Papers

Vision-language model (VLM) agents can control robots through visual feedback and action primitives, but repeated LLM calls and redundant observations incur substantial token overhead. We introduce PyRUA-Lean, an interactive code-execution framework in which the agent composes robot primitives, including vision-language-action (VLA) policies, into Python cells that check and retry locally and return only the images and state they request. Across 700 simulated task instances from LIBERO-PRO, RoboTwin 2.0, and RoboCasa365, it raises success from 63.1% to 71.7% over tool calling with the same GPT-6 Astra planner and primitives, using 49% fewer LLM calls and 65% fewer input tokens on jointly solved instances. On RoboDojo, without any VLA policy or privileged information, it reaches 51.6% average success rate (Score 58.2), 19.4 points above the top entry of the public leaderboard. Based on the PyRUA-Lean framework, we train PyRUA-Lean Coder v0.1-4B, an open 4B model, with GRPO on 45 LIBERO-PRO instances; it improves from 18.8% to 44.1% on unseen task instances, surpassing the Qwen3.5-27B base model.

Exam-style accuracy does not establish whether large language models (LLMs) reason well over clinical records. We define clinical reasoning as integrating and updating evidence across time and sources to form, revise and justify a patient's problem representation and a defensible plan. This structured narrative review maps three literatures: medical education assessment instruments, clinical LLM benchmarks published from 2023 onwards, and general-domain methods for evaluating long-form generation. We examine six dimensions: problem representation, temporal synthesis, differential and management reasoning, counterfactual reasoning, calibrated uncertainty, and reasoning faithfulness. Preprints are included and flagged. No single instrument covers all six dimensions. Problem representation and differential or management reasoning are reasonably covered, although reliability varies by instrument and setting. TIMER-Eval targets temporal synthesis, and ER-Reason assesses sequential diagnostic belief updating. Dedicated uncertainty and counterfactual evaluations are emerging, but their applicability to longitudinal free-text reasoning remains limited. Factual completeness is well theorised in general-domain evaluation, with early clinical evidence of important omissions. Faithfulness remains the weakest dimension, with one identified clinical causal-ablation study on multiple-choice questions. Existing tools should be combined through binary rubric items, separate completeness and correctness scores, case-specific importance weighting with non-compensable safety caps, temporal order-consistency checks, and chance-corrected reliability reporting. Further design work is needed for calibrated uncertainty, counterfactual reasoning and faithfulness over longitudinal free-text records. This review provides a design rationale, not a validated instrument.

Topology is, by its nature and design, suited to structure that is nonlinear, multiscale, and nonstationary - however, within machine learning, its use remains largely confined to topological data analysis. We advocate that tools from low-dimensional topology which have remained almost exclusively contained within the domain of pure mathematics (such as Morse theory) offer a strong, complementary, and yet virtually unexplored perspective on the hidden structure of data-generating processes and learning tasks built upon them. Here we introduce concepts from cobordism theory and harness tools from discrete Morse theory to improve the performance of graph diffusion models through our pipeline MG-Diff. Further, we derive theoretical guarantees and sufficient conditions so that under a positive decision-gap, the Morse-theoretic tools and their application for induced diffusion guidance are stable under small perturbations. Finally, we illustrate the utility of discrete Morse theory in application to graph diffusion models for spatio-temporal graph forecasting and graph regeneration, and argue that these applications are only a small window into the part of what low-dimensional topology can offer to the field of machine learning.

Large Language Models (LLMs) have demonstrated remarkable fluency and versatility across natural language tasks but remain fundamentally limited by their static knowledge and susceptibility to hallucinations, especially in domains requiring up to date or attribute grounded information. Retrieval Augmented Generation (RAG) addresses these challenges by integrating external retrieval mechanisms with generative models, enabling dynamic, context aware generation grounded in verifiable data sources. This survey presents a comprehensive examination of RAG as a modular and evolving paradigm that enhances factual reliability, adaptability, and task alignment in LLM based systems. We formalize the RAG framework through its three foundational components retrieval, generation, and augmentation and survey state of the art methods spanning dense and sparse retrievers, fusion strategies, embedding optimizations, and reinforcement learning based retrieval policies. Anchored around four emerging axes efficiency, security, user centric interactivity, and complex reasoning we categorize recent innovations and highlight their implications for scalability, robustness, and personalization. The paper also reviews advances in evaluation protocols, domain specific applications, and architectural variants such as Naïve RAG, Advanced RAG, and Modular RAG. Finally, we identify persistent challenges and outline future directions aimed at advancing the integration of retrieval with LLMs for more grounded, interpretable, and controllable generation.

Text, images, and other rich covariates are increasingly compressed into AI-learned representations and then used as controls in causal analysis. We study when this approach is valid and develop a practical framework for causal inference with learned representations. For a broad class of estimands, an imperfect representation distorts the target causal parameter by the product of two representation errors: one in the outcome regression and one in the balancing weight (or Riesz representer). This yields three constructive results. First, cross-fitted double machine learning (DML) provides valid Wald inference for the representation-dependent target. When representation errors are small, the same interval covers the causal parameter, and it can even attain the semiparametric efficiency bound. Second, fold-wise representation learning (or fine-tuning) is compatible with DML inference for the causal parameter. To this end, we develop convex- and star-aggregation pipelines for learning and combining representations. Third, when representation errors are substantial, we can provide interpretable sensitivity regions and root-$n$ inference for their endpoints. In a multi-modal demand application, seven representation-specific estimates and their star aggregate all imply a negative near-unit elasticity for rank-based price response, and the result remains robust over the reported sensitivity grid.

Hypernetworks have recently shown success in dynamically adapting the parameters of Large Language Models (LLMs) at runtime based on signals such as task descriptions or additional demostrations. Here we ask: how much adaptation signal can be obtained using only the input query to an LLM?. To answer this, we study query-conditioned Hypernetworks for LoRA estimation. Further, we introduce distributional Hypernetworks, able to produce not only point estimates of parameter adaptors, but also a distribution over possible LoRAs. For this we propose a simple end-to-end loss using a differentiable Monte Carlo approximation and explore multiple distribution parametrizations including regression and convex combination variants. Results show that even using the mean of the learned distribution can outperform deterministic hypernetworks. Crucially, the learned distribution enables a different form of test-time scaling: instead of spending additional compute only by sampling more token sequences from a fixed model, we sample weight updates, yielding multiple adapted models for the same query. Performance improves as more weight samples are considered and remains stronger than corresponding token-sampling adaptation baselines. Finally, we find that generated updates can transfer across queries, suggesting that the hypernetwork learns reusable structure in how the model should adapt. Together, these results show that query-conditioned distributions over weight updates can support both adaptation and test-time scaling.

Inference-time scaling adapts pretrained diffusion models to new sampling tasks without additional training. Existing methods rely primarily on Monte Carlo sampling with more particles, yet are premised on the pretrained model being exact. In practice, data and training limitations make the model imperfect, and these methods inherit its error. More particles reduce Monte Carlo error but cannot remove the mismatch between the endpoint and the desired target or the error in tracking the prescribed probability path. We introduce the Energy-based Feynman-Kac Corrector (EBFKC), a framework for energy-based diffusion models that corrects these errors on the fly given a reference energy. We first derive Feynman-Kac dynamics that track a prescribed path exactly in the continuous-time population limit even when the model is imperfect, and approximate these dynamics using sequential Monte Carlo with variance-controlling guidance. To remove the endpoint mismatch, we use the pretrained energy as a surrogate along the diffusion path and progressively incorporate the discrepancy between the learned and target terminal energies. Experiments on Gaussian mixture models, particle systems, alanine dipeptide, and alanine tetrapeptide show that our method closely matches target distributions and molecular free-energy profiles under annealing and reward tilting, whereas standard inference-time scaling baselines retain substantial sampling errors.

We propose a novel approach to state preparation by embracing the full power of open quantum systems. Instead of working with the whole unitary group in $n$ qubits, we identify a small Lie group $G$ in $(n+3)$ qubits. The dimension of its Lie algebra is $\text{poly}(n)$. Every state can be approximated starting from any fixed input state by repeatedly using partial trace and state preparation in the environment alongside group operations. Moreover, we also identify a single (open system magic) interaction Hamiltonian whose unitary group can be combined with open system quantum technology mentioned above to achieve transitivity on the space of all densities. Similarly, we show that all channels on $n$ qubits can be approximated by using a larger environment, achieving channel universality. The interesting small groups we identify are derived from Lie algebras of local Pauli strings and lead to a new landscape of cheap and expensive states, densities, and channels.

Accurate stellar parameters underpin much of astrophysics, from exoplanetary systems to Galactic archaeology. For isolated field stars, masses and especially ages are usually inferred from stellar evolution models, yet these inferences depend on the chosen model grid, and the systematic differences between grids can rival the statistical uncertainties of any single pipeline. We present LACHESIS, a Python package that determines stellar masses, radii, and ages from photometry and spectroscopy while explicitly accounting for the systematic uncertainty from the choice of model grid. We interpolate five independent isochrone grids in [Fe/H], log age, and equivalent evolutionary phase conditioned on the data, sample each with nested sampling, and combine the per-grid posteriors by Bayesian model averaging, weighting each grid by its Bayesian evidence. We validate LACHESIS against a benchmark of Kepler asteroseismic dwarfs and subgiants. Masses are recovered with a robust scatter of ~4% and radii to ~2%, with negligible radius bias (<1%) and a small ~3% mass systematic. Age is recovered with a robust scatter of 30% that depends strongly on evolutionary state; the grid-to-grid systematic contributes a ~9% floor, a subdominant but real term that a single-grid fit omits. Model averaging is better calibrated than selecting a single grid: its credible intervals approach nominal coverage once the reference uncertainty is included, and cover near the nominal rate in injection tests even when the generating model lies outside the ensemble. By marginalizing over an ensemble of stellar model grids, LACHESIS folds the choice-of-grid systematic directly into the posterior, so the reported uncertainties no longer reflect only within-grid data noise. This delivers homogeneous, well-characterized masses, radii, and ages for exoplanet hosts and stellar population studies.

Ultrasound computed tomography is emerging as a promising diagnostic imaging tool. 2D geometries suffer from notorious out-of-plane scattering artifacts. Image reconstruction can be achieved with frequency-domain full waveform inversion and it can be further accelerated by randomly phase-encoding the elementary sources. In this manuscript, we extend our previous results for the 2D geometry of a ring-array to the 3D cylindrical geometry of multiple rings. In particular, we consider the cases of an elementary source described by a single array element (quasi-omni-directional transmission), a line source and a focused transmission respectively. With differences in image quality, we prove that a fully randomized frequency-domain inversion in 3D is capable to reconstruct portions of a human breast surrounded by the cylindrical geometry and detect mm-size masses of varying contrast in dense breast, in reasonable computing times, thus opening the concrete possibility to the design of 3D imaging devices with high sensitivity levels. The methods are applicable to multiple tomographic geometries in 3D and, in principle, can be integrated into next generation medical ultrasound scanners.

We present an analysis of pulsar spin noise based on physically-motivated two-component models of spin wandering. We focus on two models, distinguished by their total angular momentum dynamics: a singular, nonstationary model with a diffusive total angular momentum, and a minimal, stationary model anchored on a conserved total angular momentum. We develop scalable Gaussian process methods and, using mock data, show that the two models are fully testable and distinguishable with full-state observations, i.e., simultaneous independent data on the crust and the superfluid rotational states. This paves a path to testing stationarity in pulsar spin noise, potentially achievable with joint continuous gravitational wave observations and radio timing of pulsars. However, robust inferences and predictions are harder to achieve, and depend on the priors and data quality, when only one component is observationally accessible.

We study regularised optimal transport with subquadratic or entropic regularisation. Under $C^{1,\alpha}$-assumptions on the marginals and the domains of their support, we prove global gradient-Lipschitz bounds on the potentials, uniform in the regularisation parameter. Moreover, we show that, up to the boundary, the support of the conditional supports $\text{supp}\,\pi_\varepsilon(\cdot|x)$ behaves like balls of radius $\varepsilon^\frac 4 {d(p-1)+2}$. Our work was carried out concurrently and independently of [9]. We obtain the results of [9,Theorem 1.2] under weaker assumptions and with a different proof.

Feedforward foundation models have recently shown remarkable 3D reconstruction capabilities. However, existing models exhibit large tracking drift in long-context streaming reconstruction due to error accumulation. In this paper, we revisit loop closure with streaming reconstruction foundation models to enable accurate, drift-free, kilometer-scale reconstruction. Specifically, our method detects loop candidates through global descriptor retrieval, and constructs loop-conditioned windows to estimate the relative poses between looped frames. Given the observation that our adopted streaming reconstruction backbone produces a globally consistent scale, we optimize all frame poses on the SE(3) manifold with sequential and loop closure constraints, avoiding the pose graph optimization on the Sim(3) or higher-dimensional SL(4) manifolds employed in prior works. Extensive experiments show that our method reduces drift and produces consistent geometry on kilometer-scale sequences, significantly outperforming the state of the art. Code is available at https://github.com/MoyangLi00/CLoSeR.git.

Increasing depth of transformer models improves recognition, but it comes at a substantial cost. Each additional layer requires more parameters, which makes the process computationally inefficient. We ask whether additional processing can focus on integrating features already computed. Looped Audio Spectrogram Transformer (LAST) first processes all tokens, then reuses the same blocks to refine only the class token over fixed audio features, thereby making later passes inexpensive. On AudioSet, ten-pass LAST achieves 0.345 mean average precision, exceeding a twelve-layer sequential transformer by 2.1% relative with 49.4% fewer parameters, 42% fewer multiply-accumulate operations, and 9.8% higher measured throughput. Across separately trained models, increasing the pass count from two to ten improves accuracy while adding only 1.2% computation. Further evaluations show improved robustness to temporal masking and various other auditory augmentations, with better generalization on classification tasks with music, environmental, and event sounds.

Thermal axion production in the early Universe is controlled by the QCD topological-charge-density spectral function evaluated on the light-cone as a function of momentum k. We show how weighted integrals of this quantity can be computed using Euclidean lattice QCD without any analytic continuation, similar to what Harvey Meyer and collaborators have recently done for the photon production rate. We then test the approach in pure-glue QCD at T=1.5 Tc, finding that it is feasible to extract at least one and possibly two finite weighted k-integrals of the k-dependent production rate. This opens the possibility for extractions in full QCD at temperatures around and somewhat above the crossover temperature, where other methods are not reliable.

Supersingular abelian surfaces are essential in isogeny-based cryptography. Despite this, we have no efficient algorithm to verify if a given abelian surface is supersingular. In this work, we initiate this research topic by giving an efficient Monte Carlo algorithm to verify if an abelian surface over $\mathbb{F}_p$ is supersingular in $O(\log p)$ with negligible failure probability, and an efficient conclusive algorithm if the order is smooth. We derive this algorithm by a careful analysis on the structure of supersingular Jacobians over $\mathbb{F}_p$. Furthermore, we derive efficient algorithms to verify if an abelian variety of any dimension is minimal or maximal, and to verify if a Jacobian of any dimension is superspecial.

Networking matter-based quantum processing units (QPUs) offers a promising route to scaling fault-tolerant quantum computers. This requires distributed logical operations to be performed across photonic links, where noise is characteristically different from and stronger than in local QPUs owing to photon loss and probabilistic linear-optical operations. Measurement- and fusion-based quantum computing are designed to be robust against loss and probabilistic photonic operations, suggesting they could complement circuit-based error correction at the interface between networked QPUs. We use ZX calculus transformations to construct hybrid syndrome extraction protocols and apply them to distributed rotated surface code lattice surgery, demonstrating that several protocols attain a $50\%$ interface-erasure threshold when local noise is absent, including circuit-based lattice surgery using fused Bell pairs and hybrid protocols using linear cluster states. Relative to a straight Bell-pair interface geometry, the hybrid protocols increase the merge-observable interface distance from $d+1$ to $2d+1$ and restore the perpendicular-observable interface distance from $\lfloor(d+1)/2\rfloor$ to $d$. We calculate how the threshold decreases when using truncated resource states and map correctable regions under resource-state errors, local circuit noise, and fusion erasure. We then convert these erasure thresholds into photon-loss thresholds and probe subthreshold performance using fusion boosting. At circuit and resource-state error rates of $10^{-3}$, local errors largely mask the interface-distance advantage. As local noise is reduced to $10^{-4}$ and below, the linear-chain protocols improve more rapidly with decreasing erasure, suggesting that interface distance enhancements are effective in low local noise regimes.

This paper introduces tellegen, an open source framework for interactive power flow (PF) and optimal power flow (OPF) studies that run in the browser. This provides intuitive and democratized access to power system analysis tools compiled to WebAssembly. A user can drag and drop a case file, click and drag to change a nodal demand or line rating, preview the impacts via sensitivity analysis, and obtain an exact re-solve on release. User case files and results stay entirely on the device: tellegen transmits zero Critical Energy/Electric Infrastructure Information (CEII). The framework comprises a core numerical engine for PF and OPF, reusable browser components, saved studies, and structured WebMCP tools for agentic interaction. We evaluate the transmission OPF solver by comparing objectives with PGLib baselines; the distribution PF solver by comparing voltages and currents with OpenDSS; and the WebAssembly execution times by comparing with PowerModels.jl. On realistic synthetic grids, WebAssembly OPF solves take only 25-43% longer than native binary solves. The implementation shows how an engineer can distribute an executable numerical study as a URL, reducing installation and hosting requirements while keeping case data local.

Cross-lingual contrastive learning has been a core component of multilingual encoder training, but the ability to explicitly align representations is not possible in decoder-only LLMs because of varying multilingual tokenization. However, a growing amount of research suggests that even in LLMs, higher cross-lingual representational alignment leads to improved cross-lingual transfer. In this paper, we propose a novel approach to reimagine cross-lingual contrastive learning given the architectural constraints of modern LLMs. Rather than applying an auxiliary alignment loss on hidden states, we propose using the outputs of the mixture-of-experts (MoE) routers as the target for alignment. Router outputs lend themselves better to pooling over many tokens, enabling more reliable cross-lingual comparisons at the sequence-level. Controlled continual pre-training experiments on four open-source MoEs show that incorporating this routing loss also aligns the underlying hidden representations across languages. Most importantly, this loss improves multilingual performance on our diverse evaluation suite, demonstrating the potential of cross-lingual MoE router alignment.

Pneumatic conveying at moderate gas velocities can result in strongly heterogeneous particle transport, with slow particles accumulating near the bottom of the pipe while faster particles are transported above. Here, we investigate helical pipes as a passive means of homogenizing particle transport without increasing the gas velocity. CFD-DEM simulations show that the helical pipe induces swirling motion of the gas and particles, continuously redistributing particles across the pipe cross-section. This transforms the heterogeneous transport observed in a straight pipe into a more homogeneous state with a narrower particle-velocity distribution. The results demonstrate that helical pipes can maintain efficient pneumatic conveying at moderate gas velocities while reducing excessive particle velocities.

We study combinatorial Banach spaces, i.e. Banach spaces induced by families of finite sets, of uncountable density. We prove some general theorems, for example we characterize when such spaces contain isomorphic copies of long $c_0$ and long $\ell_1$. We discuss the existence of Banach spaces which are (complementedly) universal in the category of Banach spaces with unconditional basis of fixed cardinality. We present some examples, in particular we show that the Banach spaces generated by Suslin trees do not contain uncountable equilateral sets.

The timing behavior of a mapped circuit depends on both its logic implementation and the physical environment in which that implementation is realized. Revisiting mapping decisions after placement therefore requires a search procedure that accounts for surrounding timing constraints, fanout loads, and interconnect effects. We study local remapping in this setting and develop a framework that couples discrete mapping search with physical implementation feedback. Timing-critical regions are isolated through bounded windows whose interfaces retain the context of the surrounding circuit. Within each window, a mixed-integer formulation jointly selects logic cuts, signal polarities, and library cells under a delay model informed by estimated locations and interconnect parasitics. A continuous relaxation filters the search space before discrete optimization produces alternative implementations with similar modeled timing and different structural choices. These implementations are reconstructed and assessed through legalization, routing-based parasitic estimation, and timing analysis. Physically validated improvements are incorporated into the design, and the updated context guides subsequent searches. The framework provides a systematic way to revisit local logic implementations while accounting for their interaction with an existing placement.

Latent visual reasoning equips vision–language models with continuous intermediate states that can process visual evidence without explicit textual reasoning traces or repeated image operations. However, existing methods often allow multiple latent tokens to access the same visual evidence through shared value projections, providing no mechanism for them to extract complementary visual information; simply increasing the latent budget can therefore yield redundant latent representations. We argue that effective latent reasoning should encourage different latent tokens to extract complementary visual information, and thereby act as specialized visual experts. Based on this insight, we propose MoLE, a Mixture of Latent Experts framework that controls both what visual evidence each latent visual expert observes and how it transforms that evidence. MoLE isolates latent visual experts during evidence extraction and uses dedicated latent summary experts to aggregate the complementary representations of latent visual experts. A two-stage training pipeline first forces visual evidence through this latent pathway and then restores direct visual access, requiring neither predefined expert roles nor intermediate visual targets. Across five visual reasoning benchmarks, MoLE achieves an average score of 78.6, outperforming data-matched supervised fine-tuning by 4.9 and the strongest evaluated latent visual reasoning baseline at the same latent budget by 3.6. Representation analyses show lower latent-state similarity and more diverse visual attention, while masking the latent pathway reduces average performance by 9.2. These results demonstrate that specializing latent computation is more effective than merely increasing the number of latent tokens.

Let $K\subset\mathbb{R}^n$ be a convex body, let $X$ and $Y$ be independent uniform points of $K$, and set $R_K=(|K|/\omega_n)^{1/n}$. We consider the distribution function \[ D_K(\rho)=\mathbb P\{|X-Y|/R_K<\rho\}, \] and its homogeneous counterpart \begin{equation*} \mathcal J_\rho(K)=\iint_{K\times K}\mathbf{1}_{\{|x-y|<\rho R_K\}}dxdy. \end{equation*} Thus $D_K(\rho)$ is the proportion of ordered pairs of points of $K$ whose distance is less than $\rho R_K$, or equivalently the radial distribution function of the normalized covariogram of $K$. The functional satisfies a Brunn–Minkowski inequality, with equality precisely for homothetic bodies, and its differential gives a first Minkowski inequality. We characterize the Borel measures that occur as Wulff differentials: they are exactly the nonzero finite positive measures with zero centroid which are not concentrated on a great subsphere, and the corresponding body is unique up to translation. We further prove fixed-volume and stationary rigidity of the ball in the subcritical range, together with regularity for the resulting equation. At saturation, the equation is the classical Minkowski equation.

We compute the Cuntz semigroup of a graph C*-algebra associated to a finite graph by combining Hong and Szymański's realisation of the entire ideal lattice of a graph C*-algebra, Ara, Moreno and Pardo's computation of the Murray-von Neumann semigroup of a unital graph C*-algebra and various Cuntz semigroup techniques. We also investigate certain properties of such Cuntz semigroups, and demonstrate that at the level of the Cuntz semigroup, there is no obstruction to the conjecture that the nuclear dimension of any graph C*-algebra is at most one.

Decompositional scene reconstruction aims to reconstruct high-quality objects and background, yet existing methods still struggle with the level of quality under heavy occlusions. While generative priors offer a potential solution, 2D image-based priors often suffer from multi-view inconsistency due to a lack of 3D awareness. Conversely, 3D-native priors provide stronger structural inductive biases but frequently lead to spatial drift and misalignment within complex scenes. To address these issues, we propose DecomVoxel, formulating object completion as a guided in-situ denoising optimization that bridges 3D-native priors with neural scene reconstruction. Our framework introduces a reformulated epsilon-based distillation loss to ensure stable latent refinement, alongside adaptive spatial guidance that utilizes occupied and vacant anchors with temporal annealing to suppress generative hallucinations and mitigate spatial drift. Experiments on Replica and ScanNet++ show that DecomVoxel significantly outperforms state-of-the-art methods while faithfully preserving the original spatial layout, structural fidelity, and style-consistent texture. Our method pushes the boundary of decompositional reconstruction by delivering high-quality textured meshes with clean topology, geometry, and appearance, providing a robust solution for the decompositional reconstruction of complex real-world scenes. Code is available at https://github.com/DecomVoxel/DecomVoxel.

Let $W_{\geq -1} = \mathbb{C}[t]\partial$ and $W = \mathbb{C}[t, t^{-1}]\partial$ be the Witt algebra of algebraic vector fields on $\mathbb{C}$ and $\mathbb{C}^*$ respectively. In this paper, we make significant progress toward the open conjecture that the enveloping algebras $\mathrm{U}(W_{\geq -1})$ and $\mathrm{U}(W)$ satisfy the ascending chain condition (ACC) on two-sided ideals. We show that all homomorphic images of $\mathrm{U}(W_{\geq -1})$ and $\mathrm{U}(W)$ under the family of “orbit homomorphisms” of arbitrary Gelfand-Kirillov dimension satisfy ACC on ideals. These orbit homomorphisms were the key ingredient allowing us to “lift” the Dixmier map from finite-dimensional solvable settings to infinite-dimensional settings of the Witt and Virasoro algebras in our recent work [Pham, 2025, arXiv:2504.14670]. As a result, we completely classify the prime and primitive spectra of these homomorphic images. As these images approximate $\mathrm{U}(W_{\geq -1})$ better as their GK-dimension increases, this classification sheds new light on the two-sided and prime ideal structures of $\mathrm{U}(W_{\geq -1})$. Finally, we discuss several applications of our results to the Dixmier map for $W_{\geq -1}$.

We give a necessary and sufficient condition for a combinatorial D-class of a finite semigroup to have a permutation matching: that its structure matrix admit a balanced weighting, that is, non-negative weights with constant row sums and constant column sums. This weakens the regularity hypothesis, the role of Hall's lemma being taken up by the Birkhoff–von Neumann theorem. We then apply it to show that the monoid of all order-preserving self-maps of an n-element chain, has a permutation matching.

The standard quadratic program (StQP) minimizes a quadratic form over nonnegative variables that sum to one. We compose classical graph reductions with regularized Motzkin–Straus clique formulations to express discrete optimization problems in this continuous domain. The graph matrix has diagonal entries $\tau$, zeros on edges, and ones on nonedges. For $0<\tau<1$, its minimum is $\tau/\omega(G)$, where $\omega(G)$ is the clique number. Its strict local minimizers are precisely the uniform distributions on maximal cliques, and its global minimizers encode maximum cliques. At $\tau=1/2$, integer scaling gives coefficients in $\{0,1,2\}$ and minimum $1/\omega(G)$, yielding an NP-complete StQP threshold problem with a restricted coefficient alphabet. We give explicit formulations for satisfiability, coloring, Hamiltonian cycles, independent set, vertex cover, set packing, three-dimensional matching, and graph isomorphism. A regularized weighted clique formulation combined with local-state compatibility graphs gives an exact compiler for finite-domain factor models specified by complete local tables, including QUBO, with at most four simplex coordinates per binary pair factor. The catalog covers Karp's 21 problems: twelve use direct graph formulations, and nine use factor-state formulations, including six obtained through binary-linear feasibility. For each route we record dimensions, coefficient structure, and recovery rules. We analyze interaction count, coefficient range, objective separation, perturbation tolerance, support recovery, and decoding overhead. The separation bounds quantify the effects of clique size, factor weights, and offsets. In the complete factor-state construction, every assignment, including each suboptimal assignment, is a strict local minimum.

All the objects composing our world are enclosed within surfaces. Yet, most robot learning and motion generation frameworks treat surfaces as constraints ignoring their intrinsic geometry. This gap is acute for polyhedral meshes–the standard output of CAD and 3D reconstruction–whose discrete geometric structure remains unexploited. In this paper, we propose a unified discrete Riemannian framework that enables robot learning directly on polyhedral surface meshes. Using discrete differential geometry, we define logarithmic and exponential maps, parallel transport, and ambient-space projections that remain well-defined across faces, edges, and vertices. We instantiate the framework in three learning paradigms: (i) Dynamic Movement Primitives (DMPs), an improved exponential-map computation and a fixed-tangent-cone forcing-term encoding with parallel transport yield better cross-surface generalisation and stability over prior mesh-based approaches. (ii) Gaussian Process (GP), a geodesic-based kernel with practical admissibility control, enables regression at arbitrary mesh locations without smoothness assumptions. (iii) Riemannian Flow Matching (RFM), mesh-native operators improve generative quality over spectral baselines while reducing training time. The framework is validated in simulation against state-of-the-art methods and demonstrated on two real-robot scenarios: generalising user-drawn trajectories across different surfaces and planning polishing motions on RGB-D-reconstructed surfaces.