Abstract Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We define what it means for a finite dimensional representation of $\mathfrak{g}$ to be rectangular and completely classify faithful rectangular representations. As an application, we obtain new $\lambda $-independence results on the algebraic monodromy groups of compatible systems of $\lambda $-adic Galois representations of number fields.

We introduce Motion2VecSets, a 4D diffusion model for dynamic surface mesh generation from various ambiguous observations, including a sequence of RGB images, sparse and partial point clouds, and low-resolution voxel grids. While recent methods using neural field representations have shown success in modeling non-rigid objects, conventional feed-forward architectures struggle with noisy, partial, or sparse observations due to their deterministic nature. To address the inherent one-to-many mapping problem, we introduce a diffusion model that explicitly learns the shape and motion distribution of non-rigid objects through an iterative denoising process of compressed latent representations. The diffusion-based priors provide more plausible and diverse reconstructions under ambiguous conditions. Instead of relying on global latent codes, we represent 4D dynamics using latent sets. This novel 4D representation captures local shape and deformation patterns, leading to more accurate non-linear motion capture and significantly improving generalization capacity to unseen motions and identities. For temporally coherent tracking, we jointly denoise latent sets across frames and enable cross-frame information exchange. To reduce computational cost, we design an interleaved spatial-temporal attention block that alternately aggregates deformation latents along spatial and temporal dimensions. Extensive experiments on datasets of humans, animals, and articulated objects demonstrate that Motion2VecSets outperforms prior methods in reconstructing and tracking non-rigid deformations from various imperfect observations. Our implementation is available at https://vveicao.github.io/projects/Motion2VecSets/.

On the accuracy of implicit neural representations for cardiovascular anatomies and hemodynamic fields.

This paper presents a comprehensive theoretical and computational study on the sequential representation of arrays within the algebraic framework of Galois Fields (GF), with a primary focus on GF(7). By employing modular arithmetic, symmetry principles, and binary encoding techniques, the work demonstrates how discrete array elements can be systematically mapped to finite field structures to achieve efficient and fault-tolerant data representation. The proposed methodology integrates sequential indexing with field operations to ensure closure, invertibility, and predictable algebraic transformations. Through detailed mathematical formulations and computational simulations, the study validates that array differences and mappings preserve the cyclic structure intrinsic to GF(7). Furthermore, the research highlights the applicability of this representation in several domains, including digital communication systems, cryptographic primitives, coding theory, and error-resilient computation. Sequential binary encodings derived from modular residues provide an additional layer of structure suitable for hardware-level implementation and secure data processing pipelines. The study also connects these representations to Perfect Difference Sets (PDS) and Perfect Difference Networks (PDN), illustrating how modular differences support highly symmetric interconnection topologies. Overall, the framework provides a unified perspective on finite-field-based array representation, offering both analytical clarity and practical relevance for modern computational architectures and secure information systems.

Abstract. In order to study the behaviour of cloud droplets at the cloud-clear interface, the “Solving The Entrainment Puzzle” (STEP) project examined a droplet stream in the Turbulent Leipzig Aerosol Cloud Interaction Simulator (LACIS-T). LACIS-T comprises two particle free air streams, that are turbulently mixed, and during the experiment one air stream was resembling in-cloud conditions, whereas the other air stream was set to out-of-cloud conditions. A droplet stream was injected by a droplet generator into the mixing plane of the two air streams. Droplet size distributions were observed with a phase Doppler anemometer at various levels in the measurement section of LACIS-T, corresponding to different residence times of the droplets in the turbulent environment. Additionally, observations were made using different flow speeds in the two air streams to create shear flows in the wind tunnel. The experiment was accompanied by computational fluid dynamics simulations to provide a full 3D representation of meteorological fields and turbulence parameters. This manuscript provides a description of the laboratory settings and instrumentation, the experimental design, the simulations, and a general overview of the data. We invite the scientific community for joint data analysis and numerical studies using the data which is freely available from the Eurochamp Data Centre, see Table 2 in the Data availability section for details.

Molecular dynamics (MD) simulations are a powerful tool for characterizing membrane-protein dynamics, yet their predictive accuracy critically depends on the choice of force field and membrane representation. Here, we present a systematic benchmark of the AMBER 14SB and CHARMM 36 m force fields across multiple bilayer sizes, using human aquaporin-7 (aquaglyceroporin-7; hAQP7) as a representative membrane protein system. Both force fields maintained global structural integrity, but differed markedly in their dynamic profiles: CHARMM 36 m sampled a broader conformational space and produced more hydrated pore profiles, whereas AMBER 14SB favored conformations closer to the crystallographic structure. Lipid organization and packing also diverged, with CHARMM generating more compact bilayers and AMBER yielding larger areas per lipid. The membrane size exerted minimal influence on the structural or functional descriptors, supporting the use of smaller, computationally efficient membrane patches for equilibrium simulations. The hAQP7 monomers functioned independently, without detectable cooperativity under the simulated conditions. Collectively, these results highlight the substantial impact of force-field selection on aquaporin dynamics and provide practical guidance for designing accurate MD simulations of transmembrane protein channels.

This conceptual understanding article is part of a series where we analyze the recognition and conversion of representations of the electric field concept; in this article, we present the case of algebraic notation. We conducted a study with introductory and upper-division physics students taking electricity and magnetism courses in a large private Mexican university to learn how students recognize the electric field’s main characteristics in the algebraic notation of the field and how they convert to and from different representations. We refer to the theory of registers of semiotic representations as a theoretical framework and use a phenomenographic approach to analyze data. We explored students’ recognition and conversion abilities through interpretation and construction tasks for the electric field’s algebraic notation. We found that the main difficulties of interpreting and constructing the algebraic notation are related to separating the mathematical expression from the situation’s physical meaning. Sometimes, students referred only to the physical meaning without using algebraic notation. In other cases, they construct algebraic notation without explicitly describing the physical meaning. Another source of difficulty is the treatment process because some students make mistakes or misinterpretations that they carry throughout. We recommend that introductory and upper-division electricity and magnetism instructors and physics education researchers in higher education be aware of the difficulties that some interpretation and construction tasks may present to students learning the electric field concept.

Aerial remote sensing efficiently surveys large areas, but accurate direct object-level measurement remains difficult in complex natural scenes. Advancements in 3D computer vision, particularly radiance field representations such as NeRF and 3D Gaussian splatting, can improve reconstruction fidelity from posed imagery. Nevertheless, direct aerial measurement of important attributes like tree diameter at breast height (DBH) remains challenging. Trunks in aerial forest scans are distant and sparsely observed in image views; at typical operating altitudes, stems may span only a few pixels. With these constraints, conventional reconstruction methods have inaccurate breast-height trunk geometry. TreeDGS is an aerial image reconstruction method that uses 3D Gaussian splatting as a continuous scene representation for trunk measurement. After SfM–MVS initialization and Gaussian optimization, we extract a dense point set from the Gaussian field using RaDe-GS’s depth-aware cumulative-opacity integration and associate each sample with a multi-view opacity reliability score. Then, we isolate trunk points and estimate DBH using opacity-weighted solid-circle fitting. Evaluated on 10 plots with field-measured DBH, TreeDGS reaches 4.79 cm RMSE (about 2.6 pixels at this GSD) and outperforms a LiDAR baseline (7.66 cm RMSE). This shows that TreeDGS can enable accurate, low-cost aerial DBH measurement.

ABSTRACTThe medial subdivision of the inferior pulvinar (PIm) has been implicated in motion processing, visuomotor integration, and residual visual function, yet a comprehensive account of its cortical inputs remains unresolved. Previous studies often relied on indirect cortical injections or tracer deposits spanning multiple pulvinar subdivisions, limiting anatomical specificity. Here, we used MRI-guided, cytoarchitectonically restricted retrograde tracer injections to selectively target PI in the common marmoset (Callithrix jacchus) and systematically map its cortical afferents.Across four cases, retrogradely labeled neurons were widely distributed throughout occipital, temporal, parietal, and cingulate cortices, with a strong predominance in layer V, consistent with driver-like corticothalamic projections. Early and middle-tier visual areas (V1, V2, V3, V3A, V4, V6/DM) contributed substantial input, with labeling patterns corresponding to peripheral visual field representations. The middle temporal complex (MT, MTc, MST, FST) represented one of the densest sources of cortical projections. Prominent inputs also arose from posterior parietal regions, including LIP, MIP, VIP, AIP, and inferior parietal areas (e.g., PFG, OPt), linking PIm to visuospatial and action-related networks. Semi-quantitative analyses indicated that occipital cortex and the MT complex together accounted for approximately 60% of total cortical input, while parietal cortex contributed roughly 20%. Additional projections from retrosplenial and posterior cingulate cortices were observed.These findings identify PIm as a central integrative node embedded within distributed visual and visuomotor networks. Rather than functioning as a restricted visual relay, PIm appears positioned to coordinate motion, spatial, and action-relevant signals within cortico-thalamocortical circuits supporting adaptive visually-guided behavior.

Animals and humans rely on optic flow to navigate cluttered and unknown environments. While most previous studies have focused on how organisms achieve self-motion perception through optic flow information, biological neural networks for navigation based on optic flow remain unexplored. Here, we propose a biologically plausible neural network model for optic flow-based reactive navigation. The model incorporates a primary visual cortex, which is responsible for generating a cortical representation of the optic flow field; a higher-order cortex, which calculates the focus of expansion (FOE) of the optic flow field; and a cerebellum, which generates motor commands. A feedback inhibitory pathway from V1 layer VI to layer IV is introduced, enhancing heading sensitivity and enabling rapid adaptation in dynamic environments. To achieve precise obstacle localization, we propose a dual encoding strategy that combines optic flow with depth maps derived from the optic flow field, FOE, and control acceleration. This strategy mitigates distortions in depth estimation near the expansion center and ensures more reliable obstacle representation. The cerebellum outputs motor commands for heading direction and speed control based on the output of the visual cortex. Simulations and real-world experiments with an intelligent vehicle confirm that the proposed model enables collision-free navigation across diverse scenarios and outperforms classical optic flow balance strategies in complex environments. These findings demonstrate that biologically inspired neural networks provide a feasible solution for visual reactive navigation in autonomous agents.

The upper atmospheric microwave sounding channels data are important for atmospheric data assimilation and retrieval. However, radiative transfer simulation accuracy is constrained by the precise characterization of the Zeeman splitting effect. This study investigates key influencing factors in upper-atmospheric microwave radiance simulations, focusing on the geomagnetic field parameters and the Zeeman splitting absorption coefficients. A three-dimensional (3D) atmosphere-magnetic coupling dataset is constructed using the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) version 2.0 Level 2A atmospheric profiles and the International Geomagnetic Reference Field (IGRF-13) as input for the microwave Line-by-Line (LBL) model. Observations from Special Sensor Microwave Imager/Sounder (SSMIS) channels 19 and 20 are used to quantitatively compare the effects of 2D and 3D geomagnetic fields on simulations and evaluate the impact of updated Zeeman splitting coefficients. Quantitative analysis reveals that the average vertical attenuation rate of geomagnetic field strength between 50 and 0.001 hPa is 2.98%, and using 3D magnetic field parameters improves the observation and simulation bias (O-B) for SSMIS channels 19 and 20 by approximately 3.67% and 3.52%, respectively. The updated microwave LBL model, incorporating molecular self-spin interactions and higher-order Zeeman effects, reduces the mean absolute error (MAE) and root mean square error (RMSE) of the SSMIS channel 20 by approximately 2.7% and 2.25%, respectively. Experimental results indicate that the 7+ line within a 2 MHz frequency shift is sensitive to moderate magnetic field strength (0.35–0.55 Gauss), while the 1− line is sensitive to strong magnetic fields (0.5–0.7 Gauss). This study demonstrates that optimizing geomagnetic field representation and Zeeman splitting coefficients can improve upper atmospheric microwave radiance simulation accuracy by detailed comparison with observations.

Abstract The electron density distribution in the lunar ionosphere has been characterized by only a few tens of vertical profiles of detectable electron density. Reported density values are highly variable: observed vertical profiles of electron density have maximum values that differ by a factor of 100. Here we report ionospheric results from five two‐way S‐band radio occultation observations performed by the Danuri spacecraft. The observations were implemented successfully and suitable raw data were acquired, but the initial versions of the derived frequency residuals and corresponding electron density profiles contain undesirable systematic features. We hypothesize that these undesirable features may be associated with the coarse representation of the lunar gravity field (degree and order of 100 × 100) that was used to reconstruct the Danuri trajectory. Simulations using different representations of the lunar gravity field suggest that improving the resolution to 400 × 400 can change the reconstructed spacecraft trajectory enough to account for the undesirable systematic features in the frequency residuals and corresponding electron density profiles. It will be important to assess how results of single frequency radio occultation observations that were processed with a relatively coarse representation of the lunar gravity field are changed if they are reprocessed with an improved representation of the lunar gravity field. Such changes may be significant. If this issue is addressed, then it may improve understanding of the lunar ionosphere.

The work is devoted to the study and analysis of finite element (FE) calculations performed by a large cycle of computational experiments of plate deformation with a section under steady-state creep conditions, which revealed a power-law self-similar distribution of the continuity function (damage) and stress components in the immediate vicinity of the tip of the section at the second and third stages of creep in a damaged medium in a related formulation of the problem, when the continuity parameter is included in the constitutional relations. The FE computations of stress fields and continuity near the tip of the defect were carried out using the powerful SIMULIA Abaqus platform using the UMAT utility, which integrates the process of damage development into the computational scenario of the finite element method (FEM). The paper implements computer modeling of uniaxial stretching of a plate weakened by a central horizontal section or an inclined section in creep mode, in which computational algorithms include damage growth that progresses over time according to the classical mechanical model of damage growth by Kachanov–Rabotnov according to a power law for various values of exponents of the kinetic equation and the power determining equation with the concept of true tension in a related formulation. Numerical study and analysis of the obtained FE representations of stress and continuity fields in the vicinity of the crack tip for a number of material constants clearly reveals a self-similar distribution of stress fields and damage near the tip of a power-type defect. The structure of the solution is revealed and the values of the exponents in the self-similar variable and the self-similar representation of the solution are found, which can be interpreted as an intermediate self-similar solution of the second type according to the classification of G.I. Barenblatt. It is shown that the discovered self-similar property of the solution can be interpreted as self-similar asymptotics of the far field of continuity and stresses. Also, the stress dependences extracted from FEM calculations on the distance from the tip of the incision, reproduced in double logarithmic coordinates, clearly demonstrate the asymptotic behavior corresponding to the near-field stress, characterized by the complete absence of a singularity in the immediate vicinity of the tip of the incision.

Abstract Groundwater model predictions are often inaccurate due to uncertainties in model structure, heterogeneous parameters, and initial conditions, leading to error accumulation during simulations. Traditional data assimilation (DA) faces severe computational challenges in high‐dimensional systems due to the costly inversion of large covariance matrices. In addition, the inaccurate estimation of background and observation error statistics introduces further biases. To address these challenges, we develop and evaluate an integrated framework that couples a computationally efficient deep learning surrogate model for rapid prediction with Latent Data Assimilation (LDA) to accurately correct simulations. The framework employs dimensionality reduction, specifically Proper Orthogonal Decomposition (POD), to project the high‐dimensional physical state into a low‐dimensional latent space. Data assimilation is then performed in this reduced space using the Ensemble Kalman Filter (EnKF). Results demonstrate that POD provides a robust representation of simulated concentration fields and interpolated observations for dimensionality reduction. The EnKF operating in the latent space effectively reduces prediction errors. Key advantages of the LDA framework include: enabling sparse observations to effectively inform global state updates through the low‐dimensional latent variables, achieving higher accuracy comparable to traditional physical‐space DA while requiring significantly fewer observations, and inherently filtering high‐frequency noise from the initial condition during the dimensionality reduction process. Collectively, these features establish LDA as a powerful and computationally efficient methodology for enhancing predictive accuracy and managing uncertainty in complex and high‐dimensional groundwater systems.

Wire arc additive manufacturing (WAAM) enables efficient fabrication of large-scale metallic components. However, the geometry of each as-deposited layer (ADL) is highly sensitive to coupled thermofluid dynamics and process fluctuations. These complex interactions make it difficult to maintain geometric stability, underscoring the need for accurate layer-wise profile predictions to support process planning, dimensional control, and defect prevention. This study presents ILPP-Net, an inter-layer profile prediction network that leverages a sequential signed distance field (SDF) representation to model the spatio-temporal evolution of multi-layer WAAM deposition. The point cloud of each layer is converted into a sequence of SDF cross-sections, which are combined with the process metadata, including global position, wire-feeding speed, deposition speed, arc current, and lateral torch offset, to predict the next-layer contour. A temporal context block (TCB) integrates geometric and process information through parametric fusion and bidirectional long short-term memory (BiLSTM), whereas a FiLM-modulated U-Net performs spatial regression to reconstruct a continuous SDF distribution. A WAAM system integrated with a 3D vision module was developed to acquire real layer datasets for training and evaluation. ILPP-Net achieves high prediction fidelity across different deposition strategies. For early thermally unstable layers (Layer 2 - 3), the model achieves an mloU of 0.85 - 0.86 with root mean square errors (RMSEs) of 0.192 - 0.187 mm, while performance improves for the more stabilized upper layers (Layers 7–10) to 0.89–0.90 mloU and 0.165–0.141 mm RMSE. Ablation studies show that TCB contributes to an accuracy gain of ≈14.36%, highlighting the importance of temporal modeling. The proposed model was further deployed in a prediction-driven feed-forward compensation experiment, where the ILPP-based parameter adjustment significantly improved the dimensional accuracy. Relative to the open-loop deposition, the compensation strategy reduced the final height RMSE by 56.57% and the mean error by 62.4%, demonstrating the effectiveness of the model for geometry-aware adaptive control.

Fluid and Scalar Field Representations in the Brans–Dicke Theory: Cosmological Scenarios

Cryptocurrency markets such as Bitcoin, Ethereum, and Solana exhibit high volatility, making price forecasting difficult when relying solely on conventional technical analysis. This study aims to analyze cryptocurrency candlestick patterns by utilizing Gramian Angular Field (GAF) representations and to evaluate the performance of a hybrid deep learning model combining CNN–LSTM–Transformer to support investment decision-making. The proposed method involves processing daily historical Open, High, Low, and Close (OHLC) data from three major cryptocurrency assets: Bitcoin (BTC-USD), Ethereum (ETH-USD), and Solana (SOL-USD), covering the period from January 1, 2020, to September 30, 2024, obtained from Yahoo Finance. The time-series data were transformed into 64×64 pixel GAF images and used to train a baseline CNN model as well as a hybrid CNN–LSTM–Transformer model. Model evaluation was conducted across multiple forecasting horizons, including 1 day, 7 days, 30 days, 180 days, and 1 year, and was further complemented by real-time testing using the CoinGecko API in March 2025. The results indicate that the hybrid model achieved the best performance at different horizons for each asset: BTC-USD at the 30-day horizon with an R² of 0.971 and an SMAPE of 0.77%, ETH-USD at the 1-year horizon with an R² of 0.948 and an SMAPE of 0.81%, and SOL-USD at the 1-year horizon with an R² of 0.910 and an SMAPE of 4.72%. Real-time testing demonstrated that the model consistently captured the overall price movement trends despite high market volatility. It can be concluded that the integration of GAF representations and the hybrid CNN–LSTM–Transformer model has strong potential to enhance cryptocurrency candlestick analysis and can be utilized as a component of a Decision Support System for digital asset investment.

In this work, we explore the interplay between information and computation in nonlinear transform-based compression for broad classes of modern information-processing tasks. We first investigate two emerging nonlinear data transformation frameworks for image compression: implicit neural representations (INRs) and two-dimensional (2D) Gaussian splatting (GS). We analyse their representational properties, behaviour under lossy compression and convergence dynamics. Our results highlight key trade-offs between INR's compact, resolution-flexible neural field representations and GS's highly parallelizable, spatially interpretable fitting, providing insights for future hybrid and compression-aware frameworks. Next, we introduce the textual transform that enables efficient compression at ultra-low bit rates-regimes, and simultaneously enhances human perceptual satisfaction. When combined with the concept of denoising via lossy compression, the textual transform becomes a powerful tool for denoising tasks. Finally, we describe a Lempel-Ziv (LZ, specifically LZ78) transform, a universal method that, when applied to any member of a broad compressor family, produces new compressors that retain the asymptotic universality guarantees of the LZ78 algorithm. Collectively, these three transforms illuminate the fundamental trade-offs between coding efficiency and computational cost. We discuss how these insights extend beyond compression to tasks such as classification, denoising and generative AI, suggesting new pathways for using nonlinear transformations to balance resource constraints and performance. This article is part of the discussion meeting issue 'Bits, neurons and qubits for sustainable AI'.

Multiscale simulations facilitate the efficient exploration of large spatiotemporal scales in chemical and physical systems, yet particle-based simulations become prohibitively expensive at time and length scales beyond the molecular level. Field-theoretic simulations offer an attractive alternative, but most existing formulations rely on top-down approximations and are not systematically connected to atomistic interactions. Here, we present a hierarchical bottom-up framework for constructing auxiliary field representations of molecular liquids directly from microscopic models. We introduce a hierarchical coarse-graining framework that constructs field-theoretic models directly from atomistic liquids. The method first maps atomistic interactions to coarse-grained center-of-mass potentials and regularizes short-range divergences through a perturbative expansion in reciprocal space. Building on the auxiliary field formulation developed in polymer field-theoretic simulations, we then generalize the Hubbard-Stratonovich transformation to arbitrary pair potentials by separating positive and negative Fourier modes and introducing two auxiliary fields. The resulting generalized mode theory extends bottom-up field-theoretic modeling beyond positive-definite kernels and is compatible with existing field-theoretic sampling strategies. By combining formal derivations with numerical regularization and mode-truncation procedures, this work provides the theoretical foundation for scalable, bottom-up field-theoretic simulations of molecular systems.

This paper presents a novel adaptive control pattern (ACP) for real-time feedback control of flow separation over airfoils using sparse processing particle image velocimetry (SPPIV) and dielectric barrier discharge plasma actuators. The study addresses the challenge of suppressing separation in deep stall at low Reynolds numbers, where previous feedback-based methods often fail to maintain effective flow attachment. Unlike conventional feedback control methods such as single-step and multiple-step prediction, where control inputs are directly decided on the basis of state predictions, ACP determines the modulation frequency of the actuation. This is done according to threshold-based flow state detection, enabling the selection of effective actuation patterns for the estimated flow features. Experiments were conducted on a NACA0015 airfoil at an angle of attack of 18 degrees and a Reynolds number of $$\approx$$ 66,000 using a plasma actuator positioned at the leading edge. The SPPIV system acquired data with a sampling frequency of 2,000 Hz, processed PIV at optimized limited interrogation windows, and estimated the state within 20% of the time between samples. Linear models for state estimation were generated via dynamic mode decomposition with control of the flow field representations from proper orthogonal decomposition. Results demonstrate that ACP control successfully achieves flow reattachment at 6 kV actuation voltage where other methods fail, whereas at higher voltages, ACP control combines the fast and reliable reattachment speeds of lower actuation burst frequencies of open-loop control with stable quasisteady attached conditions of higher burst frequencies. It shows that the optimal actuation frequency for driving the reattachment process is not the same as that for maintaining a reattached condition, and that guiding the actuation frequency as the reattachment process evolves can provide substantial improvements in control authority. This breakthrough brings the plasma actuator one step closer to practical control applications, with highly effective yet robust control parameters.