Search Results (453)

In this study, theorems and proofs related to spherical and focal curves are presented in the BCV-Sasakian space. An approximate solution to the differential equation characterizing spherical curves in the BCV-Sasakian manifold ${\mathfrak{M}}^3$ is obtained using the Taylor matrix collocation method. The general equations of canal and tubular surfaces are provided within this geometric framework. Additionally, the curvature properties of the tubular surface constructed around a non-vertex focal curve are computed and analyzed. All of these results are presented for the first time in the literature within the context of the BCV-Sasakian geometry. Thus, this study makes a substantial contribution to the differential geometry of contact metric manifolds by extending classical concepts into a more generalized and complex geometric structure. Full article

This paper explores novel inequalities for statistical submanifolds within the framework of the Norden golden-like statistical manifold. By leveraging the intrinsic properties of statistical manifolds and the structural richness of Norden golden geometry, we establish fundamental relationships between the intrinsic and extrinsic invariants of submanifolds. The methodology involves deriving generalized Chen-type and $\delta (2,2)$ curvature inequalities using curvature tensor analysis and dual affine connections. A concrete example is provided to verify the theoretical framework. The novelty of this work lies in extending classical curvature inequalities to a newly introduced statistical structure, thereby opening new perspectives in the study of geometric inequalities in information geometry and related mathematical physics contexts. Full article

The current work establishes the geometrical bearing for hypersurfaces in a Golden Riemannian manifold with constant golden sectional curvature with respect to k-almost Newton-conformal Ricci solitons. Moreover, we extensively explore the immersed r-almost Newton-conformal Ricci soliton and determine the sufficient conditions for total geodesicity with adequate restrictions on some smooth functions using mathematical operators. Furthermore, we go over some natural conclusions in which the gradient k-almost Newton-conformal Ricci soliton on the hypersurface of the Golden Riemannian manifold becomes compact. Finally, we establish a Schur’s type inequality in terms of k-almost Newton-conformal Ricci solitons immersed in Golden Riemannian manifolds with constant golden sectional curvature. Full article

This paper presents a CFD-based method for the aerodynamic design of a high-pressure compressor rotor blisk, taking into account manufacturing constraints. Focus is placed on the influence of geometric deviations caused by the dynamic constraints of the milling machine. Special attention is given to the leading edge region of the blade, where high curvature results in increased sensitivity in both aerodynamic behavior and manufacturability. The generic blisk geometry on which this study is based is characterized by an elliptical leading edge. For the optimization, the leading edge is described by Bézier curves that transition smoothly to the suction and pressure sides with continuous curvature and a non-dimensional length ratio. In steady-state RANS parameter studies, the length ratio is systematically varied while the chord length is kept constant. For the aerodynamic evaluation of the design’s key performance parameters such as blade pressure distribution, total pressure loss and compressor efficiency are considered. To evaluate the machine dynamics for a given design, compliance with the nominal feed rate and the deviation between the planned and actual tool tip positions were used as evaluation parameters. Compared to the reference geometry with an elliptical leading edge, the purely aerodynamic optimization achieved an isentropic efficiency improvement of +0.24 percentage points in the aerodynamic design point and a profile deviation improvement of $-3$$\mu$m in the 99th quantile. The interdisciplinary optimization achieved an improvement of +0.20 percentage points and $-30$$\mathrm{\mu }$m, respectively. This comparative study illustrates the potential of multidisciplinary design approaches that balance aerodynamic performance goals with manufacturability via a novel approach for Design-to-Manufacture-to-Design. Full article

In this study, We explore for Minkowski 3-space ${\mathbb{E}}_1^3$ harmonic surfaces’ geometric features by employing a common tangent vector field along a curve situated on the surface. Our analysis is grounded in the rotation minimizing (RM) Darboux frame, which offers a robust alternative to the classical Frenet frame particularly valuable in the Lorentzian setting, where singularities frequently arise. The RM Darboux frame, tailored to curves lying on surfaces, enables the expression of fundamental invariants such as geodesic curvature, normal curvature, and geodesic torsion. We derive specific conditions that characterize harmonic surfaces based on these invariants. We also clarify the connection between the components of the RM Darboux frame and thesurface’s mean curvature vector. This formulation provides fresh perspectives on the classification and intrinsic structure of harmonic surfaces within Minkowski geometry. To support our findings, we present several illustrative examples that demonstrate the applicability and strength of the RM Darboux approach in Lorentzian differential geometry. Full article

This paper investigates the vibration characteristics of tensioned umbrella-shaped membrane structures with complex curvature under heavy rainfall. To solve the geometrical problem of the complex curvature of a membrane surface, we set the rule of segmentation and simplify the shape by dividing it into multi-segment conical membranes. The generatrix becomes a polyline with a constant surface curvature in each segment, simplifying calculations. The equivalent uniform load of different rainfall intensity is determined by the theory of the stochastic process. The governing equations of the isotropic damped nonlinear forced vibration of membranes are established by using the theory of large deflection by von Karman and the principle of d’Alembert. The equations of the forced vibration of the membrane are solved by using Galerkin’s method and the small-parameter perturbation method, and the displacement function, vibration frequency, and acceleration of the membrane are obtained. At last, the influence of the height–span ratio, number of segments, pretension and load on membrane displacement, vibration frequency, and acceleration of the membrane surface are analyzed. Based on the above data, the general law of deformation of the umbrella-shaped membrane under heavy rainfall is obtained. Data and methods are provided for the design and construction of the membrane structure as a reference. Moreover, we propose methods to enhance calculation accuracy and streamline the computational process. Full article

Lithological identification of outcrops in complex geological settings plays a crucial role in hydrocarbon exploration and geological modeling. To address the limitations of traditional field surveys, such as low efficiency and high risk, we proposed an intelligent lithology recognition method, SG-RFGeo, for terrestrial laser scanning (TLS) outcrop point clouds, which integrates spectral and geometric features. The workflow involves several key steps. First, lithological recognition units are created through regular grid segmentation. From these units, spectral reflectance statistics (e.g., mean, standard deviation, kurtosis, and other related metrics), and geometric morphological features (e.g., surface variation rate, curvature, planarity, among others) are extracted. Next, a double-layer random forest model is employed for lithology identification. In the shallow layer, the Gini index is used to select relevant features for a coarse classification of vegetation, conglomerate, and mud–sandstone. The deep-layer module applies an optimized feature set to further classify thinly interbedded sandstone and mudstone. Geological prior knowledge, such as stratigraphic attitudes, is incorporated to spatially constrain and post-process the classification results, enhancing their geological plausibility. The method was tested on a TLS dataset from the Yueyawan outcrop of the Qingshuihe Formation, located on the southern margin of the Junggar Basin in China. Results demonstrate that the integration of spectral and geometric features significantly improves classification performance, with the Macro F1-score increasing from 0.65 (with single-feature input) to 0.82. Further, post-processing with stratigraphic constraints boosts the overall classification accuracy to 93%, outperforming SVM (59.2%), XGBoost (67.8%), and PointNet (75.3%). These findings demonstrate that integrating multi-source features and geological prior constraints effectively addresses the challenges of lithological identification in complex outcrops, providing a novel approach for high-precision geological modeling and exploration. Full article

The offshore implementation of vertical-axis wind turbines (VAWTs) presents a promising new paradigm for advancing marine wind energy utilization, owing to their omnidirectional wind acceptance, compact structural design, and potential for lower maintenance costs. However, VAWTs still face major aerodynamic challenges, particularly due to the pitching motion, where the angle of attack varies cyclically with the blade azimuth. This leads to strong unsteady effects and susceptibility to dynamic stalls, which significantly degrade aerodynamic performance. To address these unresolved issues, this study conducts a comprehensive investigation into the dynamic stall behavior and wake vortex evolution induced by Darrieus-type pitching motion (DPM). Quasi-three-dimensional CFD simulations are performed to explore how variations in blade geometry influence aerodynamic responses under unsteady DPM conditions. To efficiently analyze geometric sensitivity, a surrogate model based on a radial basis function neural network is constructed, enabling fast aerodynamic predictions. Sensitivity analysis identifies the curvature near the maximum thickness and the deflection angle of the trailing edge as the most influential geometric parameters affecting lift and stall behavior, while the blade thickness is shown to strongly impact the moment coefficient. These insights emphasize the pivotal role of blade shape optimization in enhancing aerodynamic performance under inherently unsteady VAWT operating conditions. Full article

The primary aim of this paper is to establish two sharp geometric inequalities concerning submanifolds of S-space forms equipped with semi-symmetric metric connections (SSMCs). Specifically, we derive new inequalities involving the generalized normalized $\delta$-Casorati curvatures ${\delta }_c(t;q_1+q_2-1)$ and ${\widehat{\delta }}_c(t;q_1+q_2-1)$ for bi-slant submanifolds. The cases in which equality holds are thoroughly examined, offering a deeper understanding of the geometric structure underlying such submanifolds. In addition, we present several immediate applications that highlight the relevance of our findings, and we support the article with illustrative examples. Full article

We advance a mathematical framework for collective conviction by deriving a continuum theory from the network-based model introduced by us recently. The resulting equation governs the evolution of belief through a degenerate anisotropic logistic–diffusion process, where diffusion slows as conviction saturates. In one spatial dimension, we prove global well-posedness, demonstrate spectral front pinning that arrests the spread of influence at finite depth, and construct explicit traveling-wave solutions. In two dimensions, we uncover a geometric mechanism of curvature–induced quenching, where belief propagation halts along regions of low effective mobility and curvature. Building on this insight, we formulate a variational principle for optimal control under resource constraints. The derived feedback law prescribes how to spatially allocate repression effort to maximize inhibition of front motion, concentrating resources along high-curvature, low-mobility arcs. Numerical simulations validate the theory, illustrating how localized suppression dramatically reduces transverse spread without affecting fast axes. These results bridge analytical modeling with societal phenomena such as protest diffusion, misinformation spread, and institutional resistance, offering a principled foundation for selective intervention policies in structured populations. Full article

This study explores the dynamics of particle motion in pseudo-Galilean $3-$space $G_3^1$ by considering actions that incorporate both curvature and torsion of trajectories. We consider a general energy functional and formulate Euler–Lagrange equations corresponding to this functional under some boundary conditions in $G_3^1$. By adapting the geometric tools of the Frenet frame to this setting, we analyze the resulting variational equations and provide illustrative solutions that highlight their structural properties. In particular, we examine examples derived from natural Hamiltonian trajectories in $G_3^1$ and extend them to reflect the distinctive geometric features of pseudo-Galilean spaces, offering insight into their foundational behavior and theoretical implications. Full article

Defects arising from the 3D printing process of continuous fiber-reinforced thermoplastic composites primarily hinder their overall performance. These defects particularly include twisting, folding, and breakage of the fiber bundle, which are induced by printing trajectory errors. This study presents a follow-up theory assumption to address such issues, elucidates the formation mechanism of printing trajectory errors, and examines the impact of key geometric parameters—trace curvature, nozzle diameter, and fiber bundle diameter—on these errors. An error model for printing trajectory is established, accompanied by the proposal of a trajectory error compensation method premised on maximum printable curvature. The presented case study uses CCFRF/PA as an exemplar; here, the printing layer height is 0.1~0.3 mm, the fiber bundle radius is 0.2 mm, and the printing speed is 600 mm/min. The maximum printing curvature, gauged by the printing trajectory of a clothoid, is found to be 0.416 mm⁻¹. Experimental results demonstrate that the error model provides accurate predictions of the printed trajectory error, particularly when the printed trajectory forms an obtuse angle. The average prediction deviations for line profile, deviation kurtosis, and deviation area ratio are 36.029%, 47.238%, and 2.045%, respectively. The error compensation effectively mitigates the defects of fiber bundle folding and twisting, while maintaining the printing trajectory error within minimal range. These results indicate that the proposed method substantially enhances the internal defects of 3D printed components and may potentially be applied to other continuous fiber printing types. Full article

Near-unstable cavities hold promise for reducing thermal noise in next-generation gravitational wave detectors and for enhancing light–matter interactions in quantum electrodynamics. However, operating close to the edge of geometrical stability presents significant challenges, including increased coupling to higher-order modes and heightened sensitivity to small cavity length changes and mirror imperfections. This study employs Finesse v3 simulations to systematically investigate the modal behavior of a plano-concave cavity as it approaches instability, incorporating measured mirror surface defects and anisotropic curvature to replicate realistic conditions. The simulations highlight the degradation of beam purity and control signals as the cavity approaches instability. By validating the simulations against experimental data, we confirm Finesse’s reliability for modeling cavities while identifying critical limitations in regimes close to the edge of stability. These findings provide essential guidance for optimizing cavity designs in future gravitational wave detectors, balancing performance gains against the challenges of operating at the stability edge. Full article

This paper investigates an integrable spin system known as the Myrzakulov-XIII (M-XIII) equation through geometric and gauge-theoretic methods. The M-XIII equation, which describes dispersionless dynamics with curvature-induced interactions, is shown to admit a geometric interpretation via curve flows in three-dimensional space. We establish its gauge equivalence with the complex coupled dispersionless (CCD) system and construct the corresponding Lax pair. Using the Sym–Tafel formula, we derive exact soliton surfaces associated with the integrable evolution of curves and surfaces. A key focus is placed on the role of geometric and gauge symmetry in the integrability structure and solution construction. The main contributions of this work include: (i) a commutative diagram illustrating the connections between the M-XIII, CCD, and surface deformation models; (ii) the derivation of new exact solutions for a fractional extension of the M-XIII equation using the Kudryashov method; and (iii) the classification of these solutions into trigonometric, hyperbolic, and exponential types. These findings deepen the interplay between symmetry, geometry, and soliton theory in nonlinear spin systems. Full article

In this study, ruled surfaces are examined where the direction vectors are unit vectors derived from Smarandache curves, and the base curve is taken as an adjoint curve constructed using the integral curve of a Smarandache-type curve generated from the first and second Bishop normal vectors. The newly generated ruled surfaces will be referred to as Bishop adjoint ruled surfaces. Explicit expressions for the Gaussian and mean curvatures of these surfaces have been obtained, and their fundamental geometric properties have been analyzed in detail. Additionally, the conditions for developability, minimality, and singularities have been investigated. The asymptotic and geodesic behaviors of parametric curves have been examined, and the necessary and sufficient conditions for their characterization have been derived. Furthermore, the geometric properties of the surface generated by the Bishop adjoint curve and its relationship with the choice of the original curve have been established. The constructed ruled surfaces exhibit a notable degree of geometric regularity and symmetry, which naturally arise from the structural behavior of the associated adjoint curves and direction fields. This underlying symmetry plays a central role in their formulation and classification within the broader context of differential geometry. Finally, the obtained surfaces are illustrated with figures. Full article