Papers

In our previous papers [33, 34], we established the affine Bernstein theorem and proved the interior regularity for the affine Plateau problem in two dimensions. In this paper, we prove the interior regularity for the affine Plateau problem in all dimensions and present a smooth counterexample to the affine Bernstein problem in dimensions ten and higher. We also prove an affine Bernstein theorem in all dimensions under a growth condition.

Language models are routinely fine-tuned under a fixed context, such as a generic system prompt, persona or domain-specific instruction, yet the learned behavior sometimes stays confined to that context and sometimes broadly generalizes to unseen contexts. We propose the Persona Hierarchy Model to explain this: a shared default persona influences behavior across contexts. Under this model, fine-tuning that modifies the shared persona promotes broader transfer, whereas changes to local personas remain more context-specific. Across 120 fine-tuned models spanning four behaviors and 15 training contexts, generalization narrowness positively correlates with the similarity between the training context's persona and the default persona (Pearson's r = 0.72 for Qwen3-4B). Prior fine-tuning under the default context can broaden generalization in subsequent training under other contexts. Aligning contextual responses with default-persona responses produces stronger effects. Finally, we propose persona-preserving regularization (PPR) to confine undesired contextual generalization. In RL, PPR cuts reward hacking from 42-55% to at most 0.2% under every evaluated prompt while retaining accuracy gains. These results support the Persona Hierarchy Model as an explanation for contextual generalization and can motivate future controls on unintended generalization for better alignment of LLMs.

Sparse-view 3D coronary reconstruction commonly relies on cross-view correspondence and triangulation, which are vulnerable to vessel overlap and foreshortening, or on volumetric prediction followed by vascular-graph extraction, which does not directly provide centrelines and radii. We introduce PPCAR-Net, a projection-refined parametric coronary artery reconstruction network that directly predicts a branch-structured centreline-and-radius representation without explicit point matching, triangulation, or an intermediate volume. Given a variable number of segmented views, a coarse predictor combines frozen VGGT features with learned branch queries to estimate branch presence, B-spline centreline trajectories, and dense radius profiles. Projection-guided geometry and radius refiners then sample local evidence from the input views and apply residual corrections learned with 3D supervision. We evaluate representation fidelity and sparse-view reconstruction quantitatively and qualitatively. On simulated angiographic masks generated from CT-derived coronary anatomy, PPCAR-Net produces better connected artery reconstructions and achieves strong centreline accuracy, particularly for RCA, while maintaining competitive volumetric overlap. Coarse-to-fine inference takes 121 ms, enabling real-time reconstruction.

Scribble-based interaction provides a lightweight and intuitive way for users to specify image editing intents in interactive editing tools. However, current image editing models based on VLMs or LLMs struggle to understand and execute edits based solely on scribble inputs. To systematically study this problem, we construct a new benchmark, ScribbleEdit, that evaluates the ability of image editing models to perform image editing conditioned on scribbles. This task requires both a deep understanding of the intention of the scribble and an accurate interpretation of its spatial information. In ScribbleEdit, we design an automated data construction pipeline and introduce a dedicated evaluation protocol that explicitly measures intention alignment. Our analysis reveals that existing VLM/LLM-based editing models fail to accurately capture scribble intentions. To guide future progress on scribble-only image editing, we propose a simple yet effective soft-token baseline, which enhances the model's understanding of scribble semantics and outperforms standard image editing models on our benchmark. Our evaluation and baseline together provide a concrete foundation for assessing and improving the scribble-driven image editing.

This paper establishes a finite-aperture model for pattern-reconfigurable antennas (PRAs) and exploits its aperture-constrained Fourier representation to develop an efficient multiuser beamforming framework. First, we develop an ideal model by representing the complex radiation pattern using a Fourier series. By applying the Jacobi-Anger expansion, we relate the antenna aperture to the effective Fourier order and derive aperture-dependent power bounds for individual Fourier coefficients. Second, we employ a practical multiport network model and design three representative reconfigurable pixel antennas to verify the predicted effective Fourier orders and per-order power constraints. We then formulate weighted sum-rate maximization for multiuser downlink transmission under both models. Using bounded Fourier coefficients as auxiliary variables, we develop a coefficient-bound penalty-based weighted minimum mean-square error algorithm that separates continuous beamforming from discrete switch-state optimization. We further propose a one-bit neighborhood search that exploits the Sherman-Morrison identity to accelerate the discrete optimization. Simulation results validate the physical relevance of the ideal model and demonstrate multiuser sum-rate gains and reduced optimization runtime for densely switched antennas. The proposed model provides a physically grounded basis for PRA-enabled communication analysis and design, while the framework accommodates different beamforming objectives and PRA implementations.

Let $G_{n,d}$ be a uniformly random simple $d$-regular graph on $n$ vertices, and let $A_n$ be its adjacency matrix. For every fixed $\lambda\in(0,1/2)$, we prove that for any $\lambda(n-1)\le d\le(1-\lambda)(n-1)$, $\mathbb P(A_n\text{ is singular})\le e^{-cn}$, where $c>0$ depends only on $\lambda$, and $n$ is sufficiently large with $nd$ even.

Quasi-nonsingular relative orbital elements (qnsROE) are among the most widely used state representations for formation flying and rendezvous in near-circular inclined orbits. They admit closed-form perturbed state transition matrices (STMs) and a direct geometric mapping, which underpins eccentricity/inclination (E/I)-vector separation for passive safety. Existing eccentric extensions of the qnsROE resolve this geometry in the radial–tangential–normal frame, where a residual semimajor-axis difference makes the E/I separation margin phase dependent. This paper introduces eccentric ROE that reduce to the qnsROE as \(e\to0\) and whose first-order geometry is resolved in the velocity–normal–binormal (VNB) frame, where the secular drift is confined to the velocity direction. Their first-order position and velocity maps yield a closed-form, phase-independent binormal–normal separation bound with an explicit margin for semimajor-axis residuals. Closed-form Keplerian and secular-\(J_2\) STMs and a decoupled impulsive control input matrix are derived for the same elements, and the elements are recovered as the physically normalized integration constants of the Cartesian equations of relative motion in the VNB frame. Together, these results provide a unified ROE framework for formation flying in general elliptic orbits.

We establish sharp local-in-time dispersive estimates and global Strichartz estimates for the semiclassical Schrödinger equation in a three-dimensional cylindrical domain $\Omega\subset\mathbb{R}^{3}$ with homogeneous Neumann boundary conditions. The partially flat boundary leads to a degeneracy of curvature in the longitudinal direction and produces persistent glancing interactions that distinguish the cylindrical setting from strictly convex domains. At the spectral level, the Neumann condition is reflected in the appearance of the zeros of the Airy derivative in place of the Airy zeros associated with Dirichlet boundary conditions. Using semiclassical microlocal analysis, oscillatory integral estimates, and an Airy–Poisson summation formula adapted to Neumann reflection, we obtain uniform dispersive bounds for the localized Schrödinger propagator. The quadratic structure of the Schrödinger phase yields a uniform treatment of the low-frequency regime and, combined with the boundary-layer analysis, leads to global Strichartz estimates with derivative loss $$ \rho(q)=\frac{3}{2}\left(\frac12-\frac1q\right). $$ As an application, we prove local well-posedness in $H^{s}(\Omega)$, $s>1$, for the cubic nonlinear Schrödinger equation with either focusing or defocusing sign and homogeneous Neumann boundary conditions. These results quantify the effect of partially flat boundary geometry on Schrödinger dispersion and provide a sharp linear framework for the corresponding nonlinear dynamics.

Taxonomies are the symbolic representations through which AI systems organize evidence, aggregate patterns, and answer questions over large document collections. Over customer feedback, the category tree decides how every record is counted and routed, which problems get seen, and which team owns them. Agentic harnesses now make it easy to generate a plausible-looking hierarchy, and such trees are checked today with generic, individually scoped checks: each name fits its description, sits under the right parent, and stays distinct from its siblings. We ask a more operational question: when is a generated hierarchy actually useful as a production taxonomy? We build six taxonomies over two proprietary feedback corpora (1,940 and 5,000 records): for each corpus, a production reference and two repeated runs of the same harness under identical inputs. All six pass every generic naming and structure check, and a deeper product-coverage check even prefers the generated trees. Yet in every generated tree at least 97.7% of leaf names merely restate an ancestor's name (13.9% and 2.9% in the references), and in one, three of every four records fall under multiple top-level categories. We introduce two families of whole-tree metrics: structural discriminators test whether a tree's shape was learned from the data or imposed by its generator; team partitionability tests whether branches split feedback into groups teams can own. Trees the generic checks rate as equally correct differ by 27 percentage points in cross-branch leakage, and only one of the two beats a random split. Surface plausibility is an insufficient measure of taxonomy quality: evaluation must measure the whole tree as well as each node.

Acquiring a complete set of magnetic resonance imaging (MRI) contrasts is time-intensive and uncomfortable for patients, despite the diagnostic value of multi-contrast imaging. This motivates synthesizing missing contrasts from those already acquired, which is an inherently 3D problem requiring anatomical coherence across axial, sagittal, and coronal planes. However, fully 3D generative models are often impracti- cal under computational resources that scale cubically with volume size. We propose a unified Multi-Plane Autoregressive Diffusion (MPAD), a latent diffusion framework that achieves full-volume 3D synthesis using efficient plane-wise 2D operations while preserving volumetric coherence. A 3D autoencoder first compresses MRI scans into an isotropic 3D la- tent representation. A 2D diffusion model is then trained to reconstruct masked latent slices of the target contrast, conditioned on both source- contrast slices and unmasked target-contrast slices. During inference, we introduce plane-wise autoregressive synthesis with inter-plane priors. Slices are generated autoregressively in random order within one plane orientation to maintain intra-plane continuity, then propagated as con- ditioning priors to orthogonal plane orientations to enforce inter-plane consistency. Compared to 3D latent diffusion baselines, MPAD reduces training and inference FLOPs by 7x and 3x, respectively, while also lowering inference time and peak memory consumption. Experiments on multiple datasets demonstrate that MPAD achieves superior perfor- mance, generating high-fidelity 3D volumes and supporting one-to-many translation within a single unified model.

Organizations increasingly use frontier language models to analyze customer feedback, but answer quality also depends on how that feedback is organized and made available. We define a customer context graph as a unified model of customer and business context. Typed relationships connect customer objects (feedback, conversations, users, and accounts), operational objects (tickets, support agents, opportunities, and competitors), and analytical or action objects (taxonomy concepts, evidence, insights, work items, and outcomes). This lets an agent investigate not only what customers say, but why, who is affected, what action followed, who owns it, and whether it was resolved. For this experiment, the graph is populated from public Cursor feedback; the same architecture can support any type of feedback source. We compare Agentic RAG, a Deep Research Agent, and a Customer Context Graph-backed Agent on the same 9,432 public Cursor feedback records using 30 realistic product, incident, comparison, and metadata questions. Without exhaustive ground truth, we jointly score responses on answer quality (coverage and organization), analytical depth (specificity and decomposition), and evidence quality (citation support and traceability), using a comparative rubric calibrated on 28 of the 30 questions. We sample cited records against their claims and weight the three dimensions equally. Under this aligned rubric, the Customer Context Graph-backed Agent scores 0.961 overall, versus 0.710 for the Deep Research Agent and 0.651 for Agentic RAG, and leads the Deep Research Agent on 27 of 30 paired questions (sign-test p < 10^(-5); strictly best on 26 of 30). Its largest advantage is analytical depth (0.967 versus 0.642), reflecting more specific, hierarchically developed findings with quantified themes and traceable evidence...

Urban diagnosis integrates heterogeneous observations to identify urban problems, localize affected areas, and investigate contributing factors, informing evidence-based urban planning and management. However, its reliance on labor-intensive, case-specific expert workflows limits scalability and reuse, motivating the exploration of agent-based execution. To evaluate this capability, we introduce DUDA-Bench, a hierarchical and interactive benchmark that formalizes data-driven urban diagnosis as a multi-stage agent workflow. It comprises 86 atomic and 22 workflow tasks spanning four analytical stages, grounded in multimodal data from 12 cities covering five urban problem types. Evaluations of seven backbone models and five agent systems reveal a substantial gap between isolated analytical competence and end-to-end diagnosis, with system benefits varying across backbones. Trajectory analysis shows that unresolved evidence gaps propagate across stages, while successful recovery involves revising assumptions and actions using feedback. These findings highlight limitations in coordinating analytical capabilities across stages, particularly adaptive planning, evidence integration, and verification. More broadly, DUDA-Bench provides a framework for translating expert analytical workflows into hierarchical agent tasks and process-aware evaluation, supporting systematic assessment of end-to-end analytical capabilities.

We develop an end-to-end framework for estimating canonical thermal properties of particle-number-conserving second-quantized Hamiltonians using quantum simulations. The framework integrates initial state preparation with mixed Dicke states, particle-number-preserving Trotterized simulation, and generalized quantum signal processing to estimate canonical partition functions and observables. Building on this canonical scheme, we introduce a grand-canonical reconstruction scheme in which the same quantum measurement data can be reused for arbitrary chemical potentials through classical reweighting, thereby avoiding repeated quantum circuit executions as the chemical potential is varied. We prove the existence of system-size-independent temperature regimes in which both the canonical and reconstructed grand-canonical schemes have lower asymptotic time complexity than classical Lanczos methods, up to polynomial factors, for physically relevant Hamiltonians. We numerically validate our theoretical results for the two-dimensional Hubbard model in both weak- and intermediate-coupling regimes.

Mixture-of-Experts (MoE) layers replace the feed-forward block of a Transformer with E expert networks, and each token is routed to k of these experts. Under expert parallelism (EP) the experts are distributed across GPUs, and every MoE layer runs all-to-all collectives in the forward and backward passes to dispatch tokens to their experts and then combine the results. On a cluster with 8 AMD Instinct MI300X GPUs per node, these collectives can take 45% of the training step at EP32 with top-2 routing and 60% with top-6 routing. We find that early in pretraining routers have already learned to assign tokens to experts in correlated patterns, both within a layer and across layers. At top-2, 0.8% of the expert pairs in a layer are selected together by 42% of tokens, and the experts a token selects at one layer predict the experts it selects at the next layer. We use these correlations to keep more token–expert assignments on the token's own GPU, which reduces communication across GPUs and across nodes. Correlated expert placement puts experts that are often selected together on the same GPU. Combined with a dispatcher that sends each token to each GPU once, it removes up to 58% of dispatched rows. Token shuffling applies when sequence parallelism shards tokens across the EP group. It moves each token to the GPU predicted to hold its next-layer experts during the reduce-scatter that follows attention. On one node this raises the share of token–expert assignments served on the token's GPU from 12.5% to 59%. In Megatron-LM, across EP degrees from 8 to 64 with top-2 and top-6 routing, the two methods reduce all-to-all time by 1.16-2.63X and end-to-end step time by up to 1.41X. Neither method changes the models' underlying routing decisions or expert parameters.

Calibrated uncertainty quantification is essential to ensuring AI agents are trustworthy and reliable. However, when agents seek to maximize user engagement or revenue, confidence reports may be strategically distorted, detracting from their informativeness. We formalize this problem in the Confidence Game: a repeated signaling game with imperfect monitoring in which an agent of unknown honesty and ability reports its confidence, and a user decides whether to delegate the task or complete it herself. The agent manages the tradeoff between manipulating signals and maintaining its reputation. We characterize the Markov Perfect Bayesian Equilibria of the two-period game and show that honest reporting is not an equilibrium, inflation is the unique best response once the agent is sufficiently myopic, and under-reporting requires that the user believe honesty to be a minority. We then place an LLM in the agent role, supplying it with its true probability of success so that any gap between what it knows and what it reports is attributable to incentives rather than to miscalibration. The model claims high confidence on 56% of tasks it has been told it will probably fail. This persists on real tasks, where it must estimate its own accuracy and causes miscalibration to increase while the agent's signal becomes less informative. Furthermore, we find that the LLM agent's decisions are coherent, but it systematically underestimates both how likely the user is to delegate and how secure its reputation is, resulting in less extreme behavior. Pricing the agent's reporting rule, we find that it destroys 68% of the gains from delegation, of which 71% is information the report no longer carries and no amount of user sophistication recovers. Overall, we establish confidence reporting under delegation as a strategic problem and provide a tractable basis for modeling, analyzing, and testing agent behavior.

Let $p\ge 7$ and suppose that $p$ and $p-2$ are prime. We study $E_p:y^2=x(x-2)(x-p)$ using the classical $2$-Selmer calculation of Qiu-Zhang and the Cassels-Tate pairing. These give $2^\infty$-Selmer corank one for $p\equiv 3,5\pmod{8}$ and corank zero for $p\equiv 7\pmod{8}$. Assuming the low-corank Birch-Swinnerton-Dyer statement announced in the October 2026 OpenAI mathematics release, we deduce equality of the analytic and algebraic ranks in these cases, finiteness of the full Tate-Shafarevich group, and the exact BSD leading-term formula. The $2$-primary Tate-Shafarevich group is trivial, and the remaining factor has odd square order. We identify precisely the obstruction left in the class $p\equiv 1\pmod{8}$. An appendix gives a local descent proof of the known Selmer dimensions in the coordinates used here.

When diffusion policies were first introduced, they were expected to recover multi-modal action distributions. However, we find this expectation does not always hold, as diffusion policies often collapse to a single modality even when we guarantee the balance of dataset modalities and exact within-batch symmetry. Our analysis indicates that independent action-noise pairing contributes to this failure by increasing mixing and crossing among diffusion paths, which can produce averaged denoising responses and suppress modality-specific behavior. This issue is especially severe in robot planning, where action spaces are dense and low-dimensional, significantly increasing such mixing and crossing. To alleviate this problem, we propose Immiscible Diffusion Policy, a label-free training-time add-on to diffusion policy that uses action-noise assignment to preserve relatively distinct noise-to-action routes without modifying the policy architecture or inference procedure. Across five simulated and two real-world humanoid manipulation tasks spanning state, RGB, and point-cloud observations, our method significantly improves the policy's preservation of action modalities while maintaining strong task performance. It increases the proportion of the non-dominant modality by 6.0x-14.6x across three two-modality tasks and recovers demonstrated modalities that are entirely absent from vanilla policy rollouts on both four-modality tasks. These results demonstrate that Immiscible Diffusion Policy provides a simple yet robust approach to preserving action multi-modality in general robot learning tasks.

Motivated by an observation concerning the $S_3$-conjecture, we prove that every abelian composition factor of a finite rational group has order $2$, $3$, or $5$. Thompson previously restricted these orders to $\{2,3,5,7,11\}$; we exclude $7$ and $11$, establishing the bound that he proposed. Together with the work of Feit and Seitz on nonabelian composition factors, this completes the determination of the possible composition factors of finite rational groups.

Network coding lets the nodes of a network combine messages rather than merely forward them. In undirected networks, where the two directions of an edge share its capacity, Li and Li conjectured in 2004 that coding offers no advantage over fractional routing; the conjecture has been confirmed for many special classes but never settled in general. In this paper, we disprove it by constructing a finite connected simple undirected network with unit shared edge capacities, maximum degree three, and distinct leaf terminals, on which a binary linear block code achieves a common rate strictly above the maximum fractional routing rate. The construction turns a short completion-time code into a reversible circuit whose registers all carry independent messages, and a temporal metric on its wires yields the strict routing bound. The underlying integer-coefficient construction works over every finite field and every nontrivial finite abelian group; for deterministic fixed-schedule codes the common-rate supremum is one and the closure of the rate region is the unit cube. Amplifying the gap with a degree-preserving tensor construction and an even subdivision gives, for every $0<\beta<1$, an unbounded family of connected, simple, subcubic, bipartite networks with girth at least $n^\beta$ on which coding approaches rate one while routing is bounded by $K_\beta/(\log n)^c$, with one positive exponent $c$ independent of $\beta$. The finite counterexample and the family theorem are formalized in Lean.

This paper investigates the behavior of learning-based co-optimization for transmission expansion under strategic bidding in electricity markets. In this framework, transmission capacities are updated while market participants simultaneously learn their bidding strategies through deep reinforcement learning, resulting in coupled and non-stationary learning dynamics. We show that transient policy degradation of bidding agents can generate inconsistent cost-capacity samples, which bias the transmission-capacity update and prevent the co-optimization process from converging to the desired solution. To mitigate this, replay-based capacity updates with nearest-neighbor filtering are introduced to exclude inconsistent samples from the update data. We then analyze a new oscillatory behavior that appears when the replay memory is enlarged. Numerical results on the IEEE 30-bus system demonstrate that enlarged replay memories improve robustness against transient policy degradation but can introduce a temporal lag, leading to oscillations unless the capacity-update learning rate is appropriately reduced. These results reveal the trade-off between replay-memory size and learning rate and provide practical guidelines for stable co-optimization.

Kühn and Osthus proved that, for every positive integer $\ell$, every $2^{16}\ell^2$-connected graph $G$ has a partition $V(G)=S\cup T$ such that $G[S]$ and $G[T]$ are $\ell$-connected and $d_T(v)\geq \ell$ for every $v\in S$. And they asked whether the quadratic bound can be replaced by a linear bound. In this paper, we answer this question in the affirmative by proving that every $641\ell$-connected graph $G$ has a partition $V(G)=S\cup T$ such that $G[S]$ and $G[T]$ are $\ell$-connected and $d_T(v)\ge \ell$ for every $v\in S$. The proof integrates the Moser-Tardos resampling algorithm.

Newton-type algorithms are among the most effective methods for solving optimization problems because of their rapid local convergence. However, extending these methods to nonsmooth optimization is challenging due to the difficulty of incorporating second-order information. To address this issue, we propose a coderivative-based Newton framework for solving both unconstrained and constrained nonsmooth optimization problems using tools from variational analysis and generalized differentiation. The proposed method employs generalized Hessians, defined as coderivatives of the subgradient mapping, and is applicable to both $C ^{1,1}$ functions and convex composite optimization problems with extended-real-valued components. To enhance robustness, the algorithm incorporates an adaptive regularization strategy that enables it to handle problems whose generalized Hessians are positive semidefinite. In addition, a Hybrid Adaptive Nonmonotone (HAN) line search scheme is developed as a globalization technique to improve global convergence and practical performance. Under standard assumptions, both the exact and inexact versions of the algorithm are globally convergent and achieve local superlinear convergence when the associated subgradient mapping satisfies the semismooth* property. The proposed framework is further extended to convex composite optimization through the forward-backward envelope, allowing it to handle problems with or without strong convexity in the smooth component of the objective function. Numerical experiments on Lasso, logistic Lasso, and support vector machine (SVM) problems demonstrate the efficiency and robustness of the proposed algorithm. Comparisons with several well-established first-order and second-order methods for nonsmooth optimization show that the proposed approach is computationally competitive while maintaining strong convergence properties.

Contrails are the thin ice clouds that aircraft leave behind. They cause a large share of aviation's warming, and rerouting the few flights that produce them could avoid much of it. However, an avoided contrail only counts if a satellite can confirm that it never formed, and this check is hard: contrails are one to two pixels wide, cover only 0.18% of pixels, and look very similar to natural cirrus. We build a small diffusion model (8.4M parameters, trained on one GPU) that detects them, and we run a controlled study to find out which components matter. The model reaches 0.476 PR-AUC, compared with 0.414 for a DeepLabV3+ baseline and 0.119 for an adapted MedSegDiff. Doubling the input resolution of the CNN brings it to parity (0.499, p=0.07). Three lessons apply beyond contrails. First, check the input resolution before designing a new architecture. Second, simple flips and rotations more than double accuracy and matter more than any architectural choice we measured. Third, pretraining the model on contrail shapes is harmful: the model learns that thin strokes appear everywhere and paints them onto empty scenes. Precision collapses to 1% while recall-based metrics still rate the degraded model as excellent, and no threshold or guidance heuristic repairs this failure.

The $r$th-order Stirling cycle and subset triangles and their associated quasi-Eulerian triangles were introduced by Deb and Sokal in their study of total positivity of combinatorial triangles. They found combinatorial interpretations for the cycle case in terms of Stirling permutations, leaving the subset case open. For $r\ge 2$, we resolve this problem by introducing the notion of Stirling subset permutations along with a consecutive-descent statistic. We also prove the conjectures of Deb and Sokal on the row log-concavity of the higher-order Stirling cycle and subset triangles. Our log-concavity proofs rely on Sagan's criterion, Dey's extension, and strengthened log-concavity inequalities discovered with the assistance of ChatGPT 5.6.

Retrieval-Augmented Generation (RAG) is widely used to ground answers in documents. For complex technical documentation, however, the primary bottleneck is often not model reasoning but a mismatch between a query and the way knowledge is organized for retrieval. This mismatch is pronounced in Electronic Design Automation (EDA) documentation, where the information needed for an answer is scattered across heterogeneous yet tightly coupled artifacts. We therefore redesign the basic retrieval unit of RAG. Instead of operating on isolated chunks or binary relations, we collect typed artifacts into EDA functional units. Each unit is recorded as a hyperedge with links to its source chunks. We then train an encoder to align queries with functional units and combine unit retrieval with direct chunk retrieval. After mapping the selected units back to their sources, a unified reranker chooses the evidence given to the generator. On the newly constructed EDADocEval-QA dataset, our method improves ROUGE-L by 37.1% over Chunk RAG and 55.6% over the strongest graph baseline. On the public ORD-MMBench benchmark, it improves ROUGE-L by 30.0% over the strongest baseline. These results support function-aware evidence organization in the evaluated EDA documentation settings.

Real-world documents distribute evidence across text, tables, figures, and captions within complex page layouts. Answering complex questions over such documents therefore requires more than retrieving relevant passages: systems must recover the evidence topology that connects heterogeneous evidence units. Existing GraphRAG evaluations remain largely text-centered, while multimodal document RAG benchmarks assess cross-modal retrieval and generation without directly evaluating recovery of the intended evidence topology. We introduce TOPOGRAPHRAG-BENCH, a layout-grounded benchmark for multimodal evidence reasoning in GraphRAG, comprising 2,024 questions over 201 long, visually rich documents. Questions are constructed bottom-up from text, figure, and table evidence units under three controlled topologies: single-hop retrieval, bridge-chain reasoning, and multi-source synthesis. To ensure that questions preserve their intended structure, we apply counterfactual validation for shortcut resistance, modality necessity, and evidence necessity. We evaluate text-only GraphRAG, page-level visual retrieval, and multimodal GraphRAG systems using retrieval, generation, and topology-aware reasoning metrics. Multimodal GraphRAG systems achieve the strongest overall performance, but still fail when visual-textual evidence alignment or multi-unit composition is incomplete. Text-only GraphRAG struggles when key dependencies are grounded in figures or tables, while page-level visual retrieval lacks the fine-grained structure needed for topology recovery. These findings motivate GraphRAG systems that move beyond text-derived entity relation graphs to explicitly model document layouts, cross-modal evidence alignment, and the reasoning roles of evidence units. Code and data are available at https://richardlrc.github.io/TopoGraphRAG-Bench/.

In a standard two-conductor microwave transmission line, both the signal trace and the wide ground plane act as parasitic heat leakage paths along the propagation direction. To prevent this heat leakage, we devised a single-conductor microwave interconnect: the ground plane is removed entirely and the remaining signal trace is periodically fragmented into physically isolated metallic islands. We show that despite the galvanic isolation between adjacent metal islands, a surface confined mode of specific symmetry, the odd-symmetric spoof surface plasmon polariton, propagates through this periodic, single-conductor medium. This odd symmetry imposes an electromagnetic field null at the waveguide centerline, allowing complete removal of the metal at periodic intervals along the single-conductor waveguide without impacting EM propagation. Measurements on fabricated prototypes confirm the transmission response despite the physical fragmentation of the conductor, with de-embedded propagation loss of ~0.0031 dB/mm across a 5-6.5 GHz passband. Time-domain characterization verifies 1 Gbps amplitude-modulated data transmission with less than 145 ps group delay variation. Coupled electro-thermal simulation yields an effective thermal conductivity of 0.336 W/m*K, a 40x reduction relative to a continuous microstrip line of identical footprint.

Robots are increasingly expected to provide personalized services in everyday environments. To do so, they must ground natural-language commands such as "Where is my backpack?" or "Find my bottle" and execute them by reasoning about object instances, people, locations, and ownership. This is challenging because ownership is rarely labeled explicitly and must be inferred from long-term, behavioral evidence of human-object interactions. To address this, we present COOL, a novel robotic framework for autonomously learning object ownership from everyday observations and maintaining a long-term spatial memory of its environment. To keep its memory current, COOL uses an agent-based curiosity-driven data collection strategy that guides the robot toward the most promising locations to gain information and refresh stale observations. Offline experiments, ablation studies, and real-world evaluations show that COOL can infer ownership relations from real-world interactions and use this knowledge for ownership-conditioned navigation and task execution.

We prove positive definiteness and quantitative condition-number bounds for continuous Sobolev Gram matrices of deterministic shallow networks with Gaussian and tanh-difference activations. The prescribed feature families include the optimal Sobolev approximation constructions of Part I. On every fixed nonempty bounded Lipschitz domain in $\R^d$, $d\ge2$, and for every fixed integer $m\ge0$, the raw and column-normalized Gram matrices in the full $H^m$ inner product have spectral condition numbers bounded by $\exp(CN^{2/d}/\log(N+2))$ for sufficiently large $N$, where $N$ is the feature count and $C>0$ is independent of $N$. The upper bounds hold for midpoint and Gauss-Legendre offsets. For midpoint offsets and a suitable angular sequence, matching lower bounds on fixed ellipsoids establish sharpness of this exponent over the stated domain class for raw Gaussian matrices and both raw and normalized tanh-difference matrices. The proof combines homogeneous differential operators that isolate direction groups with one-dimensional coefficient-recovery estimates. Two-dimensional $L^2$ experiments on a square illustrate condition-number growth and normalization effects.

We prove global exponential (linear) convergence with an explicit rate in the rich scaling for wide two-layer linear networks trained with smooth Polyak-Lojasiewicz predictor losses. Gradient flow in the factors closes exactly in terms of a finite-dimensional Bures flow of the neuron law covariance, in which the predictor dynamics are preconditioned by hidden covariance blocks. Mean-field conservation laws provide uniform spectral lower bounds on the hidden preconditioning blocks when the initial covariance satisfies a spectral support gap condition. This condition encompasses positive definiteness while still allowing for singular initializations. For an initial covariance $\Sigma_0 = \sigma^2 \mathrm{Id}$, the loss converges to the global minimum with linear rate at least $4\sigma^2\kappa$, where $\kappa$ is the PL constant. We establish stability of this rate under finite-width sampling, as well as global convergence of factor gradient descent for an explicit stepsize interval depending on smoothness, the initial loss, and conserved spectral margins. Our argument extends layerwise to deep linear ResNets, subject to a residual-path bound. In the case of heavy-ball momentum, training dynamics close instead over positions and velocities in terms of a lifted phase covariance. Linear convergence holds under an explicit condition on the energy and damping, specifying a window of admissible dampings. For two-scale white initializations, this interval is nonempty for sufficiently large position scales, with a fixed initial loss gap and velocity covariance. Numerical experiments illustrate the covariance geometry and compare the predicted and observed rates.