Papers

We investigate the question of equivalence between the Einstein and Jordan frames for generic scalar-tensor and $f(R)$ gravity theories in the context of minisuperspace quantum cosmology. We consider, particularly, the minisuperspace of a homogeneous and isotropic universe with the Friedmann-Lemaître-Robertson-Walker metric. We show that the equivalence between the frames depends on the operator ordering scheme of the quantum Hamiltonian: if the quantum theory is covariant under point canonical transformations in the minisuperspace, then the equivalence between the two frames holds. We find that for a large class of operator orderings, including the Laplace-Beltrami operator in the minisuperspace, the wave functions of the universe obtained in the two frames are related by the same transformations that connect the two frames, i.e., the frames are quantum mechanically equivalent. Further, we show that adding a fluid clock with an arbitrary equation of state parameter $w$ to the system breaks the equivalence, except for the case of a radiation fluid ($w=1/3$). All our arguments are presented at a formal level in both canonical and path integral approaches, and for any scalar-tensor, $f(R)$ theories without assuming any particular model.

Robotic bin packing requires long-horizon sequential decision-making, as each object placement affects the available space for subsequent packing. Existing methods primarily rely on hand-crafted geometric heuristics that optimize predefined objectives or reinforcement learning policies learned through trial and error over predefined training configurations. Despite recent advances in multimodal large language models (MLLMs) for this task, their potential for closed-loop sequential decisions across heterogeneous packing configurations remains underexplored. To address this gap, we introduce PackLab, a comprehensive framework for developing, training, and evaluating MLLMs for closed-loop robotic bin packing. PackLab-Suite provides a physics-based simulation platform for scalable generation of diverse training packing trajectories and evaluation of their physical outcomes. PackLab-VLM is a packing-specialized MLLM that understands the evolving object and container states to jointly select objects and predict placements in a closed-loop manner. PackLab-Bench provides standardized packing scenarios at multiple difficulty levels for systematic evaluation. Extensive experiments demonstrate that, on average, PackLab-VLM outperforms conventional packing heuristics, traditional reinforcement learning methods, and general-purpose MLLMs across object sets and container configurations, highlighting the potential of MLLMs for long-horizon robotic packing. The code, model, dataset, and benchmark are available at https://github.com/Correr-Zhou/PackLab .

A recent pre-print [arXiv:2609.19214] proposed a three-dimensional (3D) strongly aperiodic monotile: a shape that tiles Euclidean space only non-periodically and which admits no symmetry of infinite order. The proof takes the 3D Chair tile identified previously by Lee and Moody, and adds geometric decorations to the faces so as to force non-periodicity (without these decorations The Chair also admits periodic tilings). Here we establish general requirements on face decorations to achieve the same end, in order to facilitate the search for physical realisations. We find that the requirements are minimal. We provide matching rules using three colours of arrow that are equivalent to the original rules, in that they force the same local and global configurations. These rules force Chairs to compose into `Superchairs' with doubled linear dimensions. In this process the matching rules themselves compose uniquely. We find that the same global structure can be forced using simpler rules based on the colours of squares, regardless of orientation. Any physical system encoding these rules (geometrically or otherwise) will force the same strongly aperiodic monotilings. We provide simple examples.

Resting-state functional magnetic resonance imaging (rs-fMRI) enables the characterization of functional interactions among distributed brain regions and has shown promise for brain disorder diagnosis. However, existing deep learning methods predominantly rely on node-centric representations, where brain regions serve as the primary learning units, potentially overlooking discriminative alterations embedded in functional connections. Here, we propose an edge-centric brain transformer (EBT) framework that reformulates rs-fMRI analysis as functional connection representation learning. Instead of modeling brain regions independently, EBT constructs edge time-series representations to capture dynamic co-fluctuation patterns of functional connections and organizes discriminative connections into a line graph for explicit connection-to-connection modeling. A structure-aware transformer is developed to learn both local dependencies among anatomically related connections and global interactions across distributed functional networks. Furthermore, an edge-level orthogonal clustering readout module is introduced to derive subject-level representations and identify latent connectivity modules associated with brain disorders. Evaluations on multiple neuroimaging datasets demonstrate that EBT consistently outperforms representative graph neural networks, brain transformers, and conventional connectivity-based approaches. Interpretability analyses further reveal stable disease-associated functional connections and connectivity modules that align with known pathological network alterations. These findings establish an edge-centric perspective for rs-fMRI-based brain disorder diagnosis and provide a promising framework for discovering interpretable connectivity biomarkers. The source code is publicly available at: https://github.com/Zdy12/Edge-centric-Brain-Transformer.

Let $\sigma_{n,d}$ denote the elementary symmetric Boolean function of $n$ variables and degree $d$. We completely determine the sensitivity, average sensitivity, and block sensitivity of $\sigma_{n,d}$ for every $1\le d\le n$. Using Lucas' theorem, we obtain a uniform binary description of the Hamming-weight value sequence, from which the sensitivity and average-sensitivity formulas follow and the computation of block sensitivity reduces to at most four explicit candidates. Combining these results with the arbitrary-degree formula for certificate complexity, we determine the exact relations among sensitivity, block sensitivity, and certificate complexity. We also prove a general result for symmetric Boolean functions: every nonconstant symmetric Boolean function $f$ satisfies \[ \bs(f)\le \max\{s(f),C(f)-1\}. \] Consequently, only the three patterns \[ s=\bs=C,\qquad s=\bs<C,\qquad s<\bs<C \] can occur for nonconstant symmetric Boolean functions. For elementary symmetric Boolean functions, we give necessary and sufficient conditions for each of these three patterns, thereby completely classifying the relations among $s(\sigma_{n,d})$, $\bs(\sigma_{n,d})$, and $C(\sigma_{n,d})$. In particular, we obtain a necessary and sufficient characterization of the full strict hierarchy \[ s(\sigma_{n,d})<\bs(\sigma_{n,d})<C(\sigma_{n,d}), \] and exhibit infinite families for which it holds.

Many real-world decisions require prioritizing high-risk cases, such as clinicians prioritizing high-risk patients before lower-risk ones. Falling rule lists (FRLs), which are ordered if–then rules with monotonically decreasing risks, provide an interpretable framework for such tasks; however, their single-path structure yields a highly restricted model class. We introduce falling trees, a new family of interpretable models that enforces the same monotonic risk constraint while permitting tree-structured branching. We present GRAVITree, a novel dynamic-programming-with-bounds algorithm for learning the Rashomon set of falling trees under depth and branching constraints. Our formulation can interpolate between rule lists and full decision trees, enabling user-desired model expressivity. In a new clinical dataset and in many public classification benchmarks, falling trees match or outperform FRLs and other interpretable baselines, often producing more sparse decisions for high-risk instances. Our results show that falling trees strike a practical balance between interpretability, expressiveness, and risk prioritization for high-stakes settings.

For integers $r\ge2$ and $n\ge1$, let $$ F_{r,n}(q)=\prod_{k=1}^{n}\frac{1-q^{rk}}{1-q^k}. $$ The coefficient of $q^j$ in \(F_{r,n}(q)\) counts partitions of $j$ into parts at most $n$, each occurring at most $r-1$ times. Hughes proved that $F_{2,n}(q)=\prod_{k=1}^{n}(1+q^k)$ is unimodal for every $n\ge 1$. This result was reproved by Stanley using an algebraic approach and Odlyzko and Richmond using an analytic approach. Almkvist conjectured that $F_{r,n}(q)$ is unimodal in the following two cases: every even $r$ and every $n\ge 1$; every odd $r$ and every $n\ge 11$. He proved that this conjecture is true for $3\le r \le 20$ and $r=100,101$. In this paper, we completely settle the conjecture.

Hydrogen lines are widely used as tracers of stellar and planetary accretion. In classical T~Tauri stars, hydrogen lines are usually interpreted as arising from magnetospheric accretion columns, whereas in lower-mass counterparts (subsolar-mass objects including brown dwarfs and gas giant planets), the post-accretion-shock region can directly emit a substantial fraction of the hydrogen-line luminosity. However, the boundary between non-shock-dominated and shock-dominated cases has remained unclear. In this study, we compare hydrogen-line profiles predicted by the shock emission model with 254 observations of 164 low-mass accreting objects ($\lesssim 0.5\,M_\odot$) in the VLT/X-Shooter archive. We simultaneously fit seven hydrogen lines (H$\beta$, H$\gamma$, H6, H8, H9, Pa$\beta$, and Br$\gamma$), testing both line profiles and flux ratios within a single framework, and introduce a phenomenological broad-component-subtracted fit for cases with mixed non-shock and shock contributions. We find that shock emission dominates the hydrogen-line emission at object masses $M\lesssim0.05\,M_\odot$ or free-fall velocities $v_\mathrm{ff}<175\,\mathrm{km\,s^{-1}}$, but becomes minor at $M \gtrsim 0.2\,M_\odot$. The inferred flow velocities at the shock front are often significantly smaller than the free-fall velocity from infinity, implying smaller truncation radii and surface dipole magnetic field strengths of sub-kG. The accretion luminosities inferred from the shock-emission fitting are systematically larger than literature values, often by orders of magnitude, likely because conventional estimates neglect line emission that is non-negligible in low-mass objects. We also confirm that H$\alpha$ is more susceptible than the other hydrogen lines to additional non-shock components.

We show that Arc Kayles is PSPACE-complete. This solves a question raised by Schaefer in 1976.

At large $N$, the analytic structure of holographic thermal correlation functions contains non-trivial information about the near-singularity region of dual black holes. In particular, complex-time singularities can be related to geodesics that probe the black hole interior and bounce off the singularity. We analyse the asymptotic behaviour of holographic OPE data entering scalar two-point functions in states dual to black branes whose blackening factors truncate at finite order in inverse powers of the radial coordinate. For this class of geometries, we find a direct relation between the power-law exponent of these bouncing singularities and the Kasner exponent associated with the time direction near the black hole singularity. We discuss the regime of validity of this relation and comment on possible extensions to the transverse spatial directions.

Obtaining the optimal action-value function in Markov decision processes is computationally intensive in large state–action spaces. In this study, we present statistically rigorous convergence results for a robust reinforcement learning algorithm warm-started by a transformer-based action-value function prediction, where natural language prompts encode task specifications. Our framework adopts the R-contamination model to characterize uncertainty in the state transition kernel, and employs conformal prediction to certify convergence via trajectory-level nonconformity scores constructed from the contracting Bellman residual. The resulting conformal quantile bounds the gap between the running and optimal action-value functions simultaneously over all iterations, thereby yielding a pre-certified stopping rule that requires little knowledge of the true transition kernel. Numerical case studies on perturbed maze environments of varying size and contamination level confirm that the transformer-based warm start measurably reduces the initial error and accelerates convergence, while the proposed conformal bounds track the true error trajectory more tightly than existing guarantees.

Probabilistic inference is generally only tractable in low-treewidth graphical models, limiting its effective applicability in high-treewidth settings. Many existing methods improve efficiency by exploiting specific parametric structure, such as symmetries. However, they typically require such structure to be explicitly present, limiting their applicability to a broader range of graphical models. To address this limitation, we propose a framework where tractable inference is controlled by latent parametric structure exploitation, rather than requiring it to be explicitly present a priori. Our approach first reparameterises a graphical model as a specific tensor network representation, which we call a base tensor network. This representation yields two key properties that allow inference tractability to be controlled by parametric structure: 1) First, the complexity of inference is mainly determined by the parametric structure of a single tensor, called the base tensor. We characterise several tractable classes of base tensors for which the entire base tensor network can be contracted efficiently. 2) Second, decomposing the base tensor yields again a collection of base tensor networks. This allows inference to be naturally reduced to decomposing the base tensor into tractable components with sufficient parametric structure. We call this procedure parametric tensor decomposition. By exploiting parametric structure within the base tensor, our framework enables a novel view on inference beyond settings where such structure is explicitly present.

Decentralised federated learning replaces server aggregation with peer-to-peer model exchange, making collaborator selection a local decision under uncertainty. Fixed probe budgets waste effort on easy choices yet fall short when peers are hard to distinguish. We propose SAGE (Sequential Anchor-Gated Exchange), an optimal-stopping peer selector under a one-model-bearing-exchange budget. A receiver scores candidate neighbours on receiver-owned anchor evidence and selects once an advantage is certified. It continues probing only while further evidence repays its cost, and otherwise falls back to random gossip. We show that the stopping problem admits an optimal rule attained at a finite stage, and that the anchor schedule is order-optimal in the peer-risk gap and the confidence level. We further show that the selector never returns a peer worse than random gossip with high probability, and prove that no such guarantee holds for selectors that commit without a certificate. A separability threshold follows, below which no probing budget improves on gossip. Experiments span two image benchmarks, two graph families and three heterogeneity levels. Selectors that always act on their evidence lose to gossip in every configuration tested. SAGE-OS matches gossip on 75.5% less evidence than a fixed budget, at half the communication overhead of two published selectors. The operative decision is not which peer to rank first, but whether the evidence justifies ranking at all.

All smooth local conservation laws, symmetries and adjoint symmetries of three-dimensional homogeneous incompressible Euler flow are classified in the primitive velocity–pressure variables. Explicit dependence on time and position and arbitrary finite orders of mixed time and spatial derivatives are allowed. Up to trivial currents, the conservation laws consist of incompressibility, linear and angular momentum, kinetic energy, helicity and generalised time-dependent momentum. Every local adjoint symmetry is represented by a multiplier of these currents. Every finite-order local symmetry is equivalent on solutions to a classical point symmetry, resolving a conjecture of Olver. An explicit local first-order Hamiltonian structure of the primitive system is also established in the geometric theory of partial differential equations. It maps the energy multiplier to time translation; its kernel on local adjoint symmetries is spanned by the helicity and incompressibility multipliers. The proof first excludes higher-order symmetries by comparing their responses to transverse and harmonic characteristic directions in compatible derivative data. The Hamiltonian map and its kernel then determine all local adjoint symmetries from the symmetry classification. Exact multiplier identities and the correspondence between multipliers and current classes complete the conservation-law classification.

We prove uniqueness, up to phase rotations and translations, of the ground state for the $L^2$-critical boson star equation \[ (\sqrt{-\Delta}+1)Q =\bigl(|x|^{-1}* |Q|^2\bigr)Q, \qquad x\in\mathbb R^3. \] We also determine the kernels of the real and imaginary linearized operators. As a consequence, we show the Lieb-Yau uniqueness conjecture holds when the mass is sufficiently close to the Chandrasekhar mass.

We consider the two-dimensional barotropic compressible nematic liquid crystal flows with a planar director in the whole space or in the half-space. Under the assumptions that the shear viscosity is a positive constant and that the bulk viscosity is given by $\lambda(\rho)=\rho^\beta$ with $\beta>4/3$, we establish the global existence and uniqueness of strong solutions. It should be mentioned that our results are obtained without any restrictions on the size of the initial data and allow for the presence of vacuum. In particular, the initial orientation field is not required to satisfy any geometric angle condition.

The disambiguation of semantically similar statutory text across jurisdictions is a retrieval problem that existing methods do not solve. This inter-context conflict can steer generative models toward confidently produced answers grounded in topically relevant but jurisdictionally incorrect sources. Tobacco and nicotine regulations vary by US jurisdiction, often sharing similar language, thus, robust reasoning requires identifying which jurisdiction's law governs a given product, not merely retrieving relevant text. Emerging products (e.g., pouches) exploit ambiguous definitions to evade regulation. State-of-the-art (SOTA) document retrieval-augmented generation (RAG) methods struggle to address this inter-context conflict, and thus struggle to connect image attributes (e.g., rich attribute captions) to the set of similar legislation texts. We introduce NicoPRISM (Nicotine Product and Regulation Image-and-Text Surveillance Multimodal), comprising 161,563 images, attribute captions, a knowledge base of product, health, and legislative documents spanning 13 US jurisdictions, and 1,495 validated question-answer pairs across two tasks: policy compliance QA and product knowledge QA. We also propose PRISM-RAG, a multimodal hypergraph RAG framework built over images, captions, and entities without any LLM calls at index time, grounding every query in a product image and routes retrieval through a jurisdiction-aware context assembly mechanism guaranteeing that statutory text from the queried jurisdiction reaches the language model by construction. PRISM-RAG retrieves passages from the correct jurisdiction in 93.9% of policy compliance queries, a 48.6 percentage point advantage over standard RAG (p<0.001), using zero LLM calls at index time and one at query time, and is competitive with or outperforms SOTA RAG frameworks across keyword, semantic, jurisdiction-, and compliance-accuracy metrics.

We study optimal stopping under dynamic risk measures with simultaneous ambiguity in the probability model and the discount rate. We introduce a paired ambiguity framework combining Girsanov model uncertainty with cash subadditive risk evaluation and characterize the stopping value by an upper reflected backward stochastic differential equation (BSDE). We establish structural properties of the resulting stopping operator and study quadratic drivers associated with entropic risk measures, obtaining explicit stopping rules in several benchmark cases. We then develop a deep learning scheme for the reflected quadratic BSDE. The convergence analysis uses discrete reflection and truncation to reduce the quadratic problem to a globally Lipschitz system and combines reflected BSDE discretization estimates with neural network approximation errors. Numerical experiments for American options illustrate the effects of discount rate and entropic ambiguity on stopping values and exercise decisions.

Purpose: Research data repositories often lack variable-level metadata. Model-choice screening, checking whether a reported model suits the values of its outcome variable, also requires summaries of those values and links to the analyses that use them. We ask how repositories can represent and acquire them as metadata with evidence and review histories. Methods: We propose a metadata application profile compatible with the Data Documentation Initiative Cross-Domain Integration (DDI-CDI) model, linking versioned variables and empirical profiles to reported analyses, estimators, and analysis roles. Extraction provenance and review decisions are recorded separately. We evaluate streaming data profiling, deterministic extraction from Stata and R code, and language-model extraction of analysis records from papers. Results: A worked export of one replication package illustrates the profile: it passes shape and interchange checks and answers four queries, one of which returns the analyses that raise no alert. Screening 4,868 replication datasets from six political-science journals links outcomes to profiled columns in 1,440 deposits and flags 965, including 14 of the 21 deposits with a hand-verified linear model on a count, proportion, binary, or ordinal outcome. AI-assisted adjudication yields 55% precision on a near-balanced sample of 119 count and proportion candidates; screening rules were developed on the same corpus. Conclusion: The profile makes analysis-variable relationships queryable while preserving evidence and review history; screening yields review candidates with context. Code, metadata, and measurements are released.

Root cause analysis at a remote site is slow: evidence is scattered across pod logs, Kubernetes events and cluster-level objects, and many operators cannot send production logs to a hosted model at all. On-premise inference removes the second constraint but raises a question live-cluster benchmarks have not addressed: when one workstation GPU fixes both the model and the context budget, how should evidence be retrieved, and what happens when the runbooks the model consults have been tampered with? We present TriFleetRCA, a pipeline running entirely on one on-premise GPU that collects evidence at one of three scopes (pod, namespace, cluster), ranks it by template de-duplication then BM25, filters runbooks through an ingest guard, and returns a root cause with the evidence lines supporting it. We evaluate on a live Kubernetes cluster into which we inject four faults, so ground truth is known by construction, across 100 analyses with Qwen2.5-14B-Instruct at temperature 0. The hit rate was 0.85, 0.90 and 0.95 at pod, namespace and cluster scope; intervals overlap, but the whole scope effect comes from the one fault whose cause is a cluster-level object, and cluster scope costs 55% more tokens. De-duplication before ranking raised the hit rate from 0.75 to 0.90 at equal token cost. A poisoned runbook telling the model to delete the namespace was rejected by the guard every run; with the guard disabled the model declined to follow it in all 20 analyses, making the guard defence in depth rather than the sole barrier. Separating citation quality from accuracy proved informative: one fault was diagnosed correctly and cited incorrectly every trial, a failure mode accuracy conceals. Median latency was 1.6 s at 2,200 prompt tokens. We release the pipeline, the fault injector and all records.

Filaments in the solar atmosphere are plasma structures held aloft by magnetic fields, counteracting the pull of gravity. Analyzing their magnetic structure is challenging due to the lack of direct observations of their magnetic fields. However, through multi-wavelength intensity observations, one can infer their morphology. We identified 35 solar filaments during the maximum period of the solar cycle 24. In order to ascertain their chirality, we analyzed H$\alpha$ intensity observations. Using coronal ultraviolet intensity images, we also determined the skewness of the related coronal arcades. Furthermore, based on adjacent active regions, we classified filaments as quiescent or active, according to strength of nearby photospheric magnetic fields. We followed every filament over time to estimates their characteristics. The hemispheric chirality preference is followed by 27 (77%) filaments across all categories, based on observed bearing sense. We examined the skewness of the coronal arcades connected to 19 filaments, and they all complied with the hemisphere helicity criterion. The morphology of 16 filaments was also ascertained by integrating the skewness and bearing sense. There were two filaments with sheared arcade morphology and fourteen with flux rope shape. Additionally, by analyzing the changes in the skewness of coronal arcades and the bearing sense of filament barbs over time, we examined the variation in filament magnetic structure. Further, we examine the temporal evolution of selected filaments, which exhibit peculiar characteristics during the observational period. Lastly, we investigated the eruptive characteristics of these filaments. We did not find filaments that are more likely to explode because they do not adhere to the hemispheric chirality preference.

In this paper, we investigate two-dimensional compressible nematic liquid crystal flows with a planar director. We assume that the shear viscosity is a positive constant and that the bulk viscosity is given by $\lambda(\rho)=\rho^\beta$ with $\beta>4/3$. In the periodic domain or in a bounded simply connected smooth domain with Navier-slip and Neumann boundary conditions, we establish the global existence and uniqueness of strong solutions without any restrictions on the size of the initial data. In particular, no geometric angle condition is imposed on the initial orientation field. Moreover, we derive a time-uniform upper bound for the density and establish exponential decay estimates for the strong solutions.

For a complex $\varepsilon$-log canonical Fano surface of Picard number one, we prove that the anticanonical volume is at least $\varepsilon^2/100$ and that every integral nef and big Weil divisor has volume at least $\varepsilon^7/1728$. Both exponents are optimal. For $\varepsilon$-log canonical log Calabi–Yau surface pairs of arbitrary Picard number, we prove a lower bound of order $\varepsilon^9/(1+\log_2(1/\varepsilon))$, without a bigness assumption or further restrictions on the boundary coefficients. We also prove $\tau(X,H)\le18\varepsilon^{-3}$ for the pseudo-effective threshold on every projective $\varepsilon$-log canonical surface, with optimal exponent three. A volume-to-birationality estimate gives a sufficient bound of order $\varepsilon^{-11/2}\sqrt{1+\log_2(1/\varepsilon)}$ in the log Calabi–Yau setting.

We show that for closed negatively curved Riemannian manifolds whose length spectrum is exponentially separated, the length spectrum is a local isometry invariant. In particular, this holds for a dense set of negatively curved metrics. The main tool used in the proof is a finitary version of the Livshits theorem with an exponential tail.

Internal voids are ubiquitous defects in manufactured structures, yet their characterization remains challenging because their geometry is hidden and can only be inferred indirectly from accessible measurements. Here we introduce GenVoid, a physics-informed generative model-based framework for identifying internal voids in complex two- and three-dimensional solids from surface displacement measurements alone. By incorporating the governing mechanics into a generative inference framework, GenVoid enables void identification across linear elastic, hyperelastic and plastic material behaviours and accommodates complex two- and three-dimensional structural geometries. Importantly, the framework explicitly accounts for uncertainty and noise in displacement measurements, producing probabilistic reconstructions of internal void geometry rather than a single deterministic estimate. We demonstrate the approach using high-fidelity synthetic datasets and experimentally measured displacement fields obtained from in-situ mechanical experiments, establishing its ability to infer hidden voids from realistic displacement measurements. To quantify the fundamental limits of such inference, we further introduce an observability measure that characterizes the sensitivity of boundary measurements to localized stiffness perturbations within the interior under an ensemble of applied loads. This framework provides a direct connection between defect location, sensor configuration and reconstruction fidelity, enabling systematic assessment of how the number and spatial distribution of boundary measurements govern void-identification accuracy. To this end, these results establish a physics-informed and uncertainty-aware approach for non-invasive characterization of hidden defects and provide a quantitative basis for designing measurement strategies for inverse problems in solid mechanics.

This paper studies the two-dimensional nonhomogeneous incompressible nematic liquid crystal flows with planar orientation fields taking values in $\mathbb{S}^1$. For the initial-boundary value problem in bounded domains with density-dependent viscosity, we establish the global existence and exponential decay of strong solutions with initial density allowing vacuum, provided that $\|\nabla \mu(\rho_0)\|_{L^q}$ is sufficiently small for some $q>2$. The smallness condition is automatically satisfied when $\mu$ is constant, and hence the result yields the global existence of strong solutions for arbitrarily large initial data in the constant viscosity case. Furthermore, for the Cauchy problem with constant viscosity and either vacuum or non-vacuum far-field density, we establish the global existence and large-time decay rates of strong solutions for arbitrarily large initial data. These results are obtained without imposing any geometric angle conditions on the initial orientation field.

Forth is a concatenative language whose words share one data stack across calls, with a depth and call effects that need not be known before execution. That leaves a meta-tracing JIT compiler with no stack location it can name: a cell is reached through the stack pointer, so its accesses stay in the compiled code, and declaring every cell of the stack array instead ties what the compiler carries to the capacity the array reserves rather than to the depth a program uses. We answer with a fixed-width window over the top of the stack, a shared spill holding every deeper cell, and a decoding function that joins the two, on top of which call-entry normalization and adaptive entry are policies. Decoding shows that all of them preserve the logical stack and that a trace exit rebuilds data-stack state bounded by the window's width, not by the stack's depth. RPyForth realizes this in an RPython interpreter covering Forth's Core word set, with two scalar fields and eight frame positions and no static stack-effect analysis, and RPyFactor realizes the same window for a subset of Factor. Exposing the window to the compiler, rather than merely caching cells in it, is what pays. With the layout and the call policy held fixed, annotating the window's fields makes eighteen Shootout kernels 1.44-1.45x faster and six Appbench applications about 1.60x faster on two x86-64 machines, and 1.42-1.56x in RPyFactor. How the window is shaped and whether calls normalize it matter much less, varying by program with no setting winning everywhere. As a complete system, RPyForth is faster than gforth-fast and SwiftForth on both suites and reaches 1.90-2.36x the speed of VFX Forth on the kernels, while the applications, whose stacks are deeper and whose calls are far more frequent, remain its weak point at 0.68-0.76x. All timings measure repeated execution of an already-loaded program.

Few-shot recognition with frozen visual features is especially fragile under domain shift and one-shot supervision, where a single labelled image is an unreliable estimate of its class. We ask how far this fragility can be reduced purely at test time, without retraining the encoder or augmenting the source domain. We present Spectral Transductive Refinement (STR), a training-free transductive inference rule that exploits the geometry of the complete support-query episode. Given frozen embeddings, STR builds a joint k-nearest-neighbour graph, maps the episode into a normalized-Laplacian spectral coordinate system, initializes class representatives from the labelled support, and iteratively refines them using pseudo-labelled queries. We evaluate STR under two protocols. A controlled component study with frozen ResNet-18 features shows that spectral refinement consistently improves over single-prototype spectral initialization across five shifted domains, with the largest gains in the one-shot regime where support estimates are weakest. We then benchmark STR against recent Cross-Domain Few-Shot Learning (CD-FSL) methods using the standard miniImageNet-pretrained ResNet-10 backbone over eight established target domains. Operating entirely at inference time, STR attains the highest 1-shot average among compared methods and remains competitive at 5-shot, rivalling approaches relying on heavy source-domain meta-training augmentations. Because STR is transductive, we report its setting explicitly. Diagnostics attribute its gains to iterative refinement in spectral coordinates rather than added prototype capacity, which remains inactive in our configuration.

We demonstrate a simple and effective approach for realizing extended volumetric optical trapping using a single linearly polarized Gaussian beam propagating through a stratified medium with engineered refractive index gradients. On modifying the gradients, the focal field undergoes axial elongation and develops multiple localized intensity maxima, enabling simultaneous trapping of particles at distinct axial planes. Both on-axis and off-axis trapping configurations are experimentally observed, forming a volumetric particle distribution without the need for complex holographic beam shaping. A rigorous theoretical framework based on the Debye Wolf diffraction formalism combined with Generalized Lorenz-Mie Theory is employed to model the focused field and the resulting optical forces. The analysis, substantiated with numerical simulations, shows that spherical aberration arising from refractive-index discontinuities plays a crucial role in redistributing optical energy, leading to the formation of multiple stable trapping regions. The axial trapping potential decreases monotonically with increasing mismatch, indicating a trade-off between multi-site trapping capability and trapping potential, which is in quantitative agreement with experimental observations. This work establishes refractive index engineering as a powerful tool for tailoring three-dimensional optical force landscapes using minimal optical complexity, thus offering a scalable alternative to conventional holographic optical tweezers towards applications in colloidal assembly, micromanipulation, and biophotonics.

Lattice surgery on the surface code is a leading route to fault-tolerant quantum computation. Compiling logical circuits into lattice surgery operations is a pivotal step that determines the resource cost of a computation. Prior compilers build on idealized assumptions, and one that works in practical settings remains to be developed. Existing works are either unscalable or route one gate at a time within a layer, so the embedding depends on the order of gates and opportunities to route many gates at once are lost. In addition, they assume magic states are always available and cost-free, even though preparing one costs more than a Clifford operation. We present FlowRouter, the first topological lattice surgery compiler that places Clifford routing and stochastic magic state cultivation inside a single 3D embedding. At compile time, FlowRouter formulates the routing of an entire circuit layer as a maximum-flow problem, embedding many gates simultaneously. At runtime, it allows magic states to be cultivated at every idling patch, works with the realistic cultivation process with multiplexing and multi-stage post-selections, and absorbs the stochasticity by introducing delay mechanisms that add as little spacetime volume as possible. With magic states assumed free, FlowRouter reduces spacetime volume by 2.3x and compiles 10.8x faster than the state-of-the-art static topological lattice surgery compiler. With the cultivation-aware setting that includes both Clifford and magic state cost, FlowRouter reduces spacetime volume by 2.3x on a distance-13 surface code when compared with the state-of-the-art runtime compiler.