Papers

Short-horizon price-trend prediction from limit order books in equity and intraday electricity markets requires models that combine predictive quality with low single-sample latency and a small serialized model size to keep pace with rapid and continuous market updates. We introduce MBOFormer, a 7,203-parameter causal transformer, and MBOFusion, a 14,371-parameter extension with a slow temporal-context branch. Both models process market-by-order (level-3) histories of individual order submissions, cancellations, and executions. We compare them with level-2-based baselines that process sampled order-book snapshots and simple statistics instead of their underlying messages. Across three markets and four prediction horizons, our models achieve the highest mean macro-F1 in eleven of the twelve settings in a comparison against other benchmark models of varying sizes. Both our models achieve sub-millisecond median inference on a single Apple M2 CPU thread and have serialized state dictionaries below 70 kB. MBOFormer is 5.1x-8.1x faster than the million-parameter baselines at comparable predictive quality. Our results show that fine-grained market event history can reduce the need for model size under tight deployment constraints for short-term equity and electricity price forecasting tasks.

The first-principles quantum simulation of the full electronic Hamiltonian of a material model of realistic size appears daunting even in a future era of fully fault-tolerant quantum computers. In contrast, classical simulations of material excited states rely on heuristic quasiparticle frameworks, such as the GW approximation and Bethe-Salpeter equation, where only a few quasiparticles are explicitly considered. Here we give a detailed description of quasiparticle quantum simulation for multi-excitonic states within the BSE. We develop an explicit first-quantized, orbital-based block encoding of the multi-exciton BSE Hamiltonian for qubitization, and reduce its cost using integral factorization and crystal symmetries. For a materials model of size $N$, and a fixed number of quasiparticles, the resulting algorithms require $\widetilde{O}(N)$ Toffoli gates and $O(\log N)$ logical qubits, or, using space-time tradeoffs, $\widetilde{O}(\sqrt{N})$ Toffoli gates and $\widetilde{O}(\sqrt{N})$ logical qubits, achieving up to a sixth power reduction in Toffoli count, or up to exponential reduction in qubit count, versus standard full Hamiltonian simulation techniques that have been explored in materials simulation. Resource estimates for the classically challenging problem of triexcitonic eigenvalue estimation across a range of semiconductor models of a realistic physical size of $O(10^3)$ atoms, show that our Toffoli-optimized implementation reduces the Toffoli count by $10^6-10^7$ relative to full Hamiltonian simulation estimates, while a qubit-optimized implementation requires only a few hundred logical qubits. More broadly, these results show that classical heuristic frameworks will be essential in the design of future fault-tolerant quantum algorithms for materials.

We consider an abstract wave-type evolution equation with a propagation speed depending only on time. We assume that this propagation speed is the product of a monotone, but possibly degenerate, shape function and an oscillating factor bounded from above and from below by positive constants. We investigate how degeneracy and oscillations interact in determining the derivative loss of solutions. After reducing the problem to a family of ordinary differential equations through spectral analysis, we derive frequency-dependent energy estimates. In the purely degenerate case, monotonicity of the shape function prevents excessive loss, and solutions lose at most one derivative. In the presence of oscillations, the resulting derivative loss is described by a minimization problem in which the degenerate and oscillatory effects genuinely interact. The proof is based on a decomposition of the time interval into frequency-dependent regions, where different adapted energies are used.

Autoregressive video models can generate minute-long videos in real time, but they produce generic subjects from text rather than specific subjects from user-provided images. Existing customization methods either require costly per-subject optimization or use pretrained conditioning networks that jointly process all video frames with bidirectional attention. Neither approach is designed for causal streaming. We present Custom Forcing, a training-free method that stores reference-based anchor frames in the persistent KV cache of a frozen autoregressive video model. However, fixed anchors face two limitations: simple conditioning allows identity to drift, and the text prompt continues to favor a generic subject. To address these problems, drift-adaptive value amplification (DVA) scales reference influence with the degree of identity drift, while anchor contrast guidance (ACG) steers generation away from the generic class prior. Over two-minute rollouts, fixed anchors fall from 0.58 to 0.42 in DINO-I, while Custom Forcing keeps it between 0.58 and 0.62 without reducing motion. Custom Forcing also achieves higher subject similarity than bidirectional customization methods and better preserves identity over 30s than causal image-to-video and reference-to-video models, while generating each frame 9.5-28.5 times faster than these long-video baselines.

For a tree $T$, let $g(T)=\lambda_1(T)-\lambda_2(T)$ be the difference between its two largest adjacency eigenvalues. A balanced double comet is obtained by attaching equally many leaves to the two endpoints of a path. Jovović, Koledin and Stanić conjectured that such a tree attains the minimum adjacency spectral gap among trees of any fixed order. We prove that every minimizing tree of order $n\ge200$ is a balanced double comet. We also show that, for any finite irreducible reversible continuous-time Markov chain, the inverse spectral gap differs from the effective resistance between two states times the stationary variance of their hitting probability by at most the inverse Dirichlet gap for killing at those states. For Perron chains, we give an exact two-vertex Schur-complement formula for this resistance–variance quantity.

We prove that the irrationality exponent of $\zeta(2)=\pi^2/6$ satisfies $\mu(\zeta(2))<5.0193784$. The best published bound is 5.095412 (Zudilin, 2014); a bound 5.0495243 with a machine-checked proof was made public in September 2026 by J. Kleid. We use Zudilin's two hypergeometric constructions at a new choice of parameters. The two families of linear forms in 1 and $\zeta(2)$ so obtained coincide for every sufficiently large index: this coincidence, an instance of an identity conjectured by Zudilin, is established by comparing an arithmetic bound for the denominators of the difference with an analytic bound for its size, without creative telescoping. All numerical constants are certified in ball arithmetic.

Methods for training language models on stale samples are judged by comparisons against importance-corrected baselines. We show that details of the experimental harness can reverse the observed ranking of methods, and we introduce PTH (Probe The Harness), a set of checks that makes the harness visible. Our case is a comparison between SAN, a behaviour-free method, and truncated importance sampling (TIS) on verl and in a single-GPU trainer, in which SAN first finished ahead in both stacks. Four details of the harness changed this comparison: the PPO ratio was taken against the learner's own recomputed probabilities, the data seed did not reach the TIS arm, the replay queue reused its first batch for 33 updates, and two loss normalisers differed from their description. In each case the logged quantity looked consistent with a working setup, while the quantity that defines the comparison went unchecked. With the harness checked, TIS matches SAN on verl, and in the trainer TIS learns steadily while SAN keeps a margin. We contribute the signature of each detail and its effect on the comparison, reference results for TIS and uncorrected GRPO under sampler lag, and the PTH checklist.

Suppressing a small set of routed experts can weaken the safety behavior of a sparse Mixture-of-Experts (MoE) language model without retraining. Which experts to suppress is therefore a security question, and the usual answer is activation frequency, but frequency measures use, not influence. We test an alternative: router-gradient sensitivity, the sensitivity of the sequence loss to the gate weights that select an expert. Across five MoE architectures, we rank experts by each signal on 500 benign and 500 malicious prompts and measure refusal on 100 held-out malicious prompts under two budgets: equal expert counts and equal nominal malicious routing traffic (1%-5%). Under each of the two budgets, router-gradient selection reduces refusals more than activation in 24 of 25 conditions, and more than a ten-trial random mean in all 25. The largest effect is in OLMoE, where refusals fall from 34 to 9 of 100 prompts (73.53% relative) with no degraded outputs, indicating substantive compliance rather than broken generation. After matching expert counts in every layer, gradient selection still produces greater refusal reduction than activation in 23 of 25 conditions, with two ties. An exploratory cross-model analysis links larger malicious-versus-benign concentration gaps to greater peak gradient effects (rho = 0.90; exact two-sided p = 0.083, n = 5). Together, the results support gradient selection under the tested budgets.

When generating programs with language models, constrained decoding can apply program analyses to exclude tokens that violate syntax, scope, or typing rules. However, there is a duplication: standard training already teaches the model to suppress the tokens rejected by these analyses. This duplication leads to the question: if we will perform some analysis to filter a set tokens out during inference anyways, can we avoid teaching the model the said analysis altogether during training, and does this externalization lead to more efficient models? This paper defines a general constraint-aware objective satisfying this externalization desideratum and formalizes the benefits of externalization into three concrete theorems about model size and data efficiency. We show, through a controlled synthetic experiment, that the theorems survive training dynamics: constraint-aware training yields lower prediction loss at a matched parameter count and data compared to ordinary cross-entropy training, motivating training objectives that incorporate the analyses used during generation.

Radio-frequency interference threatens satellite navigation, wireless communications, radar, and industrial sensing. Reliable operation therefore requires spectrum situational awareness to detect interference and determine the number, time-frequency occupancy, and characteristics of active sources. In realistic environments, multiple sources may overlap. Source-specific segmentation and characterization are therefore particularly challenging. In this paper, we propose PI-SCAN (Permutation-Invariant Segmentation, Characterization, and Assignment Network), a joint set-based learning framework for source-specific segmentation and characterization of overlapping interference signals. A logarithmic power spectrogram derived from complex IQ samples is mapped to six unordered prediction slots. For each slot, the model estimates objectness, interference type, center frequency, bandwidth, received signal strength, binary offset carrier (BOC) modulation, and a source-specific segmentation mask. Permutation-invariant Hungarian assignment aligns the predicted slots with the unordered ground-truth sources, while a weighted multi-task objective jointly optimizes the individual prediction tasks. The framework is evaluated using a ray-traced industrial environment containing mixtures of one to six simultaneously active interferers drawn from 94 waveform configurations across six interference classes. The full seven-task configuration achieves a detection F1-score of 74.8%, type and BOC accuracies of 97.3% and 98.4%, and mean absolute errors of 1.29MHz, 2.54MHz, and 3.19dB for center frequency, bandwidth, and signal strength. Using only objectness, type, and mask prediction increases the F1-score to 91.1%, highlighting the trade-off between detection and comprehensive characterization.

Residual networks execute every block for every input, yet their functional contributions need not be input independent. We formulate a trained ResNet as a set function over binary residual-branch masks and apply Möbius inversion to decompose its output exactly into individual residual corrections and higher-order interactions. For smooth residual stacks, we show that each fixed $k$-way interaction scales as $\mathcal{O}(\lambda^k)$ under residual scaling. Exhaustive analysis of ImageNet-pretrained ResNet-18 and ResNet-34 reveals that interaction mass peaks at orders five and ten, respectively, rather than at low orders. Reducing the residual scale shifts both spectra toward lower orders, but also changes model predictions. The interaction coefficients are concentrated in magnitude but not hard sparse, and prediction-preserving sparsity weakens with depth. Crucially, the dominant interactions vary across inputs and predicted classes around a shared global core, while their overall complexity changes little with sample difficulty. These results show that dense ResNets implement an implicit form of soft routing: every block is executed, but different inputs rely on different residual interaction experts. Routing can therefore emerge at the level of functional contribution without an explicit router or sparse execution.

Predicting how stochastic systems respond to changes in viscosity, reaction rates, or external forces requires costly simulations, motivating reusable learned models. Yet matching observed outcome distributions does not ensure accurate intervention responses. We introduce Tangent Schrödinger Bridge Matching (Tangent-SBM), which learns stochastic transports from endpoint observations and mechanistic sensitivities. It propagates parameter derivatives alongside trajectories and supervises them against supplied targets. For average-response targets, single-rollout squared error also penalizes response variability; our objective uses two independent rollouts to match the mean without this additional penalty. We establish conditions under which sensitivity accuracy bounds finite-change prediction error and decision regret. Across Gaussian, stochastic double-well, PDEBench reaction–diffusion, and stochastic Navier–Stokes systems, Tangent-SBM improves sensitivity and finite-change prediction over matched conditional-bridge baselines while maintaining comparable endpoint and distributional accuracy. Controls examine target correctness, response objectives, and simulator-budget allocation. To test decision usefulness, we evaluate calibrated viscosity selection in Navier–Stokes: Tangent-SBM reduces tracking error relative to taking no action on every evaluated task.

Classical reactive molecular dynamics of boron- and lithium-conditioned tungsten surfaces show that the order in which boron and oxygen are introduced - not the boron inventory - controls surface chemistry. On a thin engineered tungsten oxide, boron captures 81% of oxygen and consumes the oxide; on an engineered ~2 nm mixed WO3/WO2 film, boron captures only 20% and leaves 52% of reaction-zone tungsten in WO3-like coordination; and in a co-deposited W-B substrate later exposed to oxygen, boron captures only 28% despite a tenfold larger inventory. Adding lithium leaves the oxide-first cases unchanged but transforms the co-deposited case, where lithium becomes the dominant oxygen captor (58%) and a lithium-borate-glass-like network form. We outline the expected, divergent responses of the boron-only and boron-plus-lithium materials to deuterium irradiation.

Decentralized stability certificates allow inverter-based resources to be connected to a grid without repeating a system-wide small-signal study. Passivity provides such certificates but is violated by practical grid-forming (GFM) controls at low frequency, and its generalizations—transformation matrices, extended passivity indices, loop shifting, and gain–phase conditions—rely on multipliers that are constructed analytically or chosen heuristically. All of these certificates are linear matrix inequalities in their multipliers at every frequency, so the search for the least conservative multiplier is a convex semidefinite program (SDP) with a global optimum and a definite infeasibility verdict. This paper demonstrates how SDP can be applied to searching for optimal multipliers for some decentralized methods proposed in the literature. Then it illustrates the methodology by presenting a systematic search algorithm for the fully plug-and-play additive nodal-admittance certificate, which requires no assembled network model and holds for arbitrary interconnections. Only sampled frequency responses are needed, making the search compatible with black-box models and impedance scans. Chebyshev parameterization together with adaptive frequency splitting provides a piecewise-polynomial multiplier without manual partitioning. On the IEEE 39-bus system with ten droop-controlled GFMs, this multiplier certifies a larger region of droop gains than piecewise-constant multipliers, as confirmed by simulation of random inverter placements.

We study how to consolidate the current VITOK progress into a single multi-teacher distillation recipe that jointly preserves global recognition and dense semantics. Our starting point is an AM-RADIO-style student distilled from SigLIP2 and DINOv3-L, where SigLIP2 supplies strong global semantics and DINOv3-L supplies stronger dense features. The central empirical issue is that the same recipe does not optimize all objectives equally well: changes that improve ImageNet-1K kNN accuracy can still degrade ADE20K segmentation. We summarize a progression of modifications that make this trade-off more explicit and more manageable: split adaptor heads for CLS and patch tokens, asymmetric cosine/MSE losses, initialization from a DINOv3-L checkpoint, teacher reweighting, masked image modeling (MIM), and PHI-S feature balancing. The resulting model reaches 83.2 patch kNN and 85.2 CLS kNN, slightly surpassing the DINOv3-L teacher on ImageNet-1K kNN classification, while PHI-S restores ADE20K performance from 46.5/58.1 to 48.5/61.0 mIoU/mAcc, matching the teacher on this dense benchmark. We also summarize negative results: scaling distillation from ImageNet-1K to ImageNet22K does not consistently help, and naively adding extra teachers such as SAM3 or HOG features introduces interference. Rather than claiming a final recipe, this paper distills the current project state into a compact empirical story and a concrete set of lessons for future iterations.

Current analyses of LLMs' parametric knowledge are largely output-centric, drawing conclusions about what a model knows without verifying what it was actually trained on. This leaves fundamental questions, such as whether a correct response reflects genuine generalization or rote memorization, grounded in speculation rather than evidence. To resolve these ambiguities, we introduce LUMOS, a diagnostic framework that traces knowledge along the causal chain from training-data exposure to behavioral output, leveraging OLMo 2 with its fully transparent training corpus. By grounding analysis in verified exposure, we reveal that models internally encode rare facts with high separability (84%) yet fail to express them behaviorally (54%), though this retrieval gap narrows with scale. Furthermore, when models are asked to self-reflect on their own answers, they perform reliably on trained content (83%) but drop to random-baseline levels (49%) on unseen content. This collapse persists even under chain-of-thought prompting, which inflates confidence signals rather than improving calibration. Collectively, these findings demonstrate that incorporating the training-data axis into LLM evaluation transforms speculative diagnoses into verifiable claims, and we advocate that this axis should be a standard component of knowledge assessment in LLMs.

Consensus-based optimization (CBO) is a robust and flexible zero-order, multi-particle class of methods devised to provide provable solutions to high-dimensional global optimization problems, including truly nonconvex and/or nonsmooth cases. The algorithmic framework is grounded in a principled trade-off between stochastic exploration of the search space and deterministic contraction toward a consensus point. This thesis is concerned with three recently established variants of CBO. The $\delta$-CBO method is similar to the classic CBO method, with the difference that the diffusion function is constant. The consensus freezing (CF) scheme can be understood as a piecewise CBO method. The overall time horizon is divided into intervals in which the consensus point is fixed. The consensus hopping (CH) scheme can be understood as a deterministic iterative scheme that jumps at each iteration to a Gaussian distribution centered at the consensus point of the previous distribution. The central contribution of this thesis is the rigorous derivation of propagation of chaos results for all three schemes. For each variant, we establish that the empirical measure of the interacting particle system converges, as the number of particles $N$ approaches infinity, to the solution of the corresponding mean-field equation. Two regimes are treated: first, on a high-probability set on which the empirical measures are uniformly bounded; second, in the unrestricted setting, where moment bounds and a careful treatment of the inter-particle dependence yield convergence without imposing conditioning on favorable events. The propagation of chaos estimates are derived by the coupling method. Together, these results provide quantitative propagation of chaos estimates for all three CBO variants and establish that the particle systems faithfully approximate the idealized mean-field dynamics in the large-particle limit.

Strong-deflection lensing by a subextremal Kerr black hole exhibits a logarithmic critical divergence, whereas the exactly extremal prograde problem develops stronger power-law behavior. We construct a uniform asymptotic description of the transition between these regimes for prograde equatorial null geodesics. The distinguished joint near-critical and near-extremal limit occurs when the impact-parameter offset from the exact critical value is of order \(M\kappa\), where \(\kappa=\sqrt{1-a^2/M^2}\). In this limit, the Kerr radial potential reduces to a quadratic normal form, yielding an analytic crossover function that connects the subextremal logarithmic and extremal power-law regimes. We derive a matched next-to-leading-order correction and validate it against high-precision integrations of the exact Kerr geodesic equations, finding improvement for all 55 trajectories tested. The same asymptotic structure yields a higher-order image-branch law interpolating between algebraic and exponential accumulation, with crossover order scaling as \(\kappa^{-1}\). We also derive and numerically validate the corresponding image-branch coordinate-time delay law, whose high-order extremal-like limit approaches the horizon-period spacing \(4\pi M\).

In their seminal work, Baum-Douglas-Taylor showed that under a certain compactness assumption, local boundary conditions for first-order elliptic operators over compact manifolds with boundary define cycles for relative K-homology. Using a systematic approach to regular boundary conditions, we generalize their result to elliptic operators of arbitrary order and to non-compact manifolds with potentially non-compact boundary. In addition, we show that under stronger regularity conditions, local boundary conditions even define cycles for absolute K-homology. In contrast to the relative classes, the resulting absolute K-homology classes are in general not independent of the choice of boundary condition.

Vision-Language-Action (VLA) models have achieved remarkable advances in robotic manipulation, yet their zero-shot generalization under out-of-distribution (OOD) conditions remains limited. These models often entangle task-relevant invariant structure with environment-specific non-invariant factors, causing policies to rely on spurious appearance cues during action prediction. In this work, we propose MixVLA, a model-agnostic training framework that improves the generalization of VLA models without requiring additional OOD data or architectural modifications. The key component of MixVLA is Adaptive Mixing of Non-Invariant Information (AMI). AMI stochastically mixes non-invariant representations to regularize distribution-specific variability while preserving complementary predictive cues. The mixed non-invariant features are then fused with invariant representations for final action prediction, resulting in improved robustness without sacrificing policy expressiveness. Extensive experiments across challenging manipulation settings, including LIBERO, LIBERO-Plus, the RoboTwin perturbation suite, and real-world tasks, demonstrate that MixVLA improves overall zero-shot robustness while retaining strong in-domain performance.

A long-running assistant cannot keep everything it has seen, so it summarises. Summarising is not neutral: what is kept depends on what the record is for, and the choice is made once, before anyone knows which of the user's goals will ask. We ask what a memory should summarise for when it serves several standing goals at once. Three write policies answer differently: summarise with no goal in view, write one summary covering every goal, or write one summary per goal and read them together. We compare them with five models, two short event streams and one LLM judge, holding the read step fixed. Answers from different goal stores overlap less than answers from one store on two runs, so the goals do pull apart. Per-goal summaries read together scored highest. They beat the all-goal summary on relevance, completeness and accuracy, but the neutral one only on relevance and completeness, and at matched answer length only on relevance. The all-goal summary scored below the neutral one, which had a quarter of its budget. A single goal store served its own goal's queries better than the other goal stores, but not better than the neutral summary.

The radiometer effect is a self-thermophoretic phenomenon in which a thin body with a temperature difference between its two sides experiences a force in a rarefied gas. We numerically investigate this effect for an infinitely thin circular disk freely translating at its terminal velocity in an otherwise quiescent gas. The problem is formulated using the linearized Bhatnagar–Gross–Krook model of the Boltzmann equation with the diffuse reflection boundary condition. A major difficulty arises from the sharp disk edge, which generates discontinuities in the velocity distribution function that propagate into the gas. Accurate resolution of these discontinuities is essential for capturing the edge-localized stress associated with the radiometric force in the near-continuum regime. The discontinuities are explicitly accounted for using a characteristic-based numerical scheme, together with the appropriate far-field asymptotics for the unbounded domain. The radiometric force, the terminal velocity, and the flow and stress fields are obtained over a wide range of Knudsen numbers. The dimensionless radiometric force increases monotonically with the Knudsen number, scaling as $\mathrm{Kn}^{3/2}$ in the continuum limit and approaching a constant in the free-molecular limit. The terminal velocity exhibits a nonmonotonic dependence on the Knudsen number and slightly exceeds its free-molecular-limit value at intermediate Knudsen numbers. In the continuum limit, the magnitude of the edge-localized variation of the normal-stress difference appears to remain finite, whereas its characteristic width is of the order of the molecular mean free path. The resulting numerical data also provide reference solutions for future studies of sharp-edge kinetic problems. Finally, the computed radiometric force is independently validated using the symmetric relation for the linearized Boltzmann equation.

Quantum entanglement in molecular cavity optomechanical (COM) systems serves as a fundamental resource to explore quantum-enhanced sensing, coherent molecular dynamics, and quantum information processing, which has been studied under Markovian approximation but not yet explored in non-Markovian reservoirs. In this paper, we propose a scheme to manipulate quantum entanglement in molecular COM system coupling with degenerate optical parametric amplifier (DOPA) under Markovian approximation. We reveal that both strength and phase of DOPA can be tuned to enhance optical-vibration entanglement and suppress vibration-vibration entanglement. The optical-vibration entanglement can exist at room temperature, demonstrating strong robustness against thermal noise. Moreover, we extend results from Markovian to non-Markovian regimes, which can further enhance both types of entanglement with DOPA fixed. In particular, we find that non-Markovian effects can reverse influences of DOPA on vibration-vibration entanglement from suppression to enhancement, leading to regeneration of entanglement. The scheme provides an effective way to control and enhance quantum entanglement in molecular COM system through steering DOPA and non-Markovian effects.

Match only geometry you can identify; otherwise spread the penalty. We quantify that decision by the trust ratio tau = epsilon / gamma (estimation uncertainty over spectral separation). Under the linear-quadratic Matching response, oracle-relative drift between estimated and oracle projector matching scales as tau^2 for probes in the chosen top-r deployment subspace – O(tau^2) in the Davis-Kahan separation region tau < 1/2, with practical usefulness depending on constants. Confidence-Calibrated Matching (CCM) turns tau into a policy – directional when tau is small, progressively isotropic when not – with thresholds from calibration, not from the theorem (match sits in the separation region; soft is mostly heuristic). Experiments show both regimes, including UCI HAR embeddings where always-match is worse than abstain on every cell.

With the advance of measurement technologies and increasing computing power, large spatial data with heterogeneous structures are often collected over high-dimensional domains. Existing Gaussian process (GP) models and computational strategies are often inadequate for analyzing such datasets in multi-dimensional domains. To address these challenges, we develop a Bayesian residual tree GP methodology called ResTGP for large spatial data with potentially heterogeneous structures in multi-dimensional domains. The key idea is to decompose a Gaussian process at a cascade of resolutions along a dyadic tree through iteratively computing predictive and residual processes so that the residual process on each tree node, both interior and leaf, becomes sufficient for the finer-level dependency within that node. This allows characterization of the underlying covariance structure in a flexible, multi-scale manner while achieving divide-and-conquer on the data domain, which leads to computational efficiency. To allow efficient tree inference, we introduce a computational strategy for Bayesian inference based on recursive message passing, which scales linearly with the sample size given the tree. This paper also proves posterior consistency of the model for estimating continuous functions in a nonparametric regression framework. Extensive numerical examples and the storm surge application confirm the advantages of the proposed method.

We study how spectral averaging interacts with the logarithm in large-dimensional rank-one Harish-Chandra–Itzykson–Zuber (HCIZ) integrals. For independent bi-unitarily invariant matrices $A,B$ with uniformly bounded deterministic singular values and weakly convergent empirical spectral measures of their Gram matrices, the integral for $(BA)^*(BA)\otimes\mathbf{1}_N$ has two limiting descriptions at speed $N^2$ for small real parameters: averaging its logarithm gives free multiplicative convolution, whereas taking the logarithm after averaging gives classical multiplicative convolution. We also prove an exact finite-dimensional equality between this annealed integral and the one for $(A^*A)\otimes(B^*B)$. For every fixed number of factors $k\geq3$, we construct counterexamples showing that the analogous equality can fail even asymptotically. For tensor products with any fixed number $k$ of factors, global Haar vectors at speed $N^k$ and product-state vectors at speed $N$ yield classical and free multiplicative convolution, respectively, for small real parameters under the corresponding hypotheses. We also derive the known multiplicativity of the $S$-transform.

In this paper, we propose a highly efficient and accurate method, called “lift-and-decoupling generalized transferable neural network" (LD-GTransNet), for numerical solution of elliptic and moving interface problems. Our method makes use of the multi-layer GTransNet approach and incorporates a novel {lift-and-decoupling} strategy, in which the hidden-layer neuron parameters are preset and the physical coordinates and the added auxiliary variable are processed by separate subnetworks. Such design enables a unified global approximation of piecewise smooth solutions without explicit domain decomposition, while preserving network expressivity and avoiding the redundancy inherent in conventional {lift} approaches. The resulting formulation reduces to a linear least-squares problem, thereby eliminating the need of nonlinear training and ensuring the high computational efficiency and accuracy. Extensive numerical experiments on benchmark interface problems in two and three dimensions demonstrate that, compared with existing neural network-based and traditional numerical methods, our proposed LD-GTransNet consistently delivers superior performance, particularly for problems with high-contrast coefficients and complex interface geometries.

We propose a robust exact sliding-mode controller for reference tracking with model-based disturbance rejection, where the disturbance is generated by a linear time-invariant model driven by a bounded input and may be bidirectionally coupled with the plant. The controller embeds the internal model principle and its error dynamics mirror those of the robust exact differentiator, ensuring finite-time convergence. We develop a proper implicit discretization for sampled-data implementation that preserves stability and accuracy. In the nominal case, which contains a disturbance generated by a $p$-th order integrator chain, the method achieves a control accuracy of $T^{p+1}\bar D$, with discretization time $T$ and disturbance bound $\bar D$. This accuracy is shown to relate to the fundamental lower bound by a factor depending only on the order of the disturbance model. Simulations demonstrate the efficacy of the proposed methods.

Ultra-long gamma-ray bursts (ul-GRBs) challenge standard collapsar models because their extreme durations require central engines active for $\sim10^{3}-10^{4}$ s. While the prompt spectral evolution of typical GRBs is difficult to resolve due to their short durations, ul-GRBs provide a unique opportunity to study spectral evolution on extended timescales. We present a time-resolved spectroscopic study of GRB 220627A, one of the most luminous ul-GRBs known ($E_{\rm iso}\sim10^{54}$ erg, $T_{90}>1000$ s), consisting of two bright emission episodes separated by $\sim600$ s quiescent interval. Using a novel HOP algorithm-based segmentation method, we identify optimal time intervals for time-resolved spectroscopy. A Comptonized power-law (CPL) model well describes the prompt emission ($\sim 10$ keV-40 MeV) across all timescales. Fermi-GBM spectroscopy reveals a transition from a hard, slow-cooling synchrotron spectrum ($\alpha_{\rm I}=-0.70$) during episode-I to a softer, fast-cooling synchrotron spectrum ($\alpha_{\rm II}=-1.29$) during episode-II. This evolution closely resembles that of the magnetar-candidate GRB 091024A, highlighting a potential progenitor degeneracy based solely on spectral evolution. However, the extreme isotropic-equivalent energy exceeds the maximum energy available from magnetar spin-down, strongly favouring a blue supergiant (BSG) collapsar origin. The prolonged activity and multi-episode structure are naturally explained by sustained accretion through an extended stellar envelope. Our results demonstrate that ul-GRBs can display magnetar-like spectral evolution while being powered by BSG collapsars. GRB 220627A highlights the need for combining time-resolved spectroscopy with energetic constraints to distinguish between progenitor scenarios and understand the physical origin of ul-GRBs.

The landscape of topological insulators has expanded beyond its traditional domain of periodic lattices with short-range hopping. Related studies have explored topological phenomena under non-Euclidean geometries, disordered structures, and long-range hopping, progressively narrowing the gap between topological phases of matter and complex networks. Here we demonstrate that genuine complex networks, far beyond periodic lattices and their conventional variants, can themselves host topological insulating phases. By developing a nonconflicting design framework for multiple topological objectives, we realize topological insulators in high-degree Hofstadter lattices and across regular, small-world, and random networks. This generalization uncovers distinct network-specific features, including an expanded range of attainable topological characteristics, a degree-dependent transition from Hofstadter-type to Haldane-type Chern insulators, and, most notably, the superior robustness and reconfigurability of small-world topological insulators. Our graph-theoretic framework establishes network complexity as a resource for realizing high-capacity and reconfigurable topological insulators.