Papers

Extreme mass ratio inspirals (EMRIs) are expected to be a major source of gravitational waves (GWs) for the future Laser Interferometer Space Antenna (LISA). An EMRI occurs when a compact object, driven by the emission of gravitational radiation, slowly inspirals into a massive black hole (MBH). The classical formation channel involves repeated two-body interactions within dense nuclear star clusters. However, both EMRI event rates and the expected distribution of their orbital parameters remain highly uncertain. A major source of this uncertainty is the poorly constrained ratio between gradually inspiraling EMRIs and direct plunges, which do not produce detectable GWs. While most studies of astrophysical EMRI production focus on the Schwarzschild limit, previous work examining the Kerr metric found that the spin of the central MBH can significantly affect the EMRI-to-plunge ratio. This enhancement depends sensitively on orbital inclination, yet earlier studies held inclination fixed. In this work, we develop the mathematical framework describing inclination evolution driven by two-body encounters evolving along Kerr geodesics. By implementing 3D diffusion into Monte Carlo simulations, we show that initially prograde orbits may evolve into retrograde states, thereby converting potential EMRIs into plunges or vice versa. We estimate the EMRI-to-plunge ratio and find that substantial MBH spin enhances the EMRI rate by a factor of $\sim 2.5$. Additionally, even for a non-spinning MBH, modeling the orbits as relativistic geodesics rather than Newtonian trajectories increases the number of EMRIs by a similar factor of $\sim 2.5$. The distribution of Kerr EMRI inclinations has a substantial prograde bias that may aid in identifying the subset of loss cone EMRIs within large future EMRI samples.

We study two-point event shapes in planar $\mathcal{N}=4$ super Yang–Mills, with source, sink and detectors given in terms of half-BPS scalars and their light transforms. Akin to local correlation functions, at order by order in perturbation theory, in the large charge limit, this event shape factorizes into the product of two octagons evaluated on the unit circle. Borrowing the octagon results, we study its behavior with respect to the coupling and the angle between the detectors.

Mass distributions of binary black hole mergers inferred from gravitational-wave observations offer clues to how binaries form. These distributions conventionally label the heavier component as the primary, with mass $m_1$, and the lighter as the secondary, with mass $m_2$, such that $m_1 \geq m_2$. This ordering folds the physically labeled mass distribution along the equal-mass line, potentially reshaping the underlying population structure. In this Letter, we determine which population features can arise from mass ordering and which require genuine astrophysical structure. We show that ordering alone can produce (1) an $m_2$ distribution that declines more rapidly than the $m_1$ distribution, without intrinsic differences between the physically labeled components, and (2) an apparent concentration along equal masses, without correlations that preferentially pair black holes of equal mass. Moreover, different physically labeled populations can produce the same ordered mass distribution. Crucially, however, a preferred companion mass that remains constant as $m_1$ increases requires a characteristic mass scale already present in the physically labeled population. Applied to binary black hole mergers in the fifth Gravitational-Wave Transient Catalog, our model recovers such a scale near $m_2 \simeq 50\,\mathrm{M}_\odot$. Around this scale, the merger-rate density declines much more rapidly toward higher $m_2$ than toward lower $m_2$, ruling out an equal or weaker high-$m_2$ decline at 99.45% credibility. This structure therefore requires a characteristic mass scale in at least one physically labeled component, but masses alone cannot determine which component carries it. Any formation model reproducing the observed high-mass binary black hole merger population must contain such a scale.

We study the prospects for an independent determination of $V_{cb}$ from bottom-tagged charged-current deep-inelastic scattering at future electron-proton colliders: the EIC, LHeC, and FCC-eh. The transition $e^- c \to \nu_e b$ probes $|V_{cb}|^2$ through a process with different theoretical and experimental uncertainties from semileptonic $B$ decays. We fit ratios of $b$-tagged charged-current event counts over flavour-inclusive ones in kinematic bins, including correlated parton-distribution and flavour-tag calibration uncertainties, and floating the other CKM parameters. With projected parton distributions and tagging performance motivated by existing studies, we find a potential relative sensitivity in $|V_{cb}|$ of about $0.8\%$ at the LHeC with $1\,\rm{ab}^{-1}$ of integrated luminosity and $0.7\%$ at FCC-eh with $2\,\rm{ab}^{-1}$. The corresponding EIC sensitivity is instead at the level of $20 - 30 \%$, depending on the configuration. The LHeC could thus provide a high-energy determination competitive with low-energy ones, but with substantially different dominant systematic uncertainties, on the timescale of its proposed operation between the HL-LHC and FCC-ee. We identify precise flavour-tag calibration and control of the kinematic response as the main requirements for realising this potential sensitivity.

The James Webb Space Telescope (JWST) has been revolutionizing our understanding of early galaxy formation over its first four years of science operations. With un- precedented sensitivity, spatial resolution, and wavelength coverage from the near- to mid-infrared, JWST has enabled astronomers to identify hundreds of star-forming galaxies at redshifts z>~8 (within ~650 million years of the Big Bang) through strong spectral breaks and characterize these galaxies through sensitive spectroscopy. These capabilities have provided new insights into the stellar populations, neutral gas, nebular continuum emission, chemical enrichment, dust content, and ionization conditions of galaxies during cosmic dawn, revealing more numerous and brighter systems than anticipated from pre-JWST models. We discuss the implications of these discoveries for reionization, the buildup of stellar mass and metals, and the evolution of galaxy populations and luminosity functions. JWST is also revealing a new physical picture of early galaxies, characterized by compact, clumpy star formation, high gas fractions, and intense radiation fields. Despite much progress, many open questions remain, including the efficiency, intermittency, and stellar initial mass function (IMF) of early star formation, conditions in the interstellar medium (ISM), masses, and structure of the earliest galaxies, and the topology of reionization. Resolving these open questions will require lensing magnification, sensitive spectroscopy to > 5 micron, including with the Atacama Large Millimeter Array (ALMA), exploration to higher redshifts, and larger, more complete samples, helping to clarify how the first galaxies formed and evolved.

We study the arithmetic structure of non-invertible duality defects in 4d $U(1)$ Maxwell theory. First, we recast the problem of realizing an elliptic $SL(2,\mathbb{Q})$ transformation as a duality defect in terms of complex multiplication for two-tori. The resulting classification includes well-studied cases in the literature, but extends to infinite Diophantine families of defects at a given gauge coupling. Second, we describe their worldvolume theories explicitly, with the Chern–Simons couplings determined by using continued fractions and tridiagonal continuant matrices. Third, we work out the fusion products of these defects, extracting the condensation defects and decoupled sectors both by lattice arithmetic and from the worldvolume descriptions. Finally, we illustrate our findings with the familiar $N$-ality defects (with $N=2,3$), but also give examples of infinite order.

Compact binaries evolving in the vicinity of massive black holes may be commonplace in the Universe, and their gravitational waves carry signatures of their environment. We study the strong lensing of these signals by the massive companion, a configuration that violates the standard assumptions of the lensing literature: geometric optics and thin lenses. Using a perturbative framework, we compute the full wave optics amplification factors for face-on and edge-on binaries, including spinning lenses. In the edge-on configuration, peak amplification does not occur at perfect source–lens–observer alignment, even for a non-spinning lens. This is a consequence of the anisotropic, coherent quadrupolar emission of the stellar-mass binary breaking the axisymmetry of the system. Restricting to the equatorial plane near the optical axis, we then find that the frequency dependence of the amplification factor is universal across every configuration studied, persisting deep into the wave optics regime. Comparing against a strong-field ray tracing calculation, we find agreement at the $\sim10\%$ level for a face-on binary lensed by a Schwarzschild black hole. In the less symmetric edge-on case only the angular peak shift and fringe spacing is reproduced to this accuracy. For a Kerr lens, the face-on amplification factor is described by its Schwarzschild counterpart to the same $\sim 10 \%$ accuracy once the effect of frame dragging is corrected for. A point mass lens, by contrast, fails to reproduce the features of the system, which may have strong implications for forecasts that adopt it for lensing by massive black holes in the strong field.

Merging galaxy clusters provide an ideal laboratory for testing the self-interacting nature of dark matter. In this paper, we present a pilot study of magnetohydrodynamical cosmological zoom-in simulations of two cluster mergers selected from TNG-Cluster, with $M_{\rm 200c} \sim 9\times10^{14}~M_\odot$ and $M_{\rm 200c}\sim 2\times10^{14}~M_\odot$. The simulations are performed with both standard collisionless cold dark matter (CDM) and self-interacting dark matter (SIDM), assuming an isotropic, velocity-independent cross-section of $\sigma_{\rm T}/m_\chi = 1~\rm cm^2~g^{-1}$, with an additional $0.5~\rm cm^2~g^{-1}$ run for the low-mass system, and the mass resolution and galaxy formation model are unchanged from TNG-Cluster. We show how dark matter self-interactions produce an isothermal core in the dark matter halo, generate unique substructures in the intracluster light, and introduce variation in the multi-wavelength observables of the intracluster plasma. Using these simulations, we test the validity of observational probes for dark matter self-interactions in cluster mergers. We find that the shock-to-shock separation shows similar evolution in the self-interaction models tested here, providing a reference for distinguishing the $\sim100-200~\rm kpc$-scale difference in halo separation induced by self-interactions. Moreover, the dark matter halo centroid of the main cluster in the massive cluster merger is offset from the brightest cluster galaxy by as much as $\sim20~\rm kpc$, larger than the offset in the sub-cluster. This offset persists beyond 4 Gyr after the first pericenter passage. Our results indicate that sample selection plays a key role in testing the possible self-interacting nature of dark matter using cluster mergers, and that double radio relic cluster mergers and massive clusters with evidence of an ancient major merger offer a promising testbed for this purpose.

Gravitational perturbations of rotating black holes in higher dimensions generally obey coupled partial differential equations. We show that scalar gravitational perturbations invariant under rotations of the transverse sphere of a singly spinning Myers-Perry black hole separate, despite the obstruction to a decoupled Teukolsky equation. Two commuting, second-order angular operators organise the three gauge-invariant curvature components into joint eigenfunctions. The physical radial problem is a $2\times2$ first-order system, equivalent to a single second-order equation. This gives a separation of the sector containing both the bar-mode and axisymmetric ultraspinning instabilities. Using only ordinary differential equations, we compute the bar-mode quasinormal spectrum in $d=5,6,7$ and find instability onsets at $a/r_+=0.90498$ and $0.83228$ in $d=6,7$. We also compute the growing axisymmetric modes and their onset rotations in $d=6,\ldots,9$. The construction shows how separability can survive the coupling between gravitational polarisations.

Stabilizer Rényi entropy quantifies magic relative to a complete Pauli frame. In settings such as helicity states of massless particles, unsynchronized qubit registers, energy eigenbases, and systems under superselection rules, the computational basis may be fixed while the transverse Pauli axes remain unspecified. The stabilizer Rényi entropy then depends on the choice of those axes. We define a phase-averaged stabilizer Rényi entropy by averaging the Pauli moments before taking the logarithm. It is exactly invariant under independent local rephasings. For pure states, it vanishes precisely when the state is a stabilizer state in every phase convention. For a broad class of sparse supports, it reduces to a closed-form function of the basis-state probabilities alone. Within this class, the ceiling for $n$ channels is $\ln[n^{3}/(6n-5)]$ for $n\geq3$ and $\ln\frac{7}{5}$ for two channels. We apply the invariant to tree-level gluon scattering at four and five points, where the helicity-basis phases are unspecified. It saturates the two- and three-channel ceilings: $\ln\frac{7}{5}$ at four points and $\ln\frac{27}{13}$ in $3\to2$ scattering. In same-helicity $2\to3$ scattering, it lies between a floor of $\ln\frac{8}{7}$ and a ceiling of $\ln\frac{184}{125}$, reached in the resolved soft limit and particular collinear limits, respectively. In every configuration studied, the invariant is independent of the fixed external colors.

Quantum simulators with disordered spin Hamiltonians offer a natural route to Haar-like unitary ensembles, but face a competition. Disorder enhances randomness, yet strong disorder suppresses dynamics and localizes the system in the computational basis. This suggests an optimal intermediate disorder strength for approximating a unitary $k$-design, which has so far been identified mainly empirically. Here, we develop a perturbative theory for the moment gap $\Delta^{(k)}$, the deviation of the $k$-th moment operator from its Haar value, for the full spectrum of the interacting XXZ chain in a fixed-magnetization sector. We find the non-monotonic asymptotic scalings $\ln \Delta^{(k)}\sim -\sigma^2$ at weak disorder and $\ln \Delta^{(k)}\sim -1/\sigma$ at strong disorder, whose crossover sets an optimal disorder strength $\sigma_c\sim J_{xy}$, where $J_{xy}$ is the energy scale of the non-random interaction. We further show that a multi-quench protocol further suppresses the gap exponentially in the number of quenches. Numerical simulations confirm both scalings and this exponential suppression, and we demonstrate the usefulness of the protocol for classical shadow tomography. This work suggests that strong disorder may block both thermalization and the formation of unitary designs through a common localization mechanism.

Chiral topological order is incompatible with strictly two-dimensional local commuting-projector Hamiltonians on finite-dimensional tensor-product Hilbert spaces. Invertible subalgebras (ISAs) provide an alternative framework. We construct an ISA supporting chiral semions, together with its anti-semion commutant, on a square lattice with one qubit per edge. We give explicit non-Pauli generators for both algebras and prove that together they generate all local operators on the Hilbert space. Starting from the $\mathbb{Z}_4$ toric code, we condense the boson $e^2m^2$ and identify the semion and anti-semion algebras. We then decompose each $\mathbb{Z}_4$ qudit into two qubits and use a finite-depth circuit to disentangle one qubit per edge fixed by the condensation constraints. Pumping the semion ISA along the $z$ direction defines an explicit three-dimensional non-Clifford quantum cellular automaton (QCA). We show that pumping eight times is equivalent to a translation followed by a finite-depth quantum circuit with ancillary qubits. We also give an explicit finite-depth circuit relating this construction to the semion QCA previously obtained from topological quantum field theory. A similar condensation construction yields a $U(1)_4$ ISA on a square lattice. Together with the semion and earlier odd-prime ISAs, finite stacking realizes every pointed Witt class of bosonic Abelian anyon theories. Finally, we extend both constructions to all spatial dimensions $d=4k{-}2$, with one qubit or $\mathbb{Z}_4$ qudit on each $(2k{-}1)$-cell for the semion or $U(1)_4$ ISA, respectively.

We extend the general relativistic Lagrangian perturbation formalism of Friedman and Schutz [Astrophys. J. 200, 204 (1975)] beyond linear order, retaining full time dependence of the perturbations. We provide general formulae to relate the Lagrangian and Eulerian perturbations to any general perturbative order. As a proof of principle, we apply the framework to tidally perturbed relativistic stars in Regge-Wheeler gauge and reduce the second-order field and conservation equations to master equations with a nonlinear source. Projecting the source onto the star's eigenmodes, we obtain a nonlinear equation of motion for the mode amplitudes and the coupling coefficients, which describe the interactions among the eigenmodes and with the external tidal field. Approximating the tidal field and the coupling coefficients, we compute the static quadratic Love number from the quadrupolar fundamental mode alone and find that it reproduces the direct metric-matching value to a reasonable accuracy of $4\%$, up to a further correction arising from the Hamiltonian and momentum constraints of the second-order tidal field.

Despite extensive work on BPS black holes in 4d flat space, there is no direct microstate counting from D-brane quantum mechanics in regimes described by Einstein gravity. We derive an exact formula counting BPS configurations of the D-brane system corresponding to black holes in $N=8$ supergravity obtained from type II strings on $T^6$. As part of the derivation, we relate the BPS equations to a solvable modified Calogero-Moser system. The result agrees with the prediction from the modified elliptic genus of the D1-D5 system, providing a highly nontrivial test of the string dualities relating the two descriptions. We emphasize that this quantum mechanical system is, as a model of quantum black holes, simpler than BFSS and propose some numerical tests to explore this.

Humanoid loco-manipulation requires coordinated whole-body interaction with the physical world, yet collecting robot demonstrations at scale remains costly. Human demonstrations offer a scalable alternative, but existing egocentric human supervision often omits locomotion and posture, while embodiment differences and controller dynamics hinder accurate end-effector execution. We introduce HumanVerse, a 500-hour dataset of open-world human loco-manipulation pairing egocentric video with synchronized body and hand motion. Our human-to-humanoid pipeline aligns camera viewpoints and arm appearance, and refines motion references and controller inputs offline to reduce end-effector errors while preserving whole-body coordination. We validate the pipeline through zero-shot transfer to Unitree G1 on representative whole-body manipulation and long-range loco-manipulation tasks, without target-task robot demonstrations. Building on this pipeline, we develop $\lambda_0$, a humanoid foundation model for whole-body loco-manipulation, with robot-aligned human demonstrations providing supervision for mid-training. Increasing whole-body human data consistently improves downstream task progress. $\lambda_0$ achieves 90.0% success on SIMPLE and 62.5% across real-world loco-manipulation tasks. We expect this study to guide scalable whole-body human data collection toward humanoid foundation models for richer interaction with the physical world. Homepage: https://lambdahumanoid.github.io/

Existing machine-learning models for perovskite solar cells struggle to predict devices made from new materials. Most treat device layers as arbitrary categorical labels and represent perovskite absorbers either through featureless composition vectors or through rigid crystal graphs that require a pre-solved unit cell. As a result, they cannot generalize to unrecorded alloys or unfamiliar cell architectures. We present the Virtual-Alloy Physics-Aware Semantic System (V-PaSS), an architecture built around the physical continuity of perovskite solid solutions. V-PaSS represents arbitrary mixed-cation, mixed-halide absorbers by interpolating relaxed crystal graphs of 23 pure $ABX_3$ endpoint compounds via Vegard's law with learned bandgap bowing. For the device stack, it couples layer sequence and cell polarity (n-i-p or p-i-n) with interfacial band offsets and encodes chemical descriptions with a scientific language model. Evaluated under an institution-grouped protocol where test devices come exclusively from unseen laboratories, an untuned default configuration of V-PaSS achieves the lowest error among fourteen tested models (2.861% MAE, a 5.2% relative reduction over a gradient-boosted baseline at 3.017%, and outperforming the best tuned benchmark at 2.912%). V-PaSS pulls furthest ahead where test devices differ most from training: reducing error by 15.7-19.0% on unseen absorber alloys, by 16.1% on mixed absorbers from new laboratories, and widening its lead over the baseline by 3.6$\times$ on devices with unobserved transport layers. Combined with tree ensembles, V-PaSS sets a new state of the art across all five benchmark splits.

Finite safety assessments cannot exclude an adversary that preserves unattacked behavior until certification and activates afterward. We characterize this limitation for embodied and tool-use AI. An adversary that matches the complete observation law until observable, irrevocable certification and retains intervention access afterward attains $v\mathbb{E}[(n-S_0)_+]$ expected post-certification events over $n$ trials, where $v$ is the active event probability and $S_0$ the unattacked certification time. The result covers adaptive and privately randomized evaluators. Switching at certification is exactly optimal for a fixed likelihood-ratio test under memoryless, event-increasing interventions without a binding intervention budget. A positive clean-certification requirement forces linear worst-case expected excess when intervention raises event rates over an authorized deployment horizon, even if certification is deferred to the deadline. In simulated RoCo collaboration, switching yields commanded-separation violations in 34 of 36 post-certification episodes. Preregistered randomized AgentDojo trials estimate 1.98 additional security violations per four-task continuation (95% interval [1.67, 2.29]). An adversarial language-model controller infers hidden certification from noisy evaluation receipts and times intervention accordingly. With permanent revocation, a mixture likelihood-ratio monitor bounds cumulative excess above a prespecified reference rate at square-root order, up to logarithmic factors, on every sample path. A single fixed alternative can permit linear expected excess under stationary attacks. Against a controller observing the monitor's rule and statistic, live mixture monitoring reduces violations from 12.00 to 3.81 per 24 assigned task opportunities. Revocation also reduces tasks executed.

Let $X:M^3\to\R^4$ be a complete, connected self-shrinker without boundary, and suppose that the squared norm $S$ of its second fundamental form is constant. We prove that $S=0$ or $S=1$. Accordingly, the image is a hyperplane through the origin or a generalized cylinder $\mathbb S^k(\sqrt{k})\times\R^{3-k}$, $1\le k\le3$, and the immersion is a Riemannian covering onto its image.

Physical-layer spoofing attacks pose a critical threat to wireless localization systems by manipulating channel state information (CSI). Detecting such attacks becomes particularly challenging when the attacker and the legitimate node are spatially proximate, as their CSI characteristics become increasingly similar. To address this challenge in indoor scenarios, we propose an adaptive Gaussian mixture model (GMM)-based detection framework that learns legitimate CSI patterns and detects spatial inconsistency through statistical anomaly detection. Using a massive multiple-input multiple-output (MIMO) experimental dataset with 64 uniform linear array (ULA) antenna and 5 mm spatial resolution, we evaluated our detector against CSI spoofing attacks with spatial displacements ranging from 50 mm to 450 mm. The adaptive GMM achieves 95-98% detection rate for displacements above 150 mm while maintaining 1% false alarm rate. We benchmark our approach against conventional GMM, local outlier factor (LOF), one-class support vector machine (SVM) and k-nearest neighbors (k-NN) classifiers, demonstrating overall superior detection performance of our proposed classifier.

Amortized Bayesian inference uses simulations to perform Bayesian estimation when likelihoods are expensive or analytically unavailable. However, learning the full joint posterior is challenging in high-dimensional models with complex dependencies. We introduce a semi-amortized method that divides parameters into smaller blocks, and uses conditional independence to identify the information needed to infer each block. We train neural estimators for blocks whose posterior updates are computationally expensive or frequently repeated. Each estimator learns from simulations of the relevant model components to approximate the block's full conditional posterior, given the relevant observations and the values of the variables on which it depends. A Markov chain then alternates direct draws from these estimators with conventional updates of the remaining variables. Each neural update uses the current conditioning values and requires neither likelihood evaluation nor an acceptance-rejection step. Where the same conditional structure recurs, one estimator can be reused across blocks and datasets, with an output dimension determined by the block rather than the full model. We illustrate the approach in high-dimensional and computationally expensive hierarchical and spatiotemporal models.

For every fixed integer $r\ge3$, we prove that the Turán density of every $r$-uniform hypergraph $F$ with $m\ge1$ edges satisfies $\pi(F)\le1-c_rm^{-(r-1)/r}$, where $c_r>0$ depends only on $r$. This improves Sidorenko's general bound, whose gap from $1$ is of order $m^{-1}$. Complete $r$-uniform hypergraphs show that the exponent $(r-1)/r$ is optimal. The proof combines a potential argument for partial embeddings with the Lovász Local Lemma.

In this paper, the relationship is studied between the Lehmer index and the reciprocal sum of the prime factors of a Lehmer number. Some upper bounds and a lower bound are proposed for the Lehmer index. Another constraint on Lehmer index is also proposed if a Lehmer number has 3 as its prime factor.

Bayesian inversion is widely used in engineering applications, including material characterisation, structural damage identification, and medical diagnostics. These inverse problems are typically ill-posed, because localised variations in material properties must be inferred from noisy observations, and they are computationally intensive, since they often require many evaluations of a high-fidelity forward solver. Quantum computing has recently attracted considerable interest, partly because some near-term algorithms are feasible on NISQ hardware. In particular, PQC combined with classical optimisation enable a hybrid quantum-classical surrogate model. A central obstacle to applying such circuits to inversion is that the discretised parameter and response fields are far too high-dimensional to be embedded in a near-term quantum register. In this work we propose a quantum-assisted latent surrogate model for the Bayesian inversion of inhomogeneous material properties, in which latent compression bridges these high-dimensional fields and the small registers accessible to near-term circuits. The low intrinsic dimensional structure of the spatially varying conductivity and potential fields is exploited by reducing the conductivity through PCA and the response through a convolutional autoencoder. A PQC is then trained to map the latent conductivity to the latent response, and stochastic variational inference is performed entirely in latent space. The methodology is demonstrated on one- and two-dimensional inverse problems under both informative and non-informative priors. We further evaluate the model on IonQ simulator and Rigetti Cepheus-1-108Q, and use a lightweight learned denoising stage that recovers the noiseless accuracy to within a small residual. (The abstract has been shortened to meet arXiv's 1,920-character limit. For the full Abstract, please download the full Article.)

Generalized Pellet's theorem is extended to analytic matrix-valued functions using Descartes' rule of signs for infinite series. Cauchy's estimate is incorporated to allow numerical computation of the bounds. To address the issue that Pellet's theorem and its generalizations may not always guarantee the existence of bounds, for the case of matrix size one, we employ the Dandelin- Lobachevsky-Graeffe method, which ensures the existence of the bounds for the roots and further improves them as the order of the transform increases. Without the need for any a priori information on the function, these theorems allow us to compute mathematically verified bounds for the roots in a certain region using an interval arithmetic implementation of the function. Numerical examples and an implementation are provided to illustrate the effectiveness of the proposed method.

Motivated by the 3d-3d correspondence, we introduce an index $\mathcal{I}_g^\phi$ for a mutation loop $\phi$ of a quiver and an integer $g \geq 0$. We prove its integrality $\mathcal{I}_g^\phi \in \mathbb{Z}$ under certain conditions, which extends the integrality conjecture of Gang–Kim–Yoon for $3$-manifolds to the broader framework of mutation loops in cluster algebras. Our index also has an interpretation as the genus-$g$ partition function of a $(1+1)$-dimensional TQFT. We provide explicit computations for mutation loops of period one. In particular, in the case of the generalized Kronecker quivers, we give an explicit formula expressing the generating function of the indices for all genera $g$ as a ratio of polynomials. Altogether, we provide a new insight to the interplay between the geometry of $3$-manifolds and cluster algebra.

This paper studies Wong–Zakai convergence and Fourier neural operator learning for stochastic Volterra integral equations (SVIEs) describing random systems with memory. A Wong–Zakai theorem establishes convergence of pathwise solutions driven by piecewise linear approximations of Brownian motion to the corresponding SVIE solution. This convergence provides a foundation for using mirror-padded Fourier neural operators to learn the solution operator mapping initial conditions and driving noise to entire solution trajectories. Numerical experiments demonstrate competitive prediction accuracy and generalization to temporal grids finer than those used for training, while assessing the influence of padding and computational cost. As a follow-up to \citet{LKP:26}, this study extends their operator-learning framework to SVIEs through separate convergence and approximation analyses tailored to convolutional memory.

Most symbolic regression methods optimize candidate equations, but improving an equation's fit does not establish whether its underlying scientific conjecture should be retained, revised, or rejected. A conjecture may admit multiple mathematical realizations, and similarly accurate equations may imply different explanations. This motivates a shift from refining individual expressions to developing and testing the conjectures that guide them. We introduce SCOPE, a framework that sustains multiple conjecture-driven research trajectories. Multiple scientist agents independently pursue conjectures through equation modeling, question formulation, and active experimentation, then exchange evidence to revise their conjectures and research directions. Their findings are consolidated into a scientific report linking models, conclusions, and evidence. Existing equation-discovery benchmarks primarily assess recovered expressions, leaving the evidence and scientific conclusions established through the investigation underexamined. We therefore introduce ClosureBench, an interactive benchmark of 60 closure-discovery tasks across five scientific domains. ClosureBench focuses on closure-discovery tasks to give equation discovery a concrete scientific purpose. Each task embeds an unknown algebraic relation in a known PDE system, where agents control experimental conditions in OpenFOAM to acquire new data for equation discovery. Beyond equation accuracy, ClosureBench proposes three metrics to evaluate the discovery process: evidence faithfulness, mechanism insight, and experimental effectiveness. Compared with diverse baselines, SCOPE achieves higher equation accuracy on both ClosureBench and a real-world scientific discovery case, while producing more accurate, reliable, and effective scientific discoveries.

Network-linked data combine a network with covariates and responses observed for its nodes. Common regression approaches either model dependence among neighboring responses using the observed network or capture similarities associated with network structure without explicitly modeling response dependence. We propose a probabilistic network-linked data analysis framework that incorporates both features while accounting for randomness in network formation. The model uses normalized connection probabilities to weight peer responses and latent positions to capture network cohesion. A two-step likelihood procedure first estimates the network structure and then fits the response model. We establish theoretical guarantees for network recovery in sparse networks with heterogeneous node effects, and prove consistency and joint asymptotic normality of the regression estimators subject to suitable conditions. The analysis also clarifies when estimating the network preserves the classical convergence rate for spatial autoregressive models. Simulation studies demonstrate the estimation performance of the method and assess its inference accuracy. An analysis of diplomatic network data illustrates its practical application.

Adenoid cystic carcinoma (ACC) of the lacrimal gland is a rare and extremely life threatening malignant tumor with a high risk of recurrence and poor long-term outcomes. On the other hand, due to its complex molecular dynamics and low incidence rate, treatment strategies are very limited. In this study, a mathematical model is developed based on the biological phenomena of the tumor microenvironment using nonlinear ordinary differential equations. It is analyzed both analytically and numerically. According to the findings, the growth of tumor cells is associated with the depletion of oxygen and nutrients, while severe hypoxia promotes the development of abnormal blood vessels and makes tumor control more difficult. Surgery and postoperative radiotherapy have been considered as treatment strategies, and we have observed that surgery alone may not be sufficient to prevent the regrowth of residual tumor cells. In addition, postoperative radiotherapy provides better control of the residual tumor burden, particularly when the remaining tumor cells have higher radiosensitivity. Furthermore, different radiotherapy interruption scenarios have been investigated, and the results indicate that longer gaps between radiotherapy doses can adversely affect the overall treatment response, leading to increased overall toxicity and a higher number of residual tumor cells at the end of the observation period. Consequently, our results suggest that postoperative radiotherapy after surgery, combined with adequate radiosensitivity of the residual tumor cells and adherence to the prescribed treatment schedule without prolonged interruptions, may reduce the risk of tumor recurrence and potentially improve long term survival.

We study PAC learning of binary classifiers when labels arrive after random delays and are censored beyond an observation window $W$. How many samples does this observation model require? We propose a distribution-dependent answer, the Delay Complexity Dimension (DCD). The DCD combines the VC dimension with the exact label-information rate of the observation channel. Every learner needs $\Omega(\mathrm{DCD}/\varepsilon^2)$ samples. An oracle pseudo-label ERM matches this rate on the instance family of the lower bound, up to constants and logarithmic factors. On that family the DCD is within a constant factor of the classical censored-ERM complexity, and whether the DCD rate is achievable where the two separate remains open. The theory also yields a quantization scheme under which delay information lowers the residual label variance, an expedition policy with a $(1-1/e)$ guarantee for its version-space objective, and a phenomenon with no analogue in the no-delay setting. Under a fixed collection horizon, a moderate window can strictly beat immediate labels. Experiments on synthetic data and the Criteo conversion logs support these predictions. A budget-matched test of a companion labeler-diversity prediction is reported as a negative result.