhoods of the solutions of \begin{equation*} \Delta u+V(x)u=0\qquad\text{in }\Omega, \qquad u=0\qquad\text{on }\partial\Omega, \end{equation*} where $\Omega\subset\mathbb R^n$ is a bounded $C^{1,\mathrm{Dini}}$ domain and $V\in L^p(\Omega)$ for some $p>n$. We first prove the explicit upper bound for the doubling index is given by $C(n,p,\Omega)(1+\|V\|_{L^p(\Omega)}^{\frac{2p}{3p-2n}})$.The analytic input is an interior volume estimate for solutions of second order elliptic equations with uniformly elliptic Dini leading coefficients and $V\in L^p$. Using the quantitative doubling index bound, boundary flattening, we show an explicit upper bound for singular sets in the neighborhood of the boundary of the $C^{1,\mathrm{Dini}}$ domain.

Small area estimation (SAE) modeling is often used to address the unreliability of estimates in areas with limited survey samples. However, standard SAE models (Fay-Herriot) can only accommodate linear patterns, whereas in real cases, including poverty, many patterns are nonlinear. This study proposes the implementation of a supervised learning model, multivariate adaptive regression splines (MARS), integrated within the SAE framework (MARS-SAE) to address the challenge of unreliable direct estimation, particularly in nonlinear cases. The hyperparameter tuning results yielded the best model with the following hyperparameter combinations: maximum basis function (Max BF) = 10, maximum interaction (MI) = 2, minimum observation (MO) = 1, penalty = 3, with general cross validation (GCV) = 6.128. MARS-SAE demonstrated very strong performance; empirically, the model proved more efficient at reducing estimation error due to its higher relative efficiency (RE) compared to the SAE Fay-Herriot model, as well as better reduction of relative standard error (RSE) than the comparison model, and it yielded a lower relative root mean squared error (RRMSE) compared to the SAE Fay-Herriot model (13.64% vs. 13.75%). MARS-SAE has been proven to capture nonlinear patterns very well while still maintaining interpretability, making it an effective tool for the predictive and diagnostic analyses of phenomena. Thus, this research can serve as an important reference for readers regionally and in Southeast Asia, given that Java Island is a strategic region for Indonesia, the largest economic power in the ASEAN region.

Modern power systems can encounter increased discrepancy between the operators' simulation model and the actual true dynamics of the grid, driven by uncertainties caused by integration of new inverter-based resources (IBRs), large loads, unmodeled dynamics, parameter drifts, etc., to name a few. All of these impact the control room operations, where some critical oscillations may not be captured during the transient studies. To circumvent these issues, we propose a learning-augmented hybrid approach where the operator simulation model is supplemented with artificial intelligence (AI)-learned residual models using the phasor measurement unit (PMU)/ point-on-wave (PoW) based sensed trajectory data. The physics-based operator model provides interpretability and structural consistency, while the learned residual captures discrepancies caused by non-idealities. The learned model employs advanced neural architectures and consists of a backbone encoder and multi-head decoder layers for heterogeneous grid channels. Subsequently, we formulated a continual learning-motivated adaptation framework such that the baseline residual AI model can also be updated when the underlying real grid model changes in future conditions. Extensive numerical simulations are performed on the IEEE 68-bus benchmark model with a diverse set of disturbances, and different state-of-the-art predictive architectures involving recurrent learners, latent neural ODEs, and transformers are explored to demonstrate both residual learning and adaptation capabilities.

We study clustering of categorical sequences using the Optimal Matching (OM) distance under finite mixtures of finite-state Markov chains. We show that the normalized OM distance between two independent chains converges almost surely to a deterministic population quantity, concentrates exponentially around its finite-horizon mean, and admits an $O(\sqrt{\log n/n})$ convergence rate when the two chains have the same transition kernel. These population quantities yield a natural separation condition: the largest within-component limit must be smaller than the smallest between-component limit. Under this condition, hierarchical clustering with any bracketed linkage and Partitioning Around Medoids consistently recover the latent mixture partition. We also propose a consistent estimator of the number of components based on empirical OM distance profiles. The results extend to finite-state hidden Markov models and multichannel categorical observations. Overall, they provide a statistical justification for standard OM-based clustering methods for categorical time series.

High-throughput first-principles property calculations are often constrained by costly post-processing and dense Brillouin-zone sampling, impeding the creation of large, internally consistent materials-property datasets and limiting AI-driven discovery workflows. Pseudo-atomic-orbital (PAO) Hamiltonians provide an exact tight-binding representation of first-principles electronic structure that enables the calculation of a wide range of electronic, optical, topological, and transport properties at negligible cost, thereby supporting scalable generation of training-quality data and AI-ready data infrastructures. In this work, we present PAOFLOW 3.0 – an open-source Python suite that automates the construction and analysis of PAO Hamiltonians from plane-wave density functional theory calculations performed with either Quantum ESPRESSO or VASP. The resulting Hamiltonians enable efficient electronic structure interpolation, Fermi surface analysis, optical and dielectric response, transport coefficients, Berry phase and topological quantities, quantum transport, and other materials properties. Compared with previous releases, PAOFLOW 3.0 substantially extends the scope of the package through the introduction of internal projections enabling support for VASP calculations, self-consistent Hubbard U and V corrections obtained using ACBN0 and eACBN0 methods, generation of environment-dependent Slater-Koster tight-binding models, Landauer–BÃŒttiker quantum transport, and calculation of quantum oscillations using the integrated PySKEAF module. The theoretical foundations and the software architecture are presented together with representative calculations illustrating the current capabilities of the package.

Reusing manipulation experience across robot embodiments is important for scaling robot learning and reducing repeated task-specific data collection. However, changes in embodiment alter visual appearance, action dimensionality and semantics, and the whole-body configurations that can realize the same tool pose. We present SkelWAM, a skeleton-guided world-action model that couples perception and control through one explicit geometric representation for single-source cross-embodiment manipulation. Arm centerline geometry, tool-center-point (TCP) pose, and parallel-jaw commands form a shared 25-D state. The same definition underlies canonical third-person and wrist observations and future whole-body action targets. Trained with predictive visual supervision, a video-action mixture of transformers predicts canonical skeleton action chunks, which embodiment-specific constrained decoders convert into joint or continuum-robot controls. This formulation requires no one-to-one joint correspondence and uses no target-task demonstrations or target policy updates. We introduce LIBERO-Cross10, a source-only cross-embodiment transfer benchmark covering ten tasks and ten target embodiments across four morphological groups. On this benchmark, Franka-trained SkelWAM achieves 43.3% success over 1,000 episodes, exceeding the best-performing evaluated baseline by 36.2 percentage points. We further deploy a JAKA mini2-trained policy on the Feagine A03 continuum robot for three tabletop manipulation tasks, illustrating the approach's potential for real-world cross-embodiment manipulation. Project page: http://www.liukepku.com/skelwam/index.html

Robot demonstration collection often produces imperfect or failed trajectories in addition to successful demonstrations. Existing methods typically exploit failed trajectories by identifying segments that still make progress toward task completion, but largely overlook failure-critical behaviors that directly lead to task failure. Here we argue that these two types of segments provide fundamentally asymmetric supervision: progressive segments should be imitated, whereas failure-critical segments should be explicitly avoided. Based on this observation, we propose CARF, a Contrastive Attraction-Repulsion of Failure-guided framework for learning from imperfect robot data. CARF introduces a progress-based importance scorer, trained solely on successful expert demonstrations and its perturbation results, to estimate step-wise contributions toward task completion and identify informative regions in failed trajectories. These scores guide a unified flow-matching objective that attracts the policy toward progressive behaviors and repels it from failure-critical ones, while excluding ambiguous segments. This enables more comprehensive utilization of imperfect data and avoids unreliable supervision from ambiguous failure segments. Extensive experiments in simulation and the real world demonstrate consistent improvements over competing baselines across diverse failure scenarios, with ablations further validating the effectiveness of the proposed scoring and attraction-repulsion mechanisms. Our website is https://zhao-sq.github.io/carf/#.

Over the past three decades, it has been established that galaxy spheroids contain supermassive black holes (BHs) whose masses correlate tightly with the properties of their host galaxies. The most predictive of these correlations is between BH mass and stellar velocity dispersion (the $M_{\rm BH}-\sigma$ relation), alongside a relation with bulge luminosity or stellar mass (the $M_{\rm BH}-M_\star$ relation). Theoretical models of galaxy evolution predict that for the most massive galaxies, whose late-time growth proceeds mainly via gas-poor `dry' mergers, $M_\star$ should become a better predictor of BH mass than $\sigma$. However, genuine deviations from the $M_{\rm BH}-\sigma$ relation have remained difficult to observe due to the extreme rarity and low central surface brightness of such galaxies. Here we report James Webb Space Telescope (JWST) measurements of central BHs in a sample of eight ultramassive galaxies. We find the largest deviations from the $M_{\rm BH}-\sigma$ relation ever reported, with most BH masses exceeding $10^{10}$ solar masses. In this extreme mass regime, the galaxies tightly follow the $M_{\rm BH}-M_\star$ relation with minimal scatter, successfully confirming theoretical predictions. Furthermore, we observe significant stochasticity: For one ultramassive galaxy, the central dispersion drops rather than peaks, revealing no evidence for a central mass; while we can formally only place an upper limit, the data are entirely consistent with no BH at all. Crucially, this physical dichotomy is already evident directly in the raw kinematic maps prior to any fitting, and is robustly confirmed by our dynamical modelling. This unexpected diversity suggests that extreme multi-body dynamical processes, such as gravitational recoil or three-body ejections, significantly disrupt BH growth in the Universe's most massive galaxies.

We present central supermassive black hole (BH) mass measurements for the JWST Ultramassive Galaxy Sample, comprising 8 galaxies selected from a complete sample of the most massive early-type galaxies in the nearby Universe ($M_\star \gtrsim 2\times10^{12} M_\odot$). Building on the MGE photometric profiles and JWST/NIRSpec stellar kinematics from Paper I, we use the Jeans Anisotropic Modelling (JAM) framework to reconstruct the nuclear dynamics. Given the unique structural properties of these dry-merger-dominated systems, our Bayesian MCMC analysis explicitly tests both oblate and prolate intrinsic geometries. By marginalizing over classical dynamical degeneracies (including orbital velocity anisotropy, mass-to-light ratios, and inclination), we extract robust BH masses. We demonstrate that JWST's exceptional resolution allows JAM to tightly constrain the central potential using nuclear kinematics alone. Crucially, not all ultramassive galaxies host ultramassive BHs: while most display a classic central velocity dispersion peak, notable exceptions emerge, including a clear central dispersion drop in one galaxy and a subdued peak in another. These kinematic differences yield distinct BH masses, suggesting BH growth at extreme mass scales is highly stochastic. This stochasticity may be driven by three-body ejections during hierarchical mergers or gravitational recoil kicks from coalescing binaries. In the latter case, the kicked BH may be displaced and wandering outside the compact NIRSpec field of view. Our derived BH masses populate the historically sparse upper envelope of the galaxy mass hierarchy, providing the critical foundation for Paper III to investigate extreme-mass BH scaling relations and test models of galaxy-BH co-evolution.

We introduce the JWST Ultramassive Galaxy Sample, comprising 8 extreme early-type galaxies ($M_\star \gtrsim 2\times10^{12}\,\mathrm{M}_\odot$) drawn as a representative subset from a comprehensive full-sky census of over 100 such objects. This program aims to dynamically measure their central supermassive black holes (BHs) to probe the extreme high-mass end of BH–galaxy scaling relations, a critical regime where the physical mechanisms governing BH growth and host-galaxy co-evolution are expected to fundamentally change. In this first paper, we present the foundational data products: detailed photometric models and spatially resolved nuclear stellar kinematics. To accurately map the stellar mass distribution from the BH sphere of influence out to the extended halo, we combine diffraction-limited JWST/NIRCam imaging with wide-field DESI Legacy Surveys data, parametrizing the surface brightness using Multi-Gaussian Expansion (MGE). We extract stellar kinematics from JWST/NIRSpec integral-field observations using the penalized pixel-fitting (pPXF) method. To fully exploit the continuous $1.66-3.17\,\mu\mathrm{m}$ wavelength coverage—avoiding the telluric gaps of ground-based empirical libraries and the computational burden of massive theoretical grids—we introduce the use of near-separable Non-Negative Matrix Factorization (NMF). This algorithm identifies an optimal, highly compressed subset of synthetic spectra from the updated BOSZ library, perfectly capturing its spanning power at a fraction of the computational cost. The resulting kinematic maps reveal that these ultramassive systems are overwhelmingly slow rotators, frequently harboring kinematically decoupled cores. These high-quality photometric and kinematic data provide the inputs required for the robust Jeans Anisotropic Modelling (JAM) of the central BH masses in Paper II.

B{\i}y{\i}ko{\u g}lu, Hordijk, Leydold, Pisanski, and Stadler conjectured that, for every $n\geq3$ and $1\leq i\leq n-2$, the hypercube $Q_n$ admits a Laplacian eigenfunction with eigenvalue $2i$ and exactly two strong nodal domains. In this paper, we confirm this conjecture.

The escaping set of a transcendental entire function consists of those points that tend to infinity under iteration. Eremenko asked whether every connected component of this set is unbounded. In arXiv:2108.10256, we constructed transcendental entire functions for which the escaping set has prescribed compact connected components; in particular, it may have a singleton component. This disproves the conjecture. In this note, we give an introduction to the problem, aimed at a general mathematical audience. We also sketch a variant of the construction in which a prescribed half-strip is an oscillating wandering domain and an escaping point lies on its boundary. The account is based on the second author's lecture at the 2026 International Congress of Basic Science.

We introduce Multiplicatively Optimistic Regret Matching (MORM), an uncoupled learning rule for finite general-sum games. Under simultaneous full-information self-play, every player achieves external regret $O(\sqrt n\log d)$ uniformly over all horizons, using only one-step optimism. The analysis combines a potential-based regret-matching argument with multiplicative stability and Hellinger control of strategy movement. A learning-rate safeguard additionally gives $O(\sqrt{T\log d})$ regret in the face of adversarial utilities.

In this paper, we study the Cauchy problem for a coupled system of damped wave equations with mixed product-type nonlinearities. We have succeeded in determining the critical curve, formulated in terms of the power exponents of nonlinearities, for the global (in time) existence of small data mild solutions and the nonexistence of global mild solutions. Moreover, the global existence result is established by means of Schauder's fixed point theorem and Banach's contraction principle, whereas the nonexistence analysis relies on a new iteration scheme adapted to the product-type structure of nonlinearities. To the best of our knowledge, this work provides the first critical threshold results for such systems.

This letter studies event-triggered linear-quadratic-Gaussian (LQG) control for multi-agent systems sharing a communication network with limited per-step capacity. Although the agent dynamics are decoupled, the communication decisions are coupled through the shared network constraint, leading to a constrained multi-agent scheduling problem. We show that the optimal control law remains certainty-equivalent and decouples across agents through independent finite-horizon Riccati recursions, whereas the transmission schedule remains globally coupled. Based on this structure, we develop a centralized receding-horizon scheduling framework and reformulate the resulting problem as a mixed-integer linear program (MILP) using a closed-form characterization of the estimation-error covariance. To improve scalability, we derive a window-based skip-pruning condition that safely fixes consecutive transmission decisions to zero before solving the MILP, and we propose an auction-inspired scheduler based on one-step transmission-benefit scores. Numerical results show that the proposed model predictive control (MPC) scheduler achieves the best trade-off between control performance and communication cost, while the auction-based scheduler attains performance close to MPC with substantially lower computational complexity.