Search Results (567)

This paper explores some frame bundles and physical implications of Killing vector fields on the two-sphere ${\mathbb{S}}^2$, culminating in a novel application to Maxwell’s equations in free space. Initially, we investigate the Killing vector fields on ${\mathbb{S}}^2$ (represented by the unit sphere of ${\mathbb{R}}^3$), which generate the isometries of the sphere under the rotation group $SO\left(3\right)$. These fields, realized as functions $K_v:{\mathbb{S}}^2\to {\mathbb{R}}^3$, defined by $K_v\left(q\right)=v\times q$ for a fixed $v\in {\mathbb{R}}^3$ and any$q\in {\mathbb{S}}^2$, generate a three-dimensional Lie algebra isomorphic to $\mathfrak{so}\left(3\right)$. We establish an isomorphism $K:{\mathbb{R}}^3\to \mathcal{K}\left({\mathbb{S}}^2\right)$, mapping vectors $v=au$ (with $u\in {\mathbb{S}}^2$) to scaled Killing vector fields $a{K}_u$, and analyze its relationship with $SO\left(3\right)$ through the exponential map. Subsequently, at a fixed point $e\in {\mathbb{S}}^2$, we construct a smooth orthonormal right-handed tangent frame $f_e:{\mathbb{S}}^2\setminus \{e,-e\}\to T{\left({\mathbb{S}}^2\right)}^2$, defined as $f_e\left(u\right)=({\widehat{K}}_e\left(u\right),u\times {\widehat{K}}_e\left(u\right))$, where ${\widehat{K}}_e$ is the unit vector field of the Killing field $K_e$. We verify its smoothness, orthonormality, and right-handedness. We further prove that any smooth orthonormal right-handed frame on ${\mathbb{S}}^2\setminus \{e,-e\}$ is either $f_e$ or a rotation thereof by a smooth map $\rho :{\mathbb{S}}^2\setminus \{e,-e\}\to SO\left(3\right)$, reflecting the triviality of the frame bundle over the parallelizable domain. The paper then pivots to an innovative application, constructing solutions to Maxwell’s equations in free space by combining spherical symmetries with quantum mechanical de Broglie waves in tempered distribution wave space. The deeper scientific significance lies in bringing together differential geometry (via $SO\left(3\right)$ symmetries), quantum mechanics (de Broglie waves in Schwartz distribution theory), and electromagnetism (Maxwell’s solutions in Schwartz tempered complex fields on Minkowski space-time), in order to offer a unifying perspective on Maxwell’s electromagnetism and Schrödinger’s picture in relativistic quantum mechanics. Full article

The Lorentz group ${\mathrm{Lor}}_{1,3}={\mathrm{SO}}_0(1,3)$ has two point fixgroups, namely $\mathrm{SO}\left(3\right)$ for time-like translations and ${\mathrm{SO}}_0(1,1)\times {\mathrm{I}\mathrm{R}}^2$ for light-like translations. However, for light-like translations, it is reasonable to consider a line fixgroup that leads to the Borel structure of the Lorentz group and provides appropriate helicities for massless particles. Therefore, whether a particle is massless or massive is not so much a physical question but rather a question of the underlying Lie group symmetry. Full article

In this paper, we present Lie symmetry analysis of a generalized (1+1)-dimensional porous medium equation characterized by parameters m and d. Through group classification, we examine how these parameters influence the Lie symmetry structure of the equation. Our analysis establishes conditions under which the equation admits either a three-dimensional or a five-dimensional Lie algebra. Using the obtained symmetry algebras, we construct optimal systems of one-dimensional subalgebras. Subsequently, we derive invariant solutions corresponding to each subalgebra, providing explicit formulas in relevant parameter regimes. These solutions deepen our understanding of the nonlinear diffusion processes modeled by porous medium equations and offer valuable benchmarks for analytical and numerical studies. Full article

The convective Cahn–Hilliard–Oono equation is analyzed under the conditions ${\mu }_1\ge 0$ and ${\mu }_3+{\mu }_4\le 0$. The Lie invariance criteria are examined through symmetry generators, leading to the identification of Lie algebra, where translation symmetries exist in both space and time variables. By employing Lie group methods, the equation is transformed into a system of highly nonlinear ordinary differential equations using appropriate similarity transformations. The extended direct algebraic method are utilized to derive various soliton solutions, including kink, anti-kink, singular soliton, bright, dark, periodic, mixed periodic, mixed trigonometric, trigonometric, peakon soliton, anti-peaked with decay, shock, mixed shock-singular, mixed singular, complex solitary shock, singular, and shock wave solutions. The characteristics of selected solutions are illustrated in 3D, 2D, and contour plots for specific wave number effects. Additionally, the model’s stability is examined. These results contribute to advancing research by deepening the understanding of nonlinear wave structures and broadening the scope of knowledge in the field. Full article

This study addresses the nonlinear estimation problem in the strapdown inertial navigation system (SINS) and Doppler velocity log (DVL) integrated navigation by proposing an improved filtering algorithm based on $S{E}_2(3)$ Lie group state representation. A dynamic model satisfying the group affine condition is established to systematically construct both left-invariant and right-invariant error state spaces, upon which two nonlinear filtering approaches are developed. Although the fast unscented transformation method is not novel by itself, its first integration with the $S{E}_2(3)$ Lie group model for SINS/DVL integrated navigation represents a significant advancement. Experimental results demonstrate that under large misalignment angles, the proposed method achieves slightly lower attitude errors compared to linear approaches, while also reducing position estimation errors during dynamic maneuvers. The 12,000 s endurance test confirms the algorithm’s stable long-term performance. Compared with conventional unscented Kalman filter methods, the proposed approach not only reduces computation time by 90% but also achieves real-time processing capability on embedded platforms through optimized sampling strategies and hierarchical state propagation mechanisms. These innovations provide an underwater navigation solution that combines theoretical rigor with engineering practicality, effectively overcoming the computational efficiency and dynamic adaptability limitations of traditional nonlinear filtering methods. Full article

In this paper, we investigate the geometric realization of $\mathrm{Spin}\left(8\right)$ triality through vector fields on the octonionic algebra $\mathbb{O}$. The triality automorphism group of $\mathrm{Spin}\left(8\right)$, isomorphic to ${\mathfrak{S}}_3$, cyclically permutes the three inequivalent 8-dimensional representations: the vector representation V and the spinor representations $S_{+}$ and $S_{-}$. While triality automorphisms are well known through representation theory, their concrete geometric realization as flows on octonionic space has remained unexplored. We construct explicit smooth vector fields $X_{\sigma }$ and $X_{{\sigma }^2}$ on $\mathbb{O}\cong {\mathbb{R}}^8$ whose flows generate infinitesimal triality transformations. These vector fields have a linear structure arising from skew-symmetric matrices that implement simultaneous rotations in three orthogonal coordinate planes, providing the first elementary geometric description of triality symmetry. The main results establish that these vector fields preserve the octonionic multiplication structure up to automorphisms in $G_2=\mathrm{Aut}\left(\mathbb{O}\right)$, demonstrating fundamental compatibility between geometric flows and octonionic algebra. We prove that the standard Euclidean metric on $\mathbb{O}$ is triality-invariant and classify all triality-invariant Riemannian metrics as conformal to the Euclidean metric with a conformal factor depending only on the isotonic norm. This classification employs Schur’s lemma applied to the irreducible $\mathrm{Spin}\left(8\right)$ action, revealing the rigidity imposed by triality symmetry. We provide a complete classification of triality-symmetric minimal surfaces, showing they are locally isometric to totally geodesic planes, surfaces of revolution about triality-fixed axes, or surfaces generated by triality orbits of geodesic curves. This trichotomy reflects the threefold triality symmetry and establishes correspondence between representation-theoretic decomposition and geometric surface types. For complete surfaces with finite total curvature, we establish global classification and develop explicit Weierstrass-type representations adapted to triality symmetry. Full article

This study introduces a data obfuscation technique that leverages the exponential map of Lie-group generators. Originating from quantum machine learning frameworks, the method injects controlled noise into these generators, deliberately breaking symmetry and obscuring the source data while retaining predictive utility. Experiments on open medical datasets show that classifiers trained on obfuscated features match or slightly exceed the baseline accuracy obtained on raw data. This work demonstrates how Lie-group theory can advance privacy in sensitive domains by providing simultaneous data obfuscation and augmentation. Full article

Matrix metalloproteinases (MMPs) are a major group of proteases known to regulate the turnover of the extracellular matrix (ECM). We observed that induced MMP-12 promotes eosinophilic inflammation-related epithelial cell mesenchymal transition (EMT), bronchial fibrosis, and airway obstruction in an allergen-exposed mouse model of chronic airway diseases in allergen-exposed mice and in airway-specific CC10-IL-13-overexpressed mice. Our histological analysis showed that the parabronchial and perivascular accumulation of eosinophils, fibroblasts, and collagen is significantly decreased in MMP-12^−/− allergen-exposed mice and airway-specific rtTA-MMP-12^−/−CC-10-IL-13-overexpressed mice compared to allergen-exposed wild-type mice and rtTA-CC10-IL-13-overexpressed mice. ELISA and Western blot analyses validated these histological findings, demonstrating that EMT and profibrotic protein levels were significantly decreased in allergen-challenged MMP-12^−/− mice and rtTA-MMP-12^−/−CC10-IL-13-overexpressed mice in comparison to the allergen-exposed wild-type mice and rtTA-CC10-IL-13-overexpressed mice. In addition, we also observed that allergen-challenged MMP-12^−/− mice have improved resistance and compliance compared to allergen-challenged wild-type mice. Most importantly, we show that treatment with MMP-12 inhibitors (PF-00356231 and MMP408) restricts the induction and progression of bronchial fibrosis and airway restrictions in allergen-exposed mice and airway-specific rtTA-CC10-IL-13 mice compared to the respective control mice. Taken together, the novelty of these findings lie in the fact that induced MMP-12 regulates eosinophilic inflammation-induced bronchial fibrosis and associated airway restriction, which may be reduced by treatment with MMP-12 inhibitors. Full article

Neural ordinary differential equations (neural ODEs) are a well-established tool for optimizing the parameters of dynamical systems, with applications in image classification, optimal control, and physics learning. Although dynamical systems of interest often evolve on Lie groups and more general differentiable manifolds, theoretical results for neural ODEs are frequently phrased on ${\mathbb{R}}^n$. We collect recent results for neural ODEs on manifolds and present a unifying derivation of various results that serves as a tutorial to extend existing methods to differentiable manifolds. We also extend the results to the recent class of neural ODEs on Lie groups, highlighting a non-trivial extension of manifold neural ODEs that exploits the Lie group structure. Full article

We present a formal reconstruction of the conventional number systems, including integers, rationals, reals, and complex numbers, based on the principle of relational finitude over a finite field ${\mathbb{F}}_{\mathsf{p}}$. Rather than assuming actual infinity, we define arithmetic and algebra as observer-dependent constructs grounded in finite field symmetries. Consequently, we formulate relational analogues of the conventional number classes, expressed relationally with respect to a chosen reference frame. We define explicit mappings for each number class, preserving their algebraic and computational properties while eliminating ontological dependence on infinite structures. For example, relationally framed rational numbers emerge from dense grids generated by primitive roots of a finite field, enabling proportional reasoning without infinity, while scale-periodicity ensures invariance under zoom operations, approximating continuity in a bounded structure. The resultant framework—that we denote as Finite Ring Continuum—aims to establish a coherent foundation for mathematics, physics and formal logic in an ontologically finite paradox-free informational universe. Full article

This paper proposes a cascaded state estimation framework based on proprioception for robots. A generalized-momentum-based Kalman filter (GMKF) estimates the ground reaction forces at the feet through joint torques, which are then input into an error-state Kalman filter (ESKF) to obtain the robot’s prior state estimate. The system’s dynamic equations on the Lie group are parameterized using canonical coordinates of the first kind, and variations in the tangent space are mapped to the Lie algebra via the inverse of the right trivialization. The resulting parameterized system state equations, combined with the prior estimates and a sliding window, are formulated as a moving horizon estimation (MHE) problem, which is ultimately solved using a parallel real-time iteration (Para-RTI) technique. The proposed framework operates on manifolds, providing a tightly coupled estimation with higher accuracy and real-time performance, and is better suited to handle the impact noise during foot–ground contact in legged robots. Experiments were conducted on the BQR3 robot, and comparisons with measurements from a Vicon motion capture system validate the superiority and effectiveness of the proposed method. Full article

Background: Numerous studies have demonstrated the beneficial effects of sodium-glucose cotransporter-2 (SGLT2) inhibitors on cardiovascular and renal outcomes in patients with heart failure and chronic kidney disease. However, whether SGLT2 inhibitors are also associated with a reduced risk of tinnitus has not been investigated. Objective: This study aimed to investigate the association between SGLT2 inhibitor therapy and the incidence of tinnitus in patients with type 2 diabetes. Methods: This retrospective cohort study was based on data from a nationally representative database of primary care practices in Germany from 2012 to 2023. Patients with type 2 diabetes who were treated with metformin and additionally received either an SGLT2 inhibitor or a dipeptidyl peptidase-4 (DPP4) inhibitor were included. Patients with a previous diagnosis of tinnitus were excluded. The primary outcome was the first tinnitus diagnosis documented by a primary care physician. The SGLT2 and DPP4 cohorts were compared for tinnitus incidence using Kaplan–Meier analysis and multivariable Cox regression. Results: 66,750 patients with SGLT2 inhibitors and 82,830 with DPP4 inhibitors were analyzed. The cumulative 5-year incidence of tinnitus was 1.9% in both groups. The multivariable regression analysis did not show a significant association between SGLT2 therapy and the occurrence of a tinnitus diagnosis (HR: 1.04; 95% CI: 0.89–1.21). Conclusion: There was no difference in tinnitus incidence between patients with SGLT2 or DPP4 inhibitors. The causes could lie in the heterogeneous, not purely vascular, etiology of tinnitus in general practitioners’ practices. Future studies should include further clinical data, including confirmed hearing impairments. Full article

Wigner’s little groups are the subgroups of the Poincaré group whose transformations leave the four-momentum of a relativistic particle invariant. The little group for a massive particle is SO(3)-like, whereas for a massless particle, it is E(2)-like. Multiple approaches to group contractions are discussed. It is shown that the Lie algebra of the E(2)-like little group for massless particles can be obtained from the SO(3) and from the SO(2, 1) group by boosting to the infinite-momentum limit. It is also shown that it is possible to obtain the generators of the E(2)-like and cylindrical groups from those of SO(3) as well as from those of SO(2, 1) by using the squeeze transformation. The contraction of the Lorentz group SO(3, 2) to the Poincaré group is revisited. As physical examples, two applications are chosen from classical optics. The first shows the contraction of a light ray from a spherical transparent surface to a straight line. The second shows that the focusing of the image in a camera can be formulated by the implementation of the focal condition to the [ABCD] matrix of paraxial optics, which can be regarded as a limiting procedure. Full article

In total, geodesic surfaces and their generalizations, namely totally umbilical and parallel surfaces, are well-known topics in Submanifold Theory and have been intensively studied in three-dimensional ambient spaces, both Riemannian and Lorentzian. In this paper, we prove the non-existence of parallel and totally umbilical (in particular, totally geodesic) surfaces for three-dimensional Lorentzian Lie groups, which admit a four-dimensional isometry group, but are neither of Bianchi–Cartan–Vranceanu-type nor homogeneous plane waves. Consequently, the results of the present paper complete the investigation of these fundamental types of surfaces in all homogeneous Lorentzian manifolds, whose isometry group is four-dimensional. As a byproduct, we describe a large class of flat surfaces of constant mean curvature in these ambient spaces and exhibit a family of examples. Full article

Privacy-sensitive electricity scene classification requires robust models under data localization constraints, making federated learning (FL) a suitable framework. Existing FL frameworks face two critical challenges in multi-label electricity scene classification: (1) Label correlations and their strengths significantly impact classification performance. (2) Electricity scene data and labels show distributional inconsistencies across regions. However, current FL frameworks lack explicit modeling of label correlation strengths, and locally trained regional models naturally capture these differences, leading to regional differences in their model parameters. In this scenario, the server’s standard single-stage aggregation often over-averages the global model’s parameters, reducing its discriminative ability. To address these issues, we propose FMMAN, a federated multi-stage attention neural network for multi-label electricity scene classification. The main contributions of this FMMAN lie in label correlation learning and the stepwise model aggregation. It splits the client–server interaction into multiple stages: (1) Clients train models locally to encode features and label correlation strengths after receiving the server’s initial model. (2) The server clusters these locally trained models into K groups to ensure that models within a group have more consistent parameters and generates K prototype models via intra-group aggregation to reduce over-averaging. The K models are then distributed back to the clients. (3) Clients refine their models using the K prototypes with contrastive group-specific consistency regularization to further mitigate over-averaging, and sends the refined model back to the server. (4) Finally, the server aggregates the models into a global model. Experiments on multi-label benchmarks verify that FMMAN outperforms baseline methods. Full article