Papers

Let \(\mathcal L\) be the Heisenberg-Virasoro type Lie superalgebra whose even part is the twisted Heisenberg-Virasoro algebra and whose odd part is an abelian ideal. We classify \(\mathbb Z_2\)-graded \(\mathcal L\)-modules that are free of super-rank \(1|1\) over \(U(\mathfrak h)=\mathbb C[L_0,F_0]\). Using polynomial shift operators, we express the defining relations as functional equations for the matrix entries of the generators. Assuming the rank one classification for the even subalgebra, we show that, up to parity reversal, every module with nontrivial odd action belongs to a triangular family \(\Omega^+(\lambda,\beta,d)\), whereas every module with trivial odd action is a direct sum of two rank-one modules for the even subalgebra. We determine the graded isomorphism classes: within the triangular family, the coupling polynomial \(d\) is invariant up to multiplication by a non-zero scalar. We also describe canonical proper submodules and show that every module in the classification is reducible. In particular, we provide a new realization of the Heisenberg-Virasoro type Lie superalgebra using $2\times2$ matrices with entries associated with polynomial shift-differential operators.

Let $F$ be a totally real field and $K/F$ a totally real quadratic extension. For a Hilbert cusp newform $f$ over $F$ with base change $f_K$ to $K,$ we determine the exact level of $f_K$ in terms of the level of $f,$ using only local representation theory: for each place $\mathfrak p$ of $F$ we compute how the conductor of the local component of $f$ changes under quadratic base change. The tame case (odd residue characteristic) follows from existing base change theory, but the case $p=2$ requires a finer $\varepsilon$-factor analysis that does not appear in the literature. We give a complete treatment of the dyadic imprimitive representations and, more delicately, the exceptional supercuspidal representations, for which no closed-form base change formula was previously known; in particular we determine, in terms of local root numbers, the base change of every exceptional supercuspidal representation of $GL(2,\mathbb Q_2)$ with trivial central character and conductor $3.$ This yields an explicit formula for the level of $f_K,$ specialized here to $F=\mathbb Q,$ along with a partial converse.

Large language models can memorize private or harmful information, motivating machine unlearning methods that remove targeted knowledge while preserving other capabilities. However, existing evaluations rely primarily on behavioral benchmarks, which assess whether a model appears to forget but provide limited insight into how unlearning changes the model or affects related knowledge. We introduce TRIAGE (Tripartite Representation-internal Introspection for Adjacency Gap Evaluation), a benchmark-agnostic evaluation framework for characterizing these changes. TRIAGE uses diagonal approximations of the Fisher information and Hessian to measure changes in parameter sensitivity and local curvature, and utilizes a Forget / Adjacent-Retain / Generic-Retain partition to quantify an adjacency gap in semantically related knowledge. Based on the magnitude and distribution of these changes, TRIAGE further classifies each algorithm's update as no-op, partially localized, collateral dominant, or globally destructive. Across 12 unlearning methods, four language models, and the WMDP, TOFU, and MUSE benchmarks, we find that methods with similar behavioral forgetting can produce substantially different internal changes and patterns of collateral damage. These signatures also vary across models and benchmarks, indicating that the effects of unlearning are not determined solely by the unlearning algorithm. TRIAGE can be applied alongside existing unlearning benchmarks to complement behavioral evaluation with a model-internal view of how unlearning reshapes the model's parameter space and affects retained knowledge.

Can the global-workspace account of transformer representations extend to state-space language models? We fit Jacobian lenses to the residual streams and recurrent states of Mamba-1, Mamba-2 and Mamba-3, using the original 1000-prompt recipe. Joint residual–state readouts improve recovery of known intermediate concepts over the residual lens on at least five of six task families in every tested Mamba checkpoint. On Mamba-2, state alone exceeds residual and logit lenses on all six families; a normalised joint readout improves on both components on five. Temporal maps and word-list experiments show earlier content remaining state-readable as residual visibility changes. We also propose sign-guarded steering, which improves target top-five success over coordinate exchange on matched verbal-report trials in five models. Recurrent state alone supports this verbal access. These gains do not extend consistently to relational answers: guarded edits often output the edited concept itself, and Mamba-3's joint edits can disrupt successful state-only redirection. Recurrent state thus provides a complementary carrier of workspace content, whose recovery, persistence and causal uses require separate measurements.

Hegerfeldt's theorem shows that the localization of a particle (or field) is not possible under very general assumptions of quantum mechanics and special relativity. This theorem, and further developments and associated theorems force penetrating questions on the proper understanding of the theory and its use. In particular, the use of the collapse postulate is called into question. The existence of particles and even localized fields are called into question. And, this in turn challenges local measurements. It generally forces one to consider the profound problems and even wrong results that arise from the collapse postulate. We give a qualitatively new example of such a wrong result. Researchers and students alike are bothered by these fundamental issues. Students' ability to learn the material, and even their interest in physics, can be compromised by such disturbing conundra that seem to make physics, or at least QFT and its QM limit, seem incomprehensible and even inconsistent. This article resolves these issues in a way accessible to undergraduates, by applying the natural (minimal) interpretation of quantum mechanics (QM) (the ensemble interpretation) and explicating and clarifying the basic principles and premises used in QM and the Hegerfeldt theorem and its application to measurements.

The neural scaling law relating longer training to better performance through a power law is central to today's large language models (LLMs), yet its origin remains debated. One recent proposal is that power laws can emerge from the strong non-linearity of a single softmax head learning peaked distributions. What happens with multiple softmax functions, as in LLMs, is unclear. Here, we show through toy models that any softmax learning peaked distributions, regardless of its position in the model, can develop logit magnitudes that grow in a power law with exponent $1/3$, becoming a training bottleneck whose loss contribution decays as a power law with the same exponent $1/3$. The overall loss therefore obeys $1/3$ scaling whenever at least one softmax learns peaked distributions. We confirm that many softmax functions in LLMs learn peaked distributions and that LLM loss scaling matches this $1/3$ prediction. Moreover, logit growth dynamics reveal that attention heads, rather than the language modeling head, are the bottleneck likely driving the $1/3$ loss scaling in LLMs. Attention trying to concentrate on specific information, which is the heart of Transformers, may therefore also be the heart of the neural scaling law of training.

The craft of weaving, where different materials are interlaced, has tremendous potential for creating active and functional systems for use in soft robots, prosthetics, wearables, exoskeletons, and more. These textile-like systems are flexible and safe for human-machine interaction; however, this inherent flexibility limits their ability to carry loads, which is essential for many robotic functions. In this work, we introduce a general framework for integrating active materials into three-dimensional (3D) woven shells to create robotic structures that combine high axial stiffness for load bearing, low bending stiffness for efficient actuation, and system-level resilience for damage tolerance. These woven robots can be modularly assembled from 'woven corners', a fundamental unit of 3D woven structures. We use eigenvalue calculations to identify load bearing and actuation mechanisms of the 3D woven structures, and use that information to make five different robots capable of locomotion. We demonstrate that these 3D woven robots can locomote carrying loads 70 times their self-weight, and can maintain repeatable performance even after being subjected to extreme compression. This work is a pathway toward the design, manufacturing, and simulation of future 3D woven robotic systems where load bearing, high stiffness, active functional deformation, locomotion, and system-level resilience are all needed.

This work presents a solution to nonconvex, game-theoretic motion planning problems subject to disturbances over long time horizons. The problem is posed as a partially-decoupled generalized Nash equilibrium problem, in which each agent's dynamics depend only on its own state and control, admitting fast solution methods for competitive multi-agent motion planning. An algorithm, WOLF, is developed that applies receding-horizon model predictive control to an open-loop differential games solver based on sequential convexification. In contrast to robust formulations that fix the uncertainty description offline, the robustness tube here is itself a dynamic state, co-optimized with the trajectory, and its thickness directly sets the tightening of the shared coupling constraints. A sufficient condition is derived under which a nominal trajectory satisfying constraints tightened against all agents' error bounds remains feasible for every admissible disturbance realization. The method is demonstrated on two adversarial on-orbit games with coupled translational-attitude dynamics: a stealthy co-orbital jamming game under an active detection-probability bound, and a sun-blocking game in which an adversary disables an evader by decreasing solar power

Extreme precision radial velocity (EPRV) spectrographs are making strides in extending the limits of radial velocity exoplanet detection. Perhaps the best evidence for this is how readily EPRV instruments are facilitating the discovery and detailed characterization of small-amplitude planetary signals around stars that have already been intensively monitored for decades with less stable instruments. In this paper, we use data from the ongoing NEID Earth Twin Survey (NETS) to detect and confirm the planet HD 126053 b (P=10.91 d), which is among the least massive planets (m sin i = 3.60 m M_\oplus) around solar-type stars within 25 pc. We then jointly characterize the known planet HD 168009 b (P=15.14 d; m sin i = 8.64 M_\oplus) orbiting a solar analog along with a previously unreported signal at 27.8 d which we tentatively attribute to rotational modulation of stellar activity (P_rot = 30.1 d). We also analyze and discuss periodic RV variations in HD 10780 arising from a combination of rotationally-modulated activity and a long-term activity cycle. We discuss HD 126053 b and HD 168009 b in the context of the broader small planet population and our sensitivity to long-period planets in these systems, and we comment on our capacity to disambiguate planet-induced and activity-induced radial velocity signals. HD 126053 is exceptionally RV-quiet, making it a promising target for continued RV monitoring in search of low-mass exoplanets in the habitable zone.

We introduce a transformer reinforcement learning algorithm that learns to solve high loop-level scattering amplitudes in planar N = 4 Super Yang-Mills theory. Our algorithm improves on previous transformer-only results by incorporating previously derived symmetries and relationships into the learning algorithm. This results in a greatly decreased fraction of the solution that needs to be known a priori to solve the entire problem. An additional benefit is that our algorithm also ensures that every output obeys the set of known relationships. Rather than predicting all coefficients independently, the model proposes assignments that are propagated through exact linear relations, while MCTS searches over assignments when propagation alone is insufficient. This is crucial to the generalization of machine learning approaches to higher loops, as without this, there is no way to overcome the factorially scaling of state sizes and compare to results derived via other methods. Using the symbology representation of the form factor, we frame the problem as learning a mapping between discrete sequences and integer coefficients.

We give a finite-temperature test for the stability of low-energy spectral subspaces. The test uses a local determinant defect for the alpha-z Renyi kernel of two Gibbs states and bounds the tracial L2 distance between their low-energy projections. The main tool is a spectral-flattening principle for bounded positive injective operators in a semifinite von Neumann algebra. It reduces the local product defect to an exact two-projection formula. The thermal result applies to lower-bounded affiliated Hamiltonians with trace-class Gibbs operators and does not require the Hamiltonians to be close in operator norm. We give an infinite-level example with unbounded Hamiltonians. For matrices, the result gives a sum of principal-angle remainders. For qubits, the defect and the thermal gaps determine the ground-state overlap exactly. The same estimate gives a uniform error bound for bounded observables in an effective low-energy model. We also prove the determinant inequalities, their equality cases, and z-monotonicity in matrix, finite, and semifinite settings.

We prove the Bleiler-Litherland conjecture: every lens space obtained by a nontrivial Dehn surgery on a hyperbolic knot in $S^3$ has order at least 18. The key ingredient is a spectral obstruction: if a hyperbolic knot $K$ of genus $g$ admits a lens space surgery with slope $\pm(4g-2)$, then $\Delta_K(t)$ has a real root outside the unit circle. The proof combines Floer-theoretic restrictions on lens space surgeries, Gabai's degeneracy-slope bound and Gabai-Oertel's persistence theorem for essential laminations, Ni's fixed-point theorem for monodromy, and a mod-2 orientability criterion, together with the relation between homological monodromy and the Alexander polynomial. As a further application of this spectral obstruction, we obtain characterizing-slope results for torus knots.

Electroencephalography (EEG) data plays an important role in understanding human brain activity, with various major applications. While current approaches, which largely build on EEG foundation models, have demonstrated strong performance across datasets and tasks, they typically rely on a fixed temporal scale. This design choice limits their ability to capture characteristics of neural dynamic behavior, often spanning across multiple timescales, which are naturally captured in EEG recordings. To address this limitation, we propose MAGE (Multi-timescale Adaptive Generalization for EEG), a foundation model that dynamically adapts to capture and integrate information across different timescales. Specifically, MAGE combines two key mechanisms: (i) a novel multi-timescale attention mechanism, capable of learning representations across multiple temporal scales and (ii) a trainable aggregation module, that dynamically identifies and adapts to the most informative timescale of the input signal. Experimental results across seven datasets demonstrate that MAGE consistently outperforms existing EEG foundation models while using substantially fewer parameters. Moreover, our results support the hypothesis that different downstream tasks benefit from distinct temporal scales. All source code will be released.

Set-theoretic types provide a rich type algebra that supports unrestricted unions, intersections, and negations, together with a decidable type constraint-solving algorithm known as tallying. These types are particularly well suited for typing dynamic languages, where functions often exhibit both generic and overloaded behavior. However, the complexity of their implementation has hindered their widespread adoption. In this paper, we introduce a modular representation for set-theoretic types and revisit the algorithms for subtyping and tallying. We compare our approach with the historical CDuce implementation and evaluate the performance impact of some optimizations and design choices.

When an object comes into view, humans can instantaneously recognize it if known to them. However, assessing how this recognition takes place remains challenging. A few studies using simple sketches have reported varying degrees of image similarity between the fixation density map (FDM), which visualizes eye movements and durations, and the psychological potential field (PPF), which visualizes part of the visual cognition process, depending on the object category. This study reports how the similarity behaves across three categories—animal, non-animal, and meaningless shape—and under rotated presentation, which is known to increase cognitive load during object recognition. The similarity was quantified through area under the curve (AUC) measures, including a baseline-adjusted AUC that isolates the early rise in similarity from stimulus-dependent baseline levels. A variability score was introduced to assess individual differences, and linear mixed models were used to evaluate category and orientation effects. Results indicated that the baseline-adjusted AUC yielded significant differences among all three categories with large effect sizes, whereas the raw AUC did not distinguish between the animal and non-animal categories due to the influence of sketch representation. No significant Category $\times$ Orientation interaction was observed, indicating that rotated presentation did not produce a category-specific effect. The meaningless shape stimuli yielded consistently low similarity and variability scores, clearly separating them from the two meaningful object categories. These results characterize the behavior of the FDM–PPF similarity under free viewing and indicate that baseline adjustment is necessary to separate category-related variation from stimulus-dependent baseline levels.

We introduce a new logarithmic digit variation for Vilenkin–Fourier partial sums and establish two-sided estimates for the corresponding Lebesgue constants with universal constants. These estimates hold for both bounded and unbounded Vilenkin systems. As an application, we show that bounded logarithmic variation yields uniform boundedness of the corresponding partial-sum operators on \(L_1\), \(L_1\)-norm convergence for every integrable function, and almost everywhere convergence.

We classify K3 surfaces with finite automorphism group in positive characteristic. The classification is the same as over the complex numbers, except when the base field is $\overline{\mathbb{F}}_p$.

This paper builds on the work of Goncharov by studying the correspondence between topological spectral covers constructed from generic $A_{m-1}$-webs with rank $m\ge2$, Hitchin spectral curves over good Hitchin loci, and the Grothendieck groups $K_0(\mathcal{D}_{\Gamma})$ associated with the quivers with potential $(Q_{\Gamma},W_{\Gamma})$ assigned to corresponding bipartite $A_{m-1}$-graphs. We show that the automorphisms on the non-frozen quiver lattices $\Lambda_{\Gamma}^{\circ}$ induced by compositions of elementary moves on ideal webs/bipartite graphs are the same as the Gauss–Manin monodromy actions on the $SL_m$ Hitchin fiber lattices $H_1(\operatorname{Prym}(\Sigma);\mathbb{Z})$ locally around discriminant loci. And the Gauss–Manin monodromy actions agree with the automorphisms on the Grothendieck groups $K_0(\mathcal{D}_{\Gamma}^{\circ})$ induced by compositions of mutations on the quivers with potential $(Q_{\Gamma},W_{\Gamma})$.

Endoscopy is widely used in medicine to visualize anatomical structures for diagnosis and treatment evaluation. However, it does not currently allow rapid, quantitative measurement of airway cross-sectional area (CSA), a key metric for diagnosing and managing respiratory pathologies. There is growing interest in AI-enabled quantitative endoscopy using depth estimation, but a fundamental challenge is the need for ground-truth depth data to train algorithms capable of quantifying depth in physical units (e.g., meters). In this study, 60,000 paired virtual endoscopy images and depth maps were created in Blender from 3D airway models built from computed tomography. A U-Net convolutional neural network was trained to predict the depth maps from the endoscopy images. Point clouds were reconstructed from the AI-generated depth maps using the Equisolid fisheye lens model, and airway perimeters were identified by intersecting a plane with the point cloud to estimate CSA. Nasopharyngeal CSA, a key metric in obstructive sleep apnea, was quantified using this AI-enabled approach with a median error of 5.3 % relative to the 3D model ground-truth. This pilot study using synthetic data provides strong evidence for the feasibility of developing AI-enabled quantitative endoscopy for airway CSA measurement.

In this article, we propose algebraic integer number formats (AINF) as a framework for highly configurable and accurate compressed computation. A key property of AINF is its straightforward reduction to purely integer arithmetic, enabling error-free multiply-accumulate operations. We develop a matrix multiplication engine that exploits this property, obtaining performance within a predictable factor of the most efficient numerical counterparts on modern hardware accelerators. Another key advantage of AIQ is the freedom to select algebraic integers tailored to the numerical characteristics of a given use case, offering optimization opportunities that are fundamentally unavailable in floating-point or fixed-point formats. Motivated by these properties, we introduce algebraic integer quantization (AIQ), a lossy compression format for large language model (LLM) inference, and provide results demonstrating that AIQ achieves competitive accuracy compared to the industry-standard MX format across a diverse set of models and benchmarks at equivalent bit-per-parameter budgets. The experiments presented in this article employ algebraic integers of degree two (quadratic integers); generalizations to higher-order and, in particular, multidimensional algebraic integers promise even greater computational advantages and remain a promising direction for future work.

Comparing parameter-efficient fine-tuning recipes under a single, shared learning rate is a common but flawed practice: when the arms being compared have very different trainable-parameter counts, a shared rate can simultaneously depress the larger arms' means and inflate their variance, manufacturing a large, seemingly multi-seed-significant advantage for the smallest arm that is not a real effect. We document this confound in a concrete setting: post-hoc SVD-based KV-cache compression, where an already-pretrained model is converted to a low-rank (multi-head-latent-attention-style) cache by factorizing its key/value weights into a down-projection ("encoder") and an up-projection ("decoder"), after which a short fine-tune ("healing") recovers the accuracy lost to truncation. Under a shared learning rate, freezing the decoder and healing only the encoder looks like a clear win over healing the decoder or both factors; once every arm is given its own tuned learning rate, that apparent advantage disappears, and encoder-only healing instead reaches parity with the alternatives, at a real, measured saving of 3x fewer trainable parameters and 3x less optimizer-state memory. We verify this parity with per-arm learning-rate tuning and three seeds per configuration on a vision-language model (Qwen2.5-VL-3B-Instruct), at the one compression ratio this protocol covers, and replicate it on a text-only testbed across two backbones. Encoder-only healing is therefore a lower-memory drop-in recipe for retrofitting low-rank KV-cache compression at training time, and the shared-learning-rate pitfall we document and correct is a cautionary result for comparing any fine-tuning recipes whose arms differ in trainable-parameter count.

This paper investigates the theory of fractional Sobolev mappings between Carnot groups, with a particular focus on the Korevaar-Schoen energy functional and interpolation properties. We establish that weak contact equations hold for fractional Sobolev mappings on Carnot groups, generalizing classical results from the Euclidean setting to the sub-Riemannian framework. Our main results include: (1) the proof that fractional Sobolev mappings satisfying weak contact equations belong to the horizontal fractional Sobolev space $W_H^{s-\lambda,q}(\Omega;G_2)$ for any sufficiently small $\lambda>0$; (2) interpolation theorems for fractional Sobolev mappings between Carnot groups, demonstrating that the limiting behavior as $s \to 1$ recovers the first-order Sobolev space; (3) a rigorous treatment of the Korevaar-Schoen energy on Carnot groups, including a characterization of its minimizers via horizontal differential forms.

Automatic Speech Recognition (ASR) systems often underperform for children and non-native speakers, while adapting adult ASR models to child speech can cause adult-speech forgetting. We study child ASR adaptation with adult retention across Arabic and English. We compare full fine-tuning, LoRA, and post-hoc weight-space merging across encoder–decoder, encoder–CTC, and AudioLLM-based ASR systems. Experiments use Arabic native and non-native child speech, English MyST child speech, and adult benchmarks from MGB-2 and LibriSpeech test-clean. We evaluate recognition quality with WER and quantify the adaptation–retention trade-off using Retention Index, Child Adaptation Gain, and Adaptation Recovery. Results show that child adaptation is necessary, especially for non-native Arabic and English child speech, but direct adaptation often reduces adult ASR performance. Bilingual adaptation is more stable than language-specific adaptation. Weight-space merging often improves the trade-off, especially for encoder–CTC, Whisper, and AudioLLM-based ASR, with LERP favoring adult retention and TIES recovering stronger child gains. For the encoder–decoder model, direct bilingual fine-tuning remains strongest in raw WER.\footnote{Code, and models are available at https://github.com/qcri/Child-ASR-Adaptation.

Full-sphere panoramic cameras let fixed monitoring systems and mobile robots track people in every direction, but a planar bounding box does not fully describe where a person is on the sphere. We introduce PanoPed, a sim-to-real benchmark for pedestrian tracking on the full sphere. PanoPed-S contains 108,000 frames from fixed, quadruped-mounted, and drone-mounted cameras, with synchronized masks, depth, camera poses, and 3D pedestrian states. PanoPed-R adds 28,002 real frames from fixed cameras, 16,247 of them densely annotated. We find that an ERP rectangle cannot uniquely determine the spherical center and angular extent of the visible person, while the detector's visual query still carries information about them. Inspired by the sextant's use of angular measurements to locate objects, we propose Sextant, a plug-and-play angular localization head with only about 0.035M parameters. It reuses a frozen detector, keeps track identities unchanged, and needs no extra image encoder. Sextant gives the best result in our PanoPed-S test comparison, raising the strongest baseline, MOTIP, from 47.30 to 49.49 HOTA, with gains on all eight test sequences. Without fine-tuning on real data, the same synthetic-trained heads improve MOTIP and HAT by 0.96-1.14 HOTA on real video, and both seeds improve every real sequence. HAT+Sextant scores best among the compared systems that add no localization image encoder.

Autonomous driving has made remarkable progress, with recent AI advances enabling commercial deployments that are reshaping urban mobility. Yet the field remains far from its universal social promise: autonomous systems that can operate robustly anywhere, anytime, for anyone. We posit that this gap is not merely a modeling problem, but a problem of the prevailing data paradigm. Current research relies heavily on a few benchmark datasets with limited spatial and scenario coverage, even though the community has collectively produced over 600 autonomous driving datasets across nearly 50 countries. However, this abundance has not translated into broad research impact: most datasets remain significantly underused due to fragmentation, limited visibility, incompatible protocols, and benchmark incentives that concentrate attention on a few dominant datasets. We therefore argue that autonomous driving research requires a collaborative, community-driven data paradigm. Such a paradigm would improve the discovery, reuse, integration, and evaluation of diverse datasets; make underexplored data easier and more rewarding to study; and lower the barrier for new contributors. We outline its key principles, illustrate an early realization, and call for collaboration across academia and industry to transform fragmented datasets into shared community infrastructure for anytime-anywhere autonomy.

We study the late time cosmological evolution of the Universe in the framework of a flat Friedmann Lemaitre Robertson-Walker spacetime in f(Q,Lm) gravity. The cosmic dynamics of the Friedmann equations are discussed using various equation of state (EoS) parameters like Chevallier Polarski Linder (CPL), Jassal Bagla Padmanabhan (JBP), Barboza Alcaniz (BA), and Logarithmic (LOG). We use joint observational datasets of Hubble (OHD), DES five year Type Ia supernovae (DES 5YR), and the Baryon Acoustic Oscillation (BAO) measurements to obtain the best fit values of model parameters, which show good agreement with current observations. These models exhibit a smooth transition from early deceleration to late time acceleration. The equation of state parameter displays quintessence like behaviour over a wide redshift range and gradually converges its behaviour from quintessence-like to phantom-like behaviour as z tends to -1, except in the BA case. Furthermore, we investigate the physical characteristics of the model in order to assess its physical viability. We also analyse the energy conditions and examine the stability of the proposed model.

Continual Graph Learning (CGL) aims to incrementally learn from graph-structured data while preserving knowledge acquired from previous tasks. A major challenge in this setting is catastrophic forgetting, where learning new tasks degrades performance on previously learned ones. Hierarchical Prototype Networks (HPNs) address this problem through a prototype-based memory mechanism that avoids storing historical data, but their reliance on linear feature extractors limits their ability to exploit graph topology, while point-based prototypes often lead to inefficient prototype growth on structurally diverse graphs. In this work, we propose HCPN-GCN, a graph-aware extension of HPN that replaces the original linear feature extractors with Graph Convolutional Networks (GCNs) and introduces cone-based prototypes with a diversity regularization objective. The proposed design produces richer graph-aware representations while compactly modeling the embedding space, reducing prototype proliferation without sacrificing discriminability. Experimental results on six continual graph learning benchmarks demonstrate that HCPN-GCN consistently improves average classification accuracy over the original HPN and representative continual learning baselines while maintaining near-zero forgetting. Furthermore, our analysis shows that the proposed model learns substantially richer class-level prototype hierarchies using approximately $30\times$ fewer atomic prototypes than the original HPN, providing a more compact and effective memory representation for continual graph learning.

Ageing bridge infrastructure is a growing global concern, yet conventional Structural Health Monitoring (SHM) systems are costly and difficult to scale, and routine visual inspections remain subjective. Drive-by, or indirect, bridge inspection, in which a sensorised vehicle recovers structural information from vehicle-bridge interaction (VBI) and vehicle-road interaction (VRI) responses, offers a scalable alternative. However, key challenges remain unresolved, including separating bridge responses from road roughness, detecting damage under normal traffic, and generalising across diverse bridge types. This paper presents a vehicle-integrated digital twin framework that unifies physics-based modelling and machine learning for continuous monitoring of bridge and road conditions. The framework comprises three pillars. First, surrogate models of VBI and VRI are constructed using a Fourier Neural Operator that learns function-to-function mappings from operating conditions to vehicle responses. Trained on both simulated and field data, these surrogates deliver millisecond-scale inference, replacing computationally intensive full-order analyses. Second, the design of a custom electric inspection vehicle, its sensor layout, and signal processing chain are optimised through Bayesian optimisation to maximise bridge information yield while suppressing road and vehicle noise. Unsupervised damage-assessment pipelines based on adversarial autoencoders, matrix profiles, and transformer architectures have been developed and validated to process the resulting vehicle data. Third, the complete workflow is validated through coordinated multi-site field trials in Australia and Japan, covering a range of bridge types, traffic conditions, and environmental settings.

We refine the density-increment argument for $k$-configurations developed by Beker. The point is not to introduce a new inverse theorem, but to retain information that is lost when the graph-counting output is compressed into a single uniform alternative. For the transitive $K_k$ appearing in the $k$-configuration problem we keep a fixed moment throughout the graph recursion and obtain three genuinely different outputs: a row-degree anomaly, a non-centred grid anomaly, and a centred grid anomaly. The centred case is treated directly through positivity of the associated Gram moments, avoiding the usual unbalancing step. Each output is then routed through the appropriate local Kelley–Meka mechanism with its own density gain, rank cost, and width cost. A weighted iteration budget prevents the most expensive local branches from being charged the maximal number of times. For a finite abelian group $G$ of odd order and $A\subseteq G$ of density $\alpha$, write $L=\log(2/\alpha)$ and $\Lambda=L+\log(2k)$. We obtain a lower bound of the form \[ \Pp_{x_1,\ldots,x_k\in G}\!\left(\frac{x_i+x_j}{2}\in A\ \text{for all }1\le i\le j\le k\right) \ge \exp\!\bigl(-C\Phi(k,L,\Lambda)\bigr), \] where \[ \Phi(k,L,\Lambda)=k^{36}L^6\Lambda^6+k^{34}L^8\Lambda^5+k^{32}L^{14}\Lambda+k^{30}L^{16}. \] Consequently, in the Erdős–Moser application one may replace the exponent $68$ in Beker's $k$-configuration route by $36$; in particular the same reduction yields the sum-free lower bound with every exponent $c<1/36$.

Confined cell migration is governed by nuclear and cellular deformation. In this work, we formulate a two-dimensional mathematical framework in which the cell and nuclear membranes are represented as evolving closed surfaces governed by force-balance laws incorporating surface tension, bending elasticity, and external mechanical interactions. Nuclear mechanics depends on several factors, including chromatin organization and compaction, which are regulated by epigenetic modifications. Molecular regulation at the nuclear periphery is described by a surface reaction-diffusion system posed on the evolving nuclear surface. We apply this coupled framework to an experimentally characterized lung cancer cell model in which the balance between the enzymes SETDB1 and SUV39H1 regulates the peripheral deposition of H3K9me3. Using experimentally measured mechanical parameters, we represent control (Ctrl) and SETDB1 knockout (SETDB1 KO) conditions, which display distinct H3K9me3 distributions, especially at the nuclear periphery. Numerical solutions obtained using the evolving surface finite element method qualitatively reproduce the changes in nuclear mechanics as well as part of the experimentally observed increase in the entry time in SETDB1 KO cells. The model therefore reproduces the direction of the experimental observations, while the residual difference between simulated and measured entry times indicates that the cellular response is multifactorial.