Search Results (122)

Metaheuristic methods for three-dimensional (3D) unmanned aerial vehicle (UAV) path planning often suffer from premature convergence and reduced accuracy in complex high-dimensional spaces, in which waypoint-based decision variables exhibit structured dependencies and segment-level regularities. In a symmetry-inspired operational sense, these regularities can be interpreted as exploitable dependency patterns across path segments and permutation invariance among homogeneous UAVs, which are often overlooked by standard algorithms. The paper proposes an enhanced dung beetle optimizer (LEDBO) that integrates interaction-aware variable handling, adaptive role regulation, and a fitness-state-driven hybrid search mechanism. Correlation-based variable grouping clusters dependent waypoints into segments to exploit statistical dependency patterns among waypoint-coordinate variables and enhance local refinement. A three-level adaptive role-regulation scheme adjusts search behaviors according to convergence status and population diversity, thereby mitigating stagnation. Meanwhile, a fitness-state-driven hybrid engine combines Nelder–Mead local refinement with Lévy-flight global exploration to balance exploitation and exploration across stages. Experiments on the CEC2017 benchmark suite and complex 3D UAV path-planning simulations demonstrate that LEDBO achieves better solution quality, convergence behavior, and robustness than representative metaheuristics, producing smoother, shorter, and safer trajectories. The results suggest that incorporating interaction-aware variable grouping and adaptive search regulation can improve UAV path planning and related high-dimensional continuous optimization tasks. Full article

A class of self-similar solutions to the model of a cylindrical shock wave in non-uniform atmosphere in the presence of monochromatic radiation and gravitation in magneto gas dynamics has been obtained by using a similarity method. The propagation of a cylindrical shock wave in an ideal gas with monochromatic radiation and gravitating effects has been discussed. Through applying similarity transformations to the system of equations, we obtained the symmetry generators of the system. By using the symmetry generators and the surface invariance condition, we obtained the group invariant solution and then, with the help of group invariant solution, we converted the given system of PDEs to the system of ODEs together with the boundary condition. The obtained system of ODEs together with boundary condition has been solved numerically by using Runge–Kutta method of order four. The flow variables are analyzed graphically behind the shock with respect to the variation of parameters. Full article

Reaction–diffusion equations provide a fundamental framework for modelling spatial population dynamics and invasion processes in mathematical biology. Among these, Fisher’s equation combines diffusion with logistic growth to describe the spread of an advantageous gene and the formation of travelling population fronts. In this work, we investigate the one-dimensional Fisher’s equation using Lie symmetry analysis to obtain a deeper analytical understanding of its wave propagation behaviour. The Lie point symmetries of the partial differential equation are derived and used to construct similarity variables that reduce Fisher’s equation to ordinary differential equations. These reduced equations are then solved by a combination of direct integration and the tanh method, yielding explicit invariant and travelling-wave solutions. Symbolic computations in MAPLE are employed to compute the symmetries, verify the reductions, and generate illustrative plots of the resulting wave profiles. The computed solutions capture sigmoidal fronts connecting stable and unstable steady states, providing clear information about wave speed and shape. Overall, this study demonstrates that Lie group methods, combined with hyperbolic-function techniques, offer a powerful and systematic approach for analysing Fisher-type reaction–diffusion models and interpreting their biologically relevant invasion dynamics. Full article

We investigated an exact solution in a conformal invariant Randall-Sundrum 5D warped brane world model on a time dependent Kerr-like spacetime. The singular points are determined by a quintic polynomial in the complex plane and fulfills Cauchy’s theorem on holomorphic functions. The solution, which is determined by a first-degree differential equation, shows many similarities with an instanton. In order to describe the quantum mechanical aspects of the black hole solution, we apply the antipodal boundary condition. The solution is invariant under time reversal and also valid in Riemannian space. Moreover, CPT invariance in maintained. The vacuum instanton solution follows from the 5D as well as the effective 4D brane equations, only when we allow the contribution of the projected 5D Weyl tensor on the brane (the KK-‘particles’). The topology of the effective 4D space of the brane is the projective $\mathbb{R}{P}^3$ (elliptic space) by identifying antipodal points on $S^3$. The 5D is completed by applying the Klein bottle embedding and the ${\mathbb{Z}}_2$ symmetry of the RS model. This model fits very well with the description of the Hawking radiation, which remains pure. We have also indicated a possible way to include fermions. Our 5D space admits a double cover of $S^3$ and after fibering to the $S^2$, we obtain the effective black hole horizon. The connection with the icosahedron discrete symmetry group is investigated. It seem that Bekenstein’s conjecture that the area of a black hole is quantized, could be applied to our model. Full article

This paper introduces and systematically develops the theory of Zappa–Szép skew braces, a novel algebraic structure that provides a unified framework for bidirectional group interactions, thereby generalizing the classical constructions of semidirect skew braces and matched-pair factorizations (ZS1–ZS4, BC1–BC2). We establish the complete axiomatic foundation for these objects, characterizing them through necessary and sufficient compatibility conditions that encode mutual actions between two digroups. Central results include a semidirect embedding theorem, explicit constructions of nontrivial examples—notably a fully mutual brace of order 12 built from $V_4$ and $C_3$—and a detailed analysis of key structural invariants such as the socle, center, and automorphism groups. The framework is further elucidated via universal properties and categorical adjunctions, positioning Zappa–Szép skew braces as fundamental objects within noncommutative algebra. Applications to representation theory, cohomology, and the construction of set-theoretic solutions to the Yang–Baxter equation are derived, demonstrating both the generality and utility of the theory. Full article

We consider the nonlinear reaction–diffusion equation on the unit sphere $u_t={\mathrm{\Delta }}_{{\mathbb{S}}^2}u+f\left(u\right)$, $f_{uu}\ne 0$, and carry out a complete Lie point symmetry analysis. Solving the associated determining system yields a rigidity theorem: for every genuinely nonlinear $f\left(u\right)$, the admitted symmetry algebra is $\mathfrak{so}\left(3\right)\oplus ⟨{\partial }_t⟩$, generated by the rotational Killing fields and time translation. We further show through a group classification that the source families that enlarge symmetries in Euclidean space do not produce any additional point symmetries on ${\mathbb{S}}^2$. From an optimal system of subalgebras, we derive curvature-adapted reductions in which the Laplace–Beltrami operator becomes a Legendre-type operator in intrinsic invariants. For the specific nonlinear source $f\left(u\right)=-e^u-2$, specific reduced ODEs admit a hidden one-parameter symmetry, yielding a first integral and explicit steady states on ${\mathbb{S}}^2$. Full article

This work establishes a complete algebraic classification of hypersurfaces with totally symmetric cubic form, including the Codazzi, parallel, and totally geodesic cases, on the 4-dimensional 3-step nilpotent Lie group $Ni{l}^4$ endowed with six left-invariant Lorentzian metrics. Combined with prior results, we achieve a complete classification of such hypersurfaces on 4-dimensional nilpotent Lie groups. The core of our approach lies in the explicit derivation and solution of the Codazzi tensor equations, which directly leads to the construction of these hypersurfaces and provides their explicit parametrizations. Our main results establish the existence of Codazzi hypersurfaces on $Ni{l}^4$, demonstrate the non-existence of totally geodesic hypersurfaces, specify the algebraic condition for a Codazzi hypersurface to become parallel, and provide their explicit parametrizations. This observation highlights fundamental differences between Lorentzian and Riemannian settings within hypersurface theory. This work thus clarifies the distinct geometric properties inherent to the Lorentzian cases on nilpotent Lie groups. Full article

This paper tackles the ill-posed inversion of initial conditions and diffusion coefficient for Burgers’ equation with a source term. Using optimal control theory combined with a finite difference discretization scheme and a dual-functional descent method (DFDM), it sets the unknown boundary function $g\left(\tau \right)$ and diffusion coefficient u as control variables to build a multi-objective functional, proving the existence of the optimal solution via the variational method. Symmetry analysis reveals the intrinsic connection between the equation’s Lie group invariances and conservation laws through Noether’s theorem, providing a natural regularization framework for the inverse problem. Uniqueness and stability are demonstrated by the adjoint equation under cost function convexity. An energy-consistent discrete scheme is created to verify the energy conservation law while preserving the underlying symmetry structure. A comprehensive error analysis reveals dual error sources in inverse problems. A multi-scale adaptive inversion algorithm incorporating symmetry considerations achieves high-precision recovery under noise: boundary error $<1\%$, energy conservation error $\approx 0.13\%$. The symmetry-aware approach enhances algorithmic robustness and maintains physical consistency, with the solution showing linear robustness to noise perturbations. Full article

Introduction: Death anxiety is a salient psychological construct across the adult lifespan; however, few studies have examined the psychometric properties of the Spanish version of the Death Anxiety Scale (DAS) in university populations spanning diverse age ranges. Objectives: To evaluate the factorial structure, model fit, and reliability of the Spanish DAS in a heterogeneous academic cohort comprising traditional (younger) and non-traditional (older) adult learners. Methods: A total of 928 participants (aged 18–93 years) from a Spanish university completed the DAS. We conducted an exploratory factor analysis (EFA; principal axis factoring with oblique rotation) to identify latent dimensions, followed by a confirmatory factor analysis (CFA) to evaluate model fit. Internal consistency was assessed using Cronbach’s alpha and McDonald’s omega, and associations with sociodemographic variables (age, religious belief) were explored. Results: EFA supported a two-factor solution comprising Fear of Death and Peacefulness/Serenity towards Death. Factor reliability was acceptable (α = 0.818 and 0.734; total α = 0.789; ω_total ≈ 0.81). CFA indicated good fit to the two-factor model (χ²(89) = 401.19, RMSEA = 0.064, 90% CI [0.058–0.071], CFI = 0.940, TLI = 0.912, SRMR = 0.063), with information criteria (AIC = 17,018.33; BIC = 17,236.77) supporting model parsimony. Age and religious belief showed small-to-moderate associations with response patterns. Conclusions: The Spanish DAS demonstrates adequate factorial validity and reliability in a university sample spanning a wide age range. The identification of a Peacefulness/Serenity dimension may enrich interpretation, although its distinctiveness should be considered provisional and warrants replication. Future research should examine measurement invariance across age groups and assess applicability in clinical and longitudinal contexts. Full article

This paper studies a mixed PDE containing the second time derivative and a quadratic nonlinearity of the Monge–Ampère type in two spatial variables, which is encountered in geophysical fluid dynamics. The Lie group symmetry analysis of this highly nonlinear PDE is performed for the first time. An invariant point transformation is found that depends on fourteen arbitrary constants and preserves the form of the equation under consideration. One-dimensional symmetry reductions leading to self-similar and some other invariant solutions that described by single ODEs are considered. Using the methods of generalized and functional separation of variables, as well as the principle of structural analogy of solutions, a large number of new non-invariant closed-form solutions are obtained. In general, the extensive list of all exact solutions found includes more than thirty solutions that are expressed in terms of elementary functions. Most of the obtained solutions contain a number of arbitrary constants, and several solutions additionally include two arbitrary functions. Two-dimensional reductions are considered that reduce the original PDE in three independent variables to a single simpler PDE in two independent variables (including linear wave equations, the Laplace equation, the Tricomi equation, and the Guderley equation) or to a system of such PDEs. A number of specific examples demonstrate that the type of the mixed, highly nonlinear PDE under consideration, depending on the choice of its specific solutions, can be either hyperbolic or elliptic. To analyze the equation and construct exact solutions and reductions, in addition to Cartesian coordinates, polar, generalized polar, and special Lorentz coordinates are also used. In conclusion, possible promising directions for further research of the highly nonlinear PDE under consideration and related PDEs are formulated. It should be noted that the described symmetries, transformations, reductions, and solutions can be utilized to determine the error and estimate the limits of applicability of numerical and approximate analytical methods for solving complex problems of mathematical physics with highly nonlinear PDEs. Full article

We study the existence of Killing vector fields for right-invariant metrics on low-dimensional Lie groups. Specifically, Lie groups of dimension two, three and four are considered. Before attempting to implement the differential conditions that comprise Killing’s equations, the metric is reduced as much as possible by using the automorphism group of the Lie algebra. After revisiting the classification of the low-dimensional Lie algebras, we review some of the known results about Killing vector fields on Lie groups and add some new observations. Then we investigate indecomposable Lie algebras and attempt to solve Killing’s equations for each reduced metric. We introduce a matrix $MM$, that results from the integrability conditions of Killing’s equations. For $n=4$, the matrix $MM$ is of size $20\times 6$. In the case where $MM$ has maximal rank, for the Lie group problem considered in this article, only the left-invariant vector fields are Killing. The solution of Killing’s equations is performed by using MAPLE, and knowledge of the rank of $MM$ can help to confirm that the solutions found by MAPLE are the only linearly independent solutions. After finding a maximal set of linearly independent solutions, the Lie algebra that they generate is identified to one in a standard list. Full article

This work investigates the stability analysis of linear control systems defined on Lie groups, with a particular focus on low-dimensional cases. Unlike their Euclidean counterparts, such systems evolve on manifolds with non-Euclidean geometry, where trajectories respect the group’s intrinsic symmetries. Stability notions, such as inner asymptotic, inner, and input–output (BIBO) stability, are studied. The qualitative behavior of solutions is shown to depend critically on the spectral decomposition of derivations associated with the drift, and on the algebraic structure of the underlying Lie algebra. We study two classes of examples in detail: Abelian and solvable two-dimensional Lie groups, and the three-dimensional nilpotent Heisenberg group. These settings, while mathematically tractable, retain essential features of non-commutativity, geometric non-linearity, and sub-Riemannian geometry, making them canonical models in control theory. The results highlight the interplay between algebraic properties, invariant submanifolds, and trajectory behavior, offering insights applicable to robotic motion planning, quantum control, and signal processing. Full article

The classification of exact solutions of Maxwell vacuum equations for pseudo-Riemannian spaces with spatial symmetry (homogeneous non-null spaces in Petrov) in the presence of electromagnetic fields invariant with respect to the action of the group of space motions is summarized. A new classification method is used, common to all homogeneous zero spaces of Petrov. The method is based on the use of canonical reper vectors and on the use of a new approach to the systematization of solutions. The classification results are presented in a form more convenient for further use. Using the previously made refinement of the classification of Petrov spaces, the classification of exact solutions of Maxwell vacuum equations for spaces with the group of motions $G_3\left(VIII\right)$ is completed. Full article

The successful integration of Terrestrial and Non-Terrestrial Networks (T/NTNs) in 6G is poised to revolutionize demanding domains like Earth Observation (EO) and Intelligent Transportation Systems (ITSs). Still, it requires Distributed Machine Learning (DML) frameworks that are scalable, private, and efficient. Existing methods, such as Federated Learning (FL) and Split Learning (SL), face critical limitations in terms of client computation burden and latency. To address these challenges, this paper proposes a novel hierarchical DML paradigm. We first introduce Federated Split Transfer Learning (FSTL), a foundational framework that synergizes FL, SL, and Transfer Learning (TL) to enable efficient, privacy-preserving learning within a single client group. We then extend this concept to the Generalized FSTL (GFSTL) framework, a scalable, multi-group architecture designed for complex and large-scale networks. GFSTL orchestrates parallel training across multiple client groups managed by intermediate servers (RSUs/HAPs) and aggregates them at a higher-level central server, significantly enhancing performance. We apply this framework to a unified T/NTN architecture that seamlessly integrates vehicular, aerial, and satellite assets, enabling advanced applications in 6G ITS and EO. Comprehensive simulations using the YOLOv5 model on the Cityscapes dataset validate our approach. The results show that GFSTL not only achieves faster convergence and higher detection accuracy but also substantially reduces communication overhead compared to baseline FL, and critically, both detection accuracy and end-to-end latency remain essentially invariant as the number of participating users grows, making GFSTL especially well suited for large-scale heterogeneous 6G ITS deployments. We also provide a formal latency decomposition and analysis that explains this scaling behavior. This work establishes GFSTL as a robust and practical solution for enabling the intelligent, connected, and resilient ecosystems required for next-generation transportation and environmental monitoring. 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