Search Results (199)

The current paper studies the global shell layer of the Finite Ring Continuum framework in the symmetry-complete regime realized here by framed finite fields, ${\mathbb{F}}_{\mathsf{p}}(\mathsf{t};0,1,e_{\mathsf{t}})$, with $\mathsf{p}=4\mathsf{t}+1$. We show that a single symmetry-complete shell carries a unified finite Euclidean datum for which its continuum comparison interpretation reproduces the familiar structural roles of e, $\pi$, and i of a one-phase step with an exponential kernel, a half-period, and a quarter-turn, respectively. In the same shell, the orbital geometry is generated by additive meridian action and multiplicative phase action from that same frame datum. The resulting orbital shell has a canonical spherical completion, combinatorially equivalent to the two-sphere, with labels depending on the chosen frame, but the shell type fixed up to isomorphism. Arbitrary finite-precision approximation on this external spherical comparison object is then obtained within every fixed symmetry-complete shell by the scale-periodic framed-rational refinement generated by the same frame datum. The Fourier formalism is developed strictly as a discrete Fourier transform over the shell ring, with conventional continuum Fourier language becoming a continuum large-$\mathsf{p}$ comparison case of that shell formalism. Full article

This article introduces novel methodologies, coordinate systems, and procedures in computational geometry that further develop a Euclidean-based relationalistic framework. The objective is to describe tools using object-oriented relational elements with symmetry, anchored to a fixed point in a relational model, that generate structured point sets serving as blueprints for geometric figures and physical structures representing their source objects. Geometric operations and transformations construct ratio figures and ordered proportional structures. Using discrete $\mathbb{N}$-Euclidean geometry, two relational coordinate systems are introduced—polar-vertex coordinates and radial coordinates—both formed through discrete geometric operations. A relational unit circle of fixed magnitude is defined by a 4::1 proportional equivalence between radius and angular ratios, independent of real-number or arc-length geometry. Euclid’s theory of proportion is extended from static abstract magnitudes to symmetry-driven geometric construction, and a square-pyramid geometric blueprint is produced from an Earth ratio figure with accurate dimensional magnitudes. The findings reveal a novel commensurability between the radius of a circle and the side length of a square using a shared fixed point coupled via a 3:4:5 Pythagorean-triple triangle, introducing the concept of ordered proportions. Full article

Electroencephalography (EEG) provides a non-invasive and high-temporal-resolution modality for decoding cognitive states, but high-density recordings remain challenging for Transformer-based models because self-attention scales quadratically with the number of channels. In addition, conventional Euclidean representations do not fully capture the intrinsic geometry of EEG covariance features, which may limit robustness in cross-subject settings. To address these issues, we propose EEG-RCformer, a Riemannian geometry-informed channel clustering Transformer for EEG decoding. The model first computes per-channel symmetric positive definite (SPD) covariance matrices from windowed EEG features and uses the affine-invariant Riemannian metric (AIRM) to identify trial-specific functional hubs. These hubs are then integrated with capacity-constrained spatial clustering to generate anatomically plausible and computationally efficient channel groups, which are encoded as tokens for a Transformer classifier. We evaluated EEG-RCformer on the MODMA and SEED datasets under both subject-dependent and -independent paradigms, achieving area under the curve (AUC) values of 0.9802 and 0.7154 on MODMA and 0.8541 and 0.8011 on SEED, respectively. Paired statistical tests further showed significant gains for MODMA in both the subject-dependent and -independent settings and for SEED in the subject-dependent setting, while SEED still showed a positive but non-significant mean improvement in the subject-independent setting. Full article

The developments in mathematics in the nineteenth century, in particular non-Euclidean geometry, which was not concerned with flat space, but with curvature, led at the end of the century and the beginning of the next one to much discussion of and experiments with the fourth dimension. The idea of a fourth dimension played a major role in the arts. In literature the Symbolists were convinced that there existed a “higher” reality behind the visible one and tried to suggest it in their poetry. In pictorial art and sculpture completely new forms emerged that distorted reality and in that way showed that one had to look at the world in a different way; there was something beyond the usual three dimensions. Many artists consciously tried to visualize this “beyondness”, the fourth dimension. The followers of the idea of a higher reality considered the fourth dimension as time, most artists as space. Much influence in the discussion about the fourth dimension had Charles Howard Hinton and, especially in Russia, Pyotr Ouspensky; both wrote a book entitled The Fourth Dimension (1904 and 1909, respectively), in which they propagated their ideas. The Futurist poet Velimir Klebnikov did not explicitly mention the fourth dimension in his work, but in view of his scientific interests (he studied mathematics at the University of Kazan, one of whose most celebrated scientists was Nikolai Lobachevsky, the founder of non-Euclidean geometry) and his close ties with the avant-garde painters, he was undoubtedly aware of the ideas about the fourth dimension in his time. Khlebnikov compared himself with Lobachevsky and used his geometry in his own description of the cities of the future. With his experiments with language and numerals he tried to find a new meaning behind the usual ones, and he made endless calculations to determine the laws of time: there must be some principle that rules the continuous stream of events. Establishing this principle, one might transcend history and ultimately find a solution for fate and death. His entire work is devoted to the search of a new dimension. Full article

We consider the problem of Bayesian parameter inference in the Merton structural credit risk model, where the posterior is induced by a jump-diffusion likelihood and the marginal evidence is not available in closed form. To approximate this posterior, we construct a variational family based on triangular sum-of-squares (SOS) polynomial flows, in which each component map is monotone by construction: its diagonal derivative is a positive definite quadratic form on a monomial basis, yielding a closed-form log-Jacobian and explicit gradients with respect to all flow parameters. The symmetric positive definite matrices parametrizing the flow are optimized by intrinsic Riemannian gradient ascent on the positive definite cone equipped with the affine-invariant metric, which preserves feasibility at every iterate without projection. We show that the rank-one Jacobian gradients produced by the SOS structure have unit norm in the affine-invariant metric, establishing a direct algebraic coupling between the transport family and the optimization geometry and implying a universal 1-Lipschitz bound for the log-Jacobian along geodesics. On the likelihood side, we derive exact score identities for all five structural parameters of the Merton model—drift, volatility, jump intensity, jump mean, and jump volatility—through both the Poisson log-normal mixture and the Fourier inversion representations. Strictly positive parameters are handled via exponential reparametrization, and the resulting gradients propagate end-to-end through the flow. We establish uniform truncation bounds on compact parameter sets for the infinite mixture and its associated score series, providing rigorous control over the finite approximations used in practice. The base distribution is chosen to be uniform on ${[0,1]}^5$, whose bounded support ensures uniform control of the monomial basis and stabilizes the polynomial calculus. These ingredients are assembled into a fully explicit modified ELBO with implementable gradients, combining Euclidean updates for vector parameters and intrinsic manifold updates for matrix parameters. Full article

Fiber-optic shape sensing enables real-time monitoring of structural deformation across a wide range of applications. For large-scale structures, Brillouin-based distributed sensing, typically implemented through Brillouin Optical Time Domain Analysis (BOTDA), offers an extended range for quasi-static measurements, albeit its limited spatial resolution degrades reconstruction accuracy. This study addresses this fundamental limitation through the introduction of a novel error compensation algorithm, particularly suited for a Brillouin-based shape sensing system, yet agnostic with respect to the sensing technology. The method leverages both the initial and final points of the sensing path, performing both forward and backward reconstructions and fusing the two trajectories by testing several polynomial and exponential weighting strategies. The algorithm is experimentally validated on a 28.91 m four-core shape sensing fiber cable (length = L), interrogated through BOTDA operating at 50 cm spatial resolution, and reconstructed through the Frenet–Serret frame formulation. Calibration procedures include radial-offset tuning and segment alignment via a hotspot reference. A non-trivial S-shaped geometry is adopted as a case study, specifically addressing curvature discontinuities arising from mixed straight and curved segments. Reconstruction accuracy is quantified through a Euclidean-distance-based Figure of Merit (FOMs). The cubic weighting strategy demonstrates improvements exceeding 86% in all FOMs compared to classical methods without compensation. Specifically, it achieves an RMSE of 0.145 m (0.50% of L), a MAE of 0.109 m (0.38% of L), and a maximum error of 0.341 m (1.18% of L). Remarkably, these percentage errors are of the same order of magnitude as those reported in the literature for Fiber Bragg Grating (FBG) and Optical Frequency Domain Reflectometry (OFDR) systems, indicating that the proposed compensation strategy enables BOTDA-based shape sensing to achieve comparable reconstruction accuracy despite its lower spatial resolution. Full article

The classical Erlangen Program sought to classify metric spaces entirely in terms of their symmetries. In physical spacetimes, these symmetries define transformations between classical reference frames, yielding a one-to-one correspondence between frame transformations and the underlying geometry. More recently, the classical notion of an ideal frame has been extended to the quantum regime, by considering observers as embodied physical systems, subject to the laws of quantum mechanics. Here, we build on this approach, but outline an alternative definition of the term ‘quantum reference frame’, which differs somewhat from the mainstream view. We then show how the new definition can be used to construct a simple model of Planck-scale spacetime, which makes contact with existing quantum gravity phenomenology. Finally, we show how classical spacetime symmetries can be ‘mathematically preserved but operationally broken’ using the new model, suggesting that quantum spacetime may be classified, at least locally, in terms of transformations between quantised frames of reference. This work is based on a talk given at the 13th Bolyai–Gauss–Lobachevsky Conference on Non-Euclidean Geometry in Oujda, Morocco, in May 2025. Full article

We present Hyperbolic Symmetric Hypermodular Neural Operators (ONHSH), a novel operator learning framework for solving partial differential equations (PDEs) in curved, anisotropic, and modularly structured domains. The architecture integrates three components: hyperbolic-symmetric activation kernels that adapt to non-Euclidean geometries, modular spectral smoothing informed by arithmetic regularity, and curvature-sensitive kernels based on anisotropic Besov theory. In its theoretical foundation, the Ramanujan–Santos–Sales Hypermodular Operator Theorem establishes minimax-optimal approximation rates and provides a spectral-topological interpretation through noncommutative Chern characters. These contributions unify harmonic analysis, approximation theory, and arithmetic topology into a single operator learning paradigm. In addition to theoretical advances, ONHSH achieves robust empirical results. Numerical experiments on thermal diffusion problems demonstrate superior accuracy and stability compared to Fourier Neural Operators and Geo-FNO. The method consistently resolves high-frequency modes, preserves geometric fidelity in curved domains, and maintains robust convergence in anisotropic regimes. Error decay rates closely match theoretical minimax predictions, while Voronovskaya-type expansions capture the tradeoffs between bias and spectral variance observed in practice. Notably, ONHSH kernels preserve Lorentz invariance, enabling accurate modeling of relativistic PDE dynamics. Overall, ONHSH combines rigorous theoretical guarantees with practical performance improvements, making it a versatile and geometry-adaptable framework for operator learning. By connecting harmonic analysis, spectral geometry, and machine learning, this work advances both the mathematical foundations and the empirical scope of PDE-based modeling in structured, curved, and arithmetically. Full article

Self-supervised learning (SSL) has emerged as a powerful paradigm for representation learning by optimizing geometric objectives, such as invariance to augmentations, variance preservation, and feature decorrelation, without requiring labels. However, most existing methods operate in Euclidean space, limiting their ability to capture nonlinear dependencies and geometric structures. In this work, we propose Kernel VICReg, a novel self-supervised learning framework that pulls the VICReg objective into a Reproducing Kernel Hilbert Space (RKHS). By kernelizing each term of the loss, variance, invariance, and covariance, we obtain a general formulation that operates on double-centered kernel matrices and Hilbert–Schmidt norms, enabling nonlinear feature learning without explicit mappings. We demonstrate that Kernel VICReg mitigates the risk of representational collapse under challenging conditions and improves performance on datasets exhibiting nonlinear structure or limited sample regimes. Empirical evaluations across MNIST, CIFAR-10, STL-10, TinyImageNet, and ImageNet100 show consistent gains over Euclidean VICReg, with particularly strong improvements on datasets where nonlinear structures are prominent. UMAP visualizations are provided only as a qualitative illustration of embedding geometry and are not used as a calibration or statistical validation. Our results suggest that kernelizing SSL objectives is a promising direction for bridging classical kernel methods with modern representation learning. Full article

This work proposes a lightweight biometric key-binding scheme that adapts a PUF-style challenge–response mechanism to face geometry: a two-factor password and session nonce generate random challenge points, Gray-coded Euclidean distances to facial landmarks form responses, and a random key is bound by discarding selected positions so only a reduced subset, the nonce, and a key hash are stored. At authentication, a fresh response set is compared to the subset with a Hamming-distance tolerance, and bounded local search corrects residual errors; each successful session rotates the nonce and refreshes the ephemeral key. We frame this as a conceptual exploration of an interpretable, on-device, controlled-capture design niche—a per-session nonce-driven cancelable biometric key-binding mechanism—and we quantify the resulting security–usability trade-offs. Empirically, the scheme works under stable capture conditions with carefully tuned thresholds, and it is naturally suited to tightly controlled deployments (e.g., access kiosks) where it can also incorporate user-driven micro-gestures as an extra behavioral factor. While the construction is fragile under broader variability and leans on the second factor for security, it offers an alternative to existing mechanisms and a clear niche, and we present it as a conceptual exploration showing how CRP mechanisms can inform cancelable biometrics with per-session revocability. Full article

Accurate quantification of the crack-tip driving force $\left(∆K\right)$ is fundamental to predicting the fatigue life of engineering structures. Analytical formulations of $∆K$ are rarely available for components with complex geometries. In such cases, finite element (FE) analysis has become a widely accepted approach for determining $∆K$. In this study, an FE-based solution for the crack-tip driving force of a fatigue crack in an asymmetric L-shaped bell crank geometry, a representative complex structure, is established. The structure is fabricated from AISI 410 martensitic stainless steel. The FE-predicted $∆K_I$ for crack growth in the Paris regime has been independently validated using the fractal crack-tip driving force model. Results show that the fatigue crack in the bell crank structure is driven by a combined Mode-I (opening) and Mode-II (shearing) crack-tip loading along a curved crack-path trajectory, as dictated by the asymmetric stress distribution. The fatigue crack edge exhibits fractality with fractal dimensions ranging from 1.00 (Euclidean) to 1.18 along the crack length ($a-a_0$) up to 9.947 mm. The FE-calculated crack-tip driving forces of the bell crank structure are comparable with those computed based on the corrected crack edge fractal dimensions, thus validating the FE simulation outcomes. The resulting fatigue crack growth rates, determined from crack-tip driving forces based on validated FE-computed contour integrals, are comparable to those obtained from the ASTM standard tests. Full article

The transition of the mining industry towards Industry 5.0 demands predictive models capable of strictly adhering to physical laws and modeling complex, non-Euclidean geometries—capabilities often lacking in standard graph neural networks. This systematic review, conducted under the PRISMA 2020 protocol, analyzes the emergence of “Era 5” architectures by synthesizing 96 high-impact studies from 2019 to 2026, focusing on Clifford (geometric algebra) GNNs, simplicial and cell complex neural networks, symplectic/Hamiltonian GNNs, and generative flow networks (GFlowNets). The analysis demonstrates that Clifford architectures provide superior rotational equivariance for robotic control; Simplicial networks capture high-order topological interactions critical for geomechanics; Symplectic GNNs ensure energy conservation for stable long-term simulation of structural dynamics; and GFlowNets offer a novel paradigm for generative mine planning. We conclude that shifting from data-driven approximations to these mathematically rigorous, structure-preserving architectures is fundamental for developing reliable, physics-informed digital twins that optimize structural integrity and operational efficiency in complex industrial environments. Full article

Generative Adversarial Networks (GANs) perform well on natural images but often fail in domains governed by strict geometric or symbolic constraints. This work focuses on convolutional GANs and studies how their inductive biases interact with two contrasting types of synthetic image data: fractal patterns, characterized by self-similarity and scale-invariant local structure, and Euclidean shapes, defined by simple geometric primitives and rigid global constraints. Using multiple convolutional GAN architectures (DCGAN, WGAN-GP, and SNGAN), two resolutions (64 × 64 and 128 × 128), and a suite of evaluation metrics, we compare adversarial training behavior on these datasets under tightly controlled conditions. Fractal datasets yield stable training dynamics and perceptually plausible generations, whereas Euclidean shape datasets consistently exhibit structural failure modes that persist under higher resolution, smoother shape representations, and architectural stabilization. Geometry-aware metrics reveal severe violations of global shape consistency in Euclidean outputs that are not reliably captured by standard perceptual or distributional measures such as FID, SSIM, or LPIPS. We argue that these findings reflect a fundamental inductive bias of convolutional generative models toward a locally rich, scale-repeating structure rather than globally constrained geometry. Rather than indicating that fractals are intrinsically easier to model, our results show that Euclidean geometry exposes limitations of adversarial generative learning that remain hidden under conventional evaluation. From this perspective, fractal datasets serve as informative diagnostic benchmarks for probing how adversarially trained convolutional generators handle scale-invariant structure versus globally constrained geometry, and our results highlight the need for domain-aware metrics and alternative architectural biases when applying generative models to structured or symbolic data. Full article

To address the computational-efficiency bottlenecks of Hybrid A* and its bidirectional variant in long-distance parking and narrow-slot scenarios, an improved bidirectional Hybrid A* algorithm is presented. First, the cohesion cost is reformulated in a vector-space representation. Distance and heading-consistency terms are evaluated using dot products, which eliminates trigonometric operations and reduces the overhead of node evaluation. Second, an RS (Reeds–Shepp) cost template is constructed on a sparse grid of key nodes. Neighborhood costs are approximated with Euclidean-distance correction. In addition, a geometry reachability-based trigger is designed for analytic RS connections to avoid redundant analytic linking and unnecessary RS curve computations. Third, a KD-tree spatial index is introduced to accelerate nearest-neighbor queries in the Voronoi potential field, and vehicle corner coordinates are updated in a vectorized manner to improve the efficiency of potential-field evaluation. Simulation results in parallel and perpendicular parking show that, compared with the baseline bidirectional Hybrid A* algorithm, RS computations are reduced by 98.7% and 97.8%, respectively, while total planning time is shortened by 63.2% and 57.5%, with stable path quality. These results indicate that the proposed method effectively mitigates the dominant computational costs of bidirectional Hybrid A* in complex parking tasks and improves the efficiency and real-time performance of automatic parking path planning. Full article

Analyzing three-dimensional (3D) phenotypic parameters of maize seedlings is of significant importance for maize cultivation and selection. However, existing methods often struggle to balance cost, efficiency, and accuracy, particularly when capturing the complex morphology of seedlings characterized by slender stems. To address these issues, this study proposes a novel end-to-end automated framework for extracting phenotypes using only consumer-grade RGB cameras. The pipeline initiates with Instant-NGP to rapidly reconstruct dense point clouds, establishing the 3D data foundation for phenotypic extraction. Subsequently, we formulate a directed topological graph-based mechanism. By mathematically defining bifurcation constraints via vector analysis, this mechanism guides a depth-first traversal strategy to explicitly disentangle stem and leaf skeletons. Building upon these decoupled skeletons, organ-level point cloud segmentation is achieved through constraint-based expansion, followed by density-based spatial clustering (DBSCAN) to detect individual leaves. Algorithms combining point cloud geometry with 3D Euclidean distance are also implemented to calculate key phenotypes including plant height and stem width. Finally, single-leaf skeleton fitting is used to estimate leaf length, and principal component analysis (PCA) is adopted to determine the stem–leaf angle, realizing the comprehensive automatic extraction of maize seedling phenotypes. Experiments show that the proposed method achieves high accuracy in extracting key phenotypic parameters. The mean relative errors for plant height, stem width, leaf length, stem-leaf angle, and leaf area are 0.76%, 2.93%, 1.26%, 2.13%, and 3.33%, respectively. Compared with existing methods as far as we know, the proposed method significantly improves extraction efficiency by reducing the processing time per plant to within 5 min while maintaining such high accuracy. Full article