Show newer

A unified quaternion-complex framework for incompressible Navier-Stokes equations: new insights and implications arxiv.org/abs/2505.22853

A unified quaternion-complex framework for incompressible Navier-Stokes equations: new insights and implications

We present a novel, unified quaternion-complex framework for formulating the incompressible Navier-Stokes equations that reveals the geometric structure underlying viscous fluid motion and resolves the Clay Institute's Millennium Prize problem. By introducing complex coordinates $z = x + iy$ and expressing the velocity field as $F = u + iv$, we demonstrate that the nonlinear convection terms decompose as $(u \cdot \nabla)F = F \cdot \frac{\partial F}{\partial z} + F^* \cdot \frac{\partial F}{\partial \bar{z}}$, separating inviscid convection from viscous coupling effects. We extend this framework to three dimensions using quaternions and prove global regularity through geometric constraints inherent in quaternion algebra. The incompressibility constraint naturally emerges as a requirement that $\frac{\partial F}{\partial z}$ be purely imaginary, linking fluid mechanics to complex analysis fundamentally. Our main result establishes that quaternion orthogonality relations prevent finite-time singularities by ensuring turbulent energy cascade remains naturally bounded. The quaternion-complex formulation demonstrates that turbulence represents breakdown of quaternion-analyticity while maintaining geometric stability, providing rigorous mathematical foundation for understanding why real fluids exhibit finite turbulent behavior rather than mathematical singularities. We prove that for any smooth initial data, there exists a unique global smooth solution to the three-dimensional incompressible Navier-Stokes equations, directly resolving the Clay Institute challenge. Applications to atmospheric boundary layer physics demonstrate immediate practical relevance for environmental modeling, weather prediction, and climate modeling.

arXiv.org

Incorporating episodic memory into quantum models of judgment and decision arxiv.org/abs/2505.21521

Modeling the Impact of Misinformation Dynamics on Antimicrobial Resistance arxiv.org/abs/2505.21540

Modeling the Impact of Misinformation Dynamics on Antimicrobial Resistance

Antimicrobial Resistance (RAM) poses a significant threat to global public health, making important medicines less useful. While the medical and biological reasons behind RAM are well studied, we still don't know enough about how false health information affects people's actions, which can speed up RAM. This study presents a new mathematical model to investigate the complex interplay between the spread of misinformation and the dynamics of RAM. We adapt a multi-strain fake news model, including distinct population compartments representing individuals susceptible to, believing in, or skeptical of various ideas related to antibiotic use. The model considers multiple "strains" of misinformation, such as the wrong belief that antibiotics are effective for viral infections or not trusting medical advice regarding prudent antibiotic prescription. Time delays are integrated to reflect the latency in information processing, behavioral change, and the manifestation of resistance. Through stability analysis and numerical simulations, this research aims to identify critical factors and parameters that influence the propagation of harmful beliefs and their consequent impact on behaviors contributing to RAM. The findings could help develop public health campaigns to reduce the negative impact of misinformation on fighting antimicrobial resistance.

arXiv.org

Fluent but Culturally Distant: Can Regional Training Teach Cultural Understanding? arxiv.org/abs/2505.21548

Fluent but Culturally Distant: Can Regional Training Teach Cultural Understanding?

Large language models (LLMs) are used around the world but exhibit Western cultural tendencies. To address this cultural misalignment, many countries have begun developing "regional" LLMs tailored to local communities. Yet it remains unclear whether these models merely speak the language of their users or also reflect their cultural values and practices. Using India as a case study, we evaluate five Indic and five global LLMs along two key dimensions: values (via the Inglehart-Welzel map and GlobalOpinionQA) and practices (via CulturalBench and NormAd). Across all four tasks, we find that Indic models do not align more closely with Indian cultural norms than global models. In fact, an average American person is a better proxy for Indian cultural values than any Indic model. Even prompting strategies fail to meaningfully improve alignment. Ablations show that regional fine-tuning does not enhance cultural competence and may in fact hurt it by impeding recall of existing knowledge. We trace this failure to the scarcity of high-quality, untranslated, and culturally grounded pretraining and fine-tuning data. Our study positions cultural evaluation as a first-class requirement alongside multilingual benchmarks and offers a reusable methodology for developers. We call for deeper investments in culturally representative data to build and evaluate truly sovereign LLMs.

arXiv.org

On the orientation of historic Christian churches of Fuerteventura: conciliating tradition, winds and topography arxiv.org/abs/2505.18161

On the orientation of historic Christian churches of Fuerteventura: conciliating tradition, winds and topography

We present the results of an analysis of the precise spatial orientation of colonial Christian churches located in the Canary Island of Fuerteventura (Spain). Our sample consists of 48 churches, most built during the period between the Castilian conquest led by the Norman Jean de Béthencourt in the 15th century and the end of the 19th century. We examine whether the standard tradition was followed regarding the orientation of the apses of historic churches eastwards. While most of the religious constructions in the sample have their main axes oriented within the solar range, the statistical analysis also reveals the presence of two different groups of churches with different possible interpretations. For the first group, mainly composed of churches located in the central part of the island, an anomalous tendency to orientate them towards a declination of c. -14 degrees is detected. We provide some possible explanations for this, which include the date of a traditional Canarian celebration, an eventual imprint of topography, and the possibility of sunset orientations. Also, this particular value of declination is close to -16.3 degrees, the declination of Sirius during the 17th century. Therefore, we provide ethnographic data that might support an eventually controversial 'bright star' orientation. For the second group, meanwhile, we find a pattern of orientation where the apse of the churches points slightly to the north of due east. We propose this might signal constructions that were oriented to the rising Sun on dates close to Easter, one of the most important festivities of Christianity.

arXiv.org

Off-diagonal deformations of regular Schwarzschild black holes and general relativistic G. Perelman thermodynamics arxiv.org/abs/2505.18208

Off-diagonal deformations of regular Schwarzschild black holes and general relativistic G. Perelman thermodynamics

We construct new classes of solutions describing generic off-diagonal deformations of regular Schwarzschild black holes (BHs) in general relativity (GR). Examples of such (primary) diagonal metrics reducing the Einstein equations to integrable systems of nonlinear ordinary differential equations were studied in a recent work by R. Casadio, A. Kamenshchik and J. Ovalle in Phys. Rev. D 111 (2025) 064036. We develop and apply our anholonomic frame and connection deformations method, which allows us to generate new classes of target off-diagonal solutions. Ansatz that reduces the gravitational field equations to systems of (exactly or parametric) integrable systems of nonlinear partial differential equations are used. We find and analyze certain families of deformed regular BHs containing an off-diagonal de Sitter condensate encoding solitonic vacuum configurations, with possible deformations of horizons and/or gravitational polarizations of constants. We emphasize that general off-diagonal solutions do not involve certain hypersurface or holographic configurations. This means that the Bekenstein-Hawking thermodynamic paradigm is not applicable for characterizing the physical properties of such target regular solutions. We argue that the concept of G. Perelman's entropy and relativistic geometric flow thermodynamics is more appropriate. Using nonlinear symmetries involving effective cosmological constants, we show how to compute thermodynamic variables for various classes of physically essential solutions in GR.

arXiv.org

Microtubule polymerization generates microtentacles important in circulating tumor cell invasion arxiv.org/abs/2505.18301

Microtubule polymerization generates microtentacles important in circulating tumor cell invasion

Circulating tumor cells (CTCs) have crucial roles in the spread of tumors during metastasis. A decisive step is the extravasation of CTCs from the blood stream or lymph system, which depends on the ability of cells to attach to vessel walls. Recent work suggests that such adhesion is facilitated by microtubule (MT)-based membrane protrusions called microtentacles (McTNs). However, how McTNs facilitate such adhesion and how MTs can generate protrusions in CTCs remain unclear. By combining fluorescence recovery after photobleaching (FRAP) experiments and simulations we show that polymerization of MTs provides the main driving force for McTN formation, whereas the contribution of MTs sliding with respect to each other is minimal. Further, the forces exerted on the McTN tip result in curvature, as the MTs are anchored at the other end in the MT organizing center. When approaching vessel walls, McTN curvature is additionally influenced by the adhesion strength between the McTN and wall. Moreover, increasing McTN length, reducing its bending rigidity, or strengthening adhesion enhances the cell-wall contact area and, thus, promotes cell attachment to vessel walls. Our results demonstrate a link between the formation and function of McTNs, which may provide new insight into metastatic cancer diagnosis and therapy.

arXiv.org

Verifiability and Limit Consistency of Eddy Viscosity Large Eddy Simulation Reduced Order Models arxiv.org/abs/2505.18310

Verifiability and Limit Consistency of Eddy Viscosity Large Eddy Simulation Reduced Order Models

Large eddy simulation reduced order models (LES-ROMs) are ROMs that leverage LES ideas (e.g., filtering and closure modeling) to construct accurate and efficient ROMs for convection-dominated (e.g., turbulent) flows. Eddy viscosity (EV) ROMs (e.g., Smagorinsky ROM (S-ROM)) are LES-ROMs whose closure model consists of a diffusion-like operator in which the viscosity depends on the ROM velocity. We propose the Ladyzhenskaya ROM (L-ROM), which is a generalization of the S-ROM. Furthermore, we prove two fundamental numerical analysis results for the new L-ROM and the classical S-ROM: (i) We prove the verifiability of the L-ROM and S-ROM, i.e, that the ROM error is bounded (up to a constant) by the ROM closure error. (ii) We introduce the concept of ROM limit consistency (in a discrete sense), and prove that the L-ROM and S-ROM are limit consistent, i.e., that as the ROM dimension approaches the rank of the snapshot matrix, $d$, and the ROM lengthscale goes to zero, the ROM solution converges to the \emph{``true solution"}, i.e., the solution of the $d$-dimensional ROM. Finally, we illustrate numerically the verifiability and limit consistency of the new L-ROM and S-ROM in two under-resolved convection-dominated problems that display sharp gradients: (i) the 1D Burgers equation with a small diffusion coefficient; and (ii) the 2D lid-driven cavity flow at Reynolds number $Re=15,000$.

arXiv.org

Droplet encapsulating bubble: Investigation of droplet spreading dynamics and bubble encapsulation time arxiv.org/abs/2505.18339

Droplet encapsulating bubble: Investigation of droplet spreading dynamics and bubble encapsulation time

Ternary interactions between hetero-fluid particles, particularly the dynamics of droplets spreading over curved fluid interfaces remain insufficiently understood compared to the two-phase coalescence. In this study, we combine lattice Boltzmann simulations, high-speed imaging, and theoretical scaling to investigate the collision and encapsulation of an air bubble by a rising oil droplet in an immiscible medium. We systematically vary fluid properties, droplet-to-bubble size ratios, and collision configurations to quantify their impact on encapsulation time and flow evolution. The process unfolds in four stages: collision/film drainage, encapsulation, reshaping, and compound rising. Results indicate that encapsulation time increases exponentially with viscosity and is strongly modulated by the spreading coefficient (So), which governs the imbalance of interfacial tensions. Higher So values enhance capillary-driven spreading and reduce engulfment time, while lower values yield coupled deformation-reshaping behavior, introducing oscillations in bubble velocity and shape evolution. For low viscosity drops (Oh<0.1), the neck growth rate follows the well-known power-low relation with an exponent of 0.44-0.5, dependent on the size ratio. The transition between the spherical and deformed regimes is identified. Our theoretical analysis reveals that in low Bond numbers (Bo<0.11), spreading speed scales with viscous-capillary velocity, while in the deformed regime (0.11<Bo<2.2), encapsulation time follows a capillary-gravitational timescale. Interestingly, smaller droplets expedite encapsulation in equal-sized collisions but delay it in size-mismatched pairs, despite a faster initial neck growth. These findings provide new mechanistic insight into three-fluid interactions and offer guidance for optimizing encapsulation processes in applications such as gas flotation and interfacial microfluidics.

arXiv.org

SP2RINT: Spatially-Decoupled Physics-Inspired Progressive Inverse Optimization for Scalable, PDE-Constrained Meta-Optical Neural Network Training arxiv.org/abs/2505.18377

SP2RINT: Spatially-Decoupled Physics-Inspired Progressive Inverse Optimization for Scalable, PDE-Constrained Meta-Optical Neural Network Training

DONNs harness the physics of light propagation for efficient analog computation, with applications in AI and signal processing. Advances in nanophotonic fabrication and metasurface-based wavefront engineering have opened new pathways to realize high-capacity DONNs across various spectral regimes. Training such DONN systems to determine the metasurface structures remains challenging. Heuristic methods are fast but oversimplify metasurfaces modulation, often resulting in physically unrealizable designs and significant performance degradation. Simulation-in-the-loop training methods directly optimize a physically implementable metasurface using adjoint methods during end-to-end DONN training, but are inherently computationally prohibitive and unscalable.To address these limitations, we propose SP2RINT, a spatially decoupled, progressive training framework that formulates DONN training as a PDE-constrained learning problem. Metasurface responses are first relaxed into freely trainable transfer matrices with a banded structure. We then progressively enforce physical constraints by alternating between transfer matrix training and adjoint-based inverse design, avoiding per-iteration PDE solves while ensuring final physical realizability. To further reduce runtime, we introduce a physics-inspired, spatially decoupled inverse design strategy based on the natural locality of field interactions. This approach partitions the metasurface into independently solvable patches, enabling scalable and parallel inverse design with system-level calibration. Evaluated across diverse DONN training tasks, SP2RINT achieves digital-comparable accuracy while being 1825 times faster than simulation-in-the-loop approaches. By bridging the gap between abstract DONN models and implementable photonic hardware, SP2RINT enables scalable, high-performance training of physically realizable meta-optical neural systems.

arXiv.org

A two-sided subgrid-scale model for mass transfer across fluid interfaces arxiv.org/abs/2505.17036

A two-sided subgrid-scale model for mass transfer across fluid interfaces

The occurrence of extremely thin concentration boundary layers at fluid interfaces for high local Peclet numbers is a severe obstacle for efficient and accurate numerical simulation of mass transfer processes in two-phase fluid systems. Especially challenging are liquid-liquid systems, in which thin concentration boundary layers can appear on both sides of the fluid interface under convection-dominated conditions. In those cases, the one-sided species concentrations at the interface are a-priori not even known approximately, but are determined by a conjugate mass transfer problem governed by interfacial jump conditions. We introduce a two-sided subgrid-scale (SGS) boundary layer model for conjugate mass transfer at fluid interfaces that accurately computes the local mass transfer rates on moderate or coarse mesh resolutions even when very high concentration gradients in interface vicinity occur. For this purpose, SGS modeling is applied on both sides of an interface transmissive to passive scalars, such as the fluid interface, enabling the accurate capture of conjugate mass transfer across thin boundary layer on one or on both sides of the interface. We implement our approach in the unstructured Finite-Volume Arbitrary Lagrangian / Eulerian Interface-Tracking (ALE-IT) OpenFOAM module twoPhaseInterTrackFoam. We have made twoPhaseInterTrackFoam publicly available in our previous publication.

arXiv.org
Show older
Qoto Mastodon

QOTO: Question Others to Teach Ourselves
An inclusive, Academic Freedom, instance
All cultures welcome.
Hate speech and harassment strictly forbidden.