Search Results (498)

In recent years, animation-based systems for human-computer interaction have attracted increasing attention. This work proposes a hybrid framework that combines mathematical modeling and bio-inspired optimization algorithms to generate motion sequences from skeletal data. The framework takes as input a complete skeletal sequence corresponding to a given action and optimizes both the number of key poses and the parameters of a homotopy-based formulation to generate transitions between consecutive poses. A homotopy-based approach is used to compute transitions between selected key poses. The homotopy parameter $\lambda$ serves as an indicator of the completeness of the transition between pairs of key poses. Four nature-inspired optimization algorithms: Genetic Algorithm, Micro Genetic Algorithm, Particle Swarm Optimization, and Ant Colony Optimization were evaluated to determine the number of key poses and homotopy parameters that enable feasible motion generation. Dynamic Time Warping (DTW) is used as an external metric to assess the similarity between generated and reference sequences. It is important to note that Dynamic Time Warping (DTW) should be considered as a sequence similarity measure, as it does not explicitly evaluate perceptual realism or biomechanical plausibility. The framework was evaluated on 18 action sequences, demonstrating its ability to generate feasible motion transitions in 16 of the 18 evaluated actions when using PSO and MicroGA. For each pair of key poses, a fixed number of intermediate frames is generated to provide a uniform temporal discretization of the motion. The results suggest that homotopy-based methods provide a feasible approach for animation-based interaction systems. Full article

In this article, we study the homotopy aspects of intersection graphs of topological semigroups. We begin by defining the top intersection graph $T{G}_{\mathbb{X}}$ and investigating how the algebraic and topological properties of a topological semigroup are reflected in the global structure of this graph. In particular, we characterize when $T{G}_{\mathbb{X}}$ is totally disconnected, bipartite, or planar in terms of the order and factorization of the underlying semigroup. We then introduce the notions of $HTG$-semigroups, graphical homomorphisms, and graphical homotopy relations, thereby developing a graphical homotopy framework. Within this setting, we study $Gr$-homotopy equivalences, $Gr$-contractible spaces, and retraction phenomena, including $DGr$-retracts and homotopy extension properties. Finally, we introduce graphical total semigroups and equip the set of $Gr$-path homotopy classes $\left[{\mathbb{X}}_{pe}\right]$ with a natural Δ-uniform topology. We show that this topology is compatible with the induced semigroup operation, yielding a topological semigroup structure. Overall, this work provides a unified algebraic, topological, and graph-theoretic perspective, and opens the door to further applications of homotopy theory in the study of intersection graphs of topological semigroups. Full article

Electromagnetic hydrodynamic (EMHD) mixed convective flow of tetra hybrid nanofluid (TeHNF) in a Darcy-Forchheimer porous medium in a vertical channel with thermal radiation is considered in the paper. The electric and magnetic fields are homogeneous, magnetic perpendicular to the walls of the channel, and electric perpendicular to the plane formed by the directions of the magnetic field and the basic current. The channel walls are impermeable, and they are at constant but different temperatures. The basic equations that describe this problem are ordinary nonlinear differential equations (ODEs), and they are transformed into dimensionless ODEs by introducing dimensionless quantities, which are analytically solved using the homotopy perturbation method (HPM). The relations for velocity and temperature distributions, Nusselt numbers and shear stresses on the channel walls were determined. These relations are functions of introduced physical parameters that characterize the observed problem. For TeHNF, where the base fluid is water and the nanoparticles are made of aluminum oxide, titanium dioxide, magnesium oxide and magnetite, a part of the obtained results is given. Velocity and temperature plots are presented in the form of graphs, and Nusselt numbers and shear stresses are presented in the form of tables. Based on the analysis of the obtained results, appropriate conclusions were drawn. It was concluded that an increase in the Hartmann number as well as an increase in the porosity factor decrease the fluid velocity and shear stress, and increase the fluid temperature and Nusselt numbers. Higher values of the Forchheimer factor and higher heat radiation correspond to lower fluid velocities, lower temperatures, lower values of shear stresses and Nusselt numbers. By increasing the value of the Grashof number, the velocity of the fluid increases, and so do the shear stresses. TeHNF shows advantages over simpler hybrid nanofluids and commercial fluids. Full article

This article proposes a novel approach for dealing with the time-fractional Navier–Stokes equations via the natural generalized Laplace transform decomposition method (NGLTDM). This hybrid method utilizes both the natural generalized Laplace transform (NGLT) and a decomposition method. The method is correct because the series solutions become more accurate when more terms are added. We establish precise theorems that verify the existence of solutions and the convergence of the series. The analysis shows that the suggested method is more general than the Homotopy Perturbation Method (HPM) and the Adomian Decomposition Method (ADM). Also, this approach can be applied to handle difficult fluid dynamics problems governed by the Navier–Stokes equations. This study enhances analytical methodologies for fractional-order flow models. Full article

In this work, we construct a homotopy theory for a class of intersection graphs arising from topological monoids. We introduce the M-intersection graph of a ${\tau }_e$-monoid, where the vertices correspond to proper ${\tau }_e$-submonoids and adjacency is defined by trivial intersection. Several structural properties of the graph, including total disconnectedness, bipartiteness and planarity, are investigated and shown to be closely related to the algebraic structure and decomposition of finite ${\tau }_e$-monoids. Based on this framework, we develop a graphical homotopy theory by introducing graphical ${\tau }_e$-monoids, graphical homomorphisms, and graphical homotopies. We study graphical homotopy equivalence, graphical contractibility, and path monoids, and examine retraction properties through graphical retracts, D-graphical retracts and graphical homotopy extension properties. Furthermore, we present an example of graphical comprehensive monoids and construct a $\theta$-homogeneous topology on the set of graphical path homotopy classes. We show that this topology is compatible with the induced monoid operation, yielding a well-behaved functorial topological monoid structure. Full article

Oscillatory motions in stratified atmospheric fluid layers significantly influence weather and climate dynamics. Shallow-water equations effectively describe these motions. This study extends the shallow-water model to the time-fractional domain using the conformable fractional derivative. This derivative preserves the local differential structure while introducing tunable time scaling in the dynamics. Approximate analytical solutions were obtained using the conformable Laplace Adomian Decomposition Method (CLADM). This method combines the conformable Laplace transform with Adomian decomposition. Numerical results for fractional orders $\vartheta \in (0, 1]$ demonstrate that the fractional parameter systematically modulates the system dynamics. The solutions at $\vartheta =1$ align well with the established Elzaki Adomian Decomposition Method (EADM), Homotopy Analysis Method (HAM), Fractional Reduced Differential Transform Method (FRDTM), and reference numerical solutions (NUM). This fractional framework offers a flexible approach to modeling atmospheric fluid-layer dynamics. Full article

This study presents a systematic comparative benchmark between two distinct paradigms for solving nonlinear partial differential equations (PDEs): the data-driven Physics-Informed Neural Networks (PINNs) and the analytical Homotopy Analysis Method (HAM). We apply both methods to a unified family of canonical PDEs, the Burgers, Huxley, Fisher, Burgers–Huxley, and Burgers–Fisher equations, under identical problem setups, domain discretization, and validation metrics. PINNs incorporate physical laws directly into neural network training by minimizing a loss function that enforces PDE residuals, yielding physically consistent solutions even for strongly nonlinear problems. HAM provides approximate analytical solutions using a unified framework, and the same initial guess, auxiliary linear operator, and auxiliary function across all equations despite their distinct nonlinearities. The controlled, consistent application of both methods enables a fair, reproducible comparison across this equation family. The results provide a quantitative performance map under identical conditions, delineating when PINNs (high accuracy, long-term stability, and generalization capability) are preferable, versus when HAM (computational speed, short-term analytic approximation, and lower memory footprint) offers advantages. While the finite radius of convergence of the truncated HAM series is theoretically expected, our controlled comparison quantifies for the first time how this degradation varies across equation types, revealing that the choice between methods depends on specific problem requirements including error tolerance, available computational resources, and temporal horizon. The novelty lies not in solving each equation individually, but in deriving a performance taxonomy that systematically connects equation features (shocks, stiffness, and reaction–diffusion coupling) to optimal solver choice—providing previously unavailable, evidence-based guidance for the scientific computing community. This study establishes the first rigorous, controlled comparative benchmark between analytic and data-driven PDE solvers across a spectrum of nonlinearities, providing a reproducible baseline for future hybrid scientific machine learning solvers. Full article

This paper presents a systematic application of the Homotopy Perturbation Method (HPM) to the nonlinear static analysis of cantilever beams subjected simultaneously to three coplanar follower loads: an axial force H, a transverse force V, and a bending moment M₁. The studied configuration introduces complex mathematical self-coupling, as the bending moment depends on the solution of the differential equation even in its boundary conditions (γ₁), transforming the problem into a nonlinear one that is resistant to standard analytical methods. The primary methodological contribution of this work is the successful extension of the HPM framework to treat, within a unified mathematical formalism, this complete loading case, which has practical applications in compliant mechanisms, micro-electromechanical systems (MEMSs), and auxetic structures. The paper provides a complete mathematical formulation and explicit derivation of the HPM solution terms up to the third order and a rigorous demonstration of the method’s convergence, with quantitative error estimates and the establishment of a practical domain of validity, γ₁ < 30°, for an accuracy below 0.5%. As a direct consequence of this analytical advancement, we derive a series of practical engineering tools: nomograms, simplified empirical formulas, interaction diagrams, and a systematic six-step design procedure, which includes an adaptive algorithm for selecting the auxiliary parameter η to optimize convergence. The solution’s structure also lends itself to AI-based optimization frameworks, demonstrating how HPM solutions can serve as a foundation for machine learning surrogates and automated multi-objective optimizations. HPM proves to be a robust and efficient alternative, providing semi-analytical solutions in the form of convergent series without requiring an explicitly small physical parameter. This enables a direct parametric understanding of the structural response and offers rapid tools for the conceptual and preliminary sizing phases, thereby complementing the intensive numerical methods used in the final design stages. Full article

In this study, we propose a homotopy-inspired prompt obfuscation framework to enhance understanding of security and safety vulnerabilities in Large Language Models (LLMs). By systematically applying carefully engineered prompts, we demonstrate how latent model behaviors can be influenced in unexpected ways. Our experiments encompassed 15,732 prompts, including 10,000 high-priority cases, across LLama, Deepseek, KIMI for code generation, and Claude to verify. The results reveal critical insights into current LLM safeguards, highlighting the need for more robust defense mechanisms, reliable detection strategies, and improved resilience. Importantly, this work provides a principled framework for analyzing and mitigating potential weaknesses, with the goal of advancing safe, responsible, and trustworthy AI technologies. Full article

This paper investigates the optimal control problem of orbital rendezvous for spacecraft in near-circular orbits with a low-thrust propulsion system. Two optimality criteria are considered: time-optimal and motor-time-optimal control. A linearized mathematical model of relative motion between the active and passive spacecraft is employed, which is formulated in dimensionless variables that characterize secular, periodic, and lateral motion components of the relative motion. By applying Pontryagin’s Maximum Principle, the equations governing the optimal relative motion of the spacecraft are derived. To address the discontinuities associated with the bang–bang switching function inherent in the motor-time-optimal problem, and the lack of a suitable initial guess, a homotopy method is adopted, in which the solution to the rendezvous time-optimal problem is used as an initial guess and is gradually deformed into the motor-time-optimal control. Considering the errors introduced by the linearization of the relative motion model, the obtained control law is validated via numerical simulations based on the original nonlinear dynamics of the system. Simulation results demonstrate that the proposed trajectory optimization methodology achieves high success rates and rapid convergence, providing valuable theoretical support and practical guidance for mission scenarios with similar trajectory design requirements. Full article

The dynamic model and nonlinear forced vibration of a laminated beam with magnetostrictive layers, embedded on a nonlinear elastic Winkler–Pasternak foundation, in the presence of an electromagnetic actuator, mechanical impact, dry friction, a longitudinal magnetic field, and van der Waals force is investigated in the present work. The dynamic equations of this complex system are established based on von Karman theory and Hamilton’s principle. Then, by means of the Galerkin–Bubnov procedure, the partial differential equations are transformed into ordinary differential equations. The Optimal Auxiliary Functions Method (OAFM) is applied to solve the nonlinear differential equation. The results obtained are validated by comparisons with numerical results given by the Runge–Kutta procedure. Local stability in the neighborhood of the primary resonance is examined by means of the homotopy perturbation method, the Jacobian matrix, and the Routh–Hurwitz criteria. Global stability is studied by introducing the control law input function and using the approximate solution obtained by the OAFM in the construction of the Lyapunov function. La Salle’s invariance principle and Potryagin’s principle complete our study. The effects of some parameters are graphically presented. Our paper reveals the immense potential of the OAFM in the study of complex nonlinear dynamical systems. Full article

The spread of computer viruses poses a critical threat to networked systems and requires accurate modeling tools. Classical integer-order approaches had failed to capture memory effects inherent in real digital environments. To address this, we developed a four-compartment fractional-order model using the Atangana–Baleanu–Caputo (ABC) derivative with Mittag-Leffler kernels. We established fundamental properties such as positivity, boundedness, existence, uniqueness, and Hyers–Ulam stability. Analytical solutions were derived via Laplace transform and homotopy series, while the Variation-of-Parameters Method and a dedicated numerical scheme provided approximations. Simulation results showed that the fractional order strongly influenced infection dynamics: smaller orders delayed peaks, prolonged latency, and slowed recovery. Compared to classical models, the ABC framework captured realistic memory-dependent behavior, offering valuable insights for designing timely and effective cybersecurity interventions. The model exhibits structural symmetries: the infection flux depends on the symmetric combination $L+I$ and the feasible region (probability simplex) is invariant under the flow. Under the parameter constraint $\delta =\theta$ (and equal linear loss terms), the system is equivariant under the involution $(L,I)\mapsto (I,L)$, which is reflected in identical Hyers–Ulam stability bounds for the latent and infectious components. Full article

In the present study, we aimed to derive analytical solutions of the homotopy analysis method (HAM) for the time-fractional Navier–Stokes equations in cylindrical coordinates in the form of a rapidly convergent series. In this work, we explore the time-fractional Navier–Stokes equations by replacing the standard time derivative with the Katugampola fractional derivative, expressed in the Caputo form. The homotopy analysis method is then employed to obtain an analytical solution for this time-fractional problem. The convergence of the proposed method to the solution is demonstrated. To validate the method’s accuracy and effectiveness, two examples of time-fractional Navier–Stokes equations modeling fluid flow in a pipe are presented. A comparison with existing results from previous studies is also provided. This method can be used as an alternative to obtain analytic and approximate solutions of different types of fractional differential equations applied in engineering mathematics. Full article

The classical fiber product in algebraic geometry provides a powerful tool for studying loci where two morphisms to a base scheme, $\varphi :X\to S$ and $\psi :Y\to S$, coincide exactly. This condition of strict equality, however, is insufficient for describing many real-world applications, such as the geometric structure of semantic spaces in modern large language models whose foundational architecture is the Transformer neural network: The token spaces of these models are fundamentally approximate, and recent work has revealed complex geometric singularities, challenging the classical manifold hypothesis. This paper develops a new framework to study and quantify the nature of approximate alignment between morphisms in the context of arithmetic geometry, using the tools of étale homotopy theory. We introduce the central object of our work, the étale mismatch torsor, which is a sheaf of torsors over the product scheme $X{\times }_{S}Y$. The structure of this sheaf serves as a rich, intrinsic, and purely algebraic object amenable to both qualitative classification and quantitative analysis of the global relationship between the two morphisms. Our main results are twofold. First, we provide a complete classification of these structures, establishing a bijection between their isomorphism classes and the first étale cohomology group $H_{ét}^1(X{\times }_{S}Y,\underline{{\pi }_1^{ét}(S)})$. Second, we construct a canonical filtration on this classifying cohomology group based on the theory of infinitesimal neighborhoods. This filtration induces a new invariant, which we term the order of mismatch, providing a hierarchical, algebraic measure for the degree of approximation between the morphisms. We apply this framework to the concrete case of generalized Howe curves over finite fields, demonstrating how both the characteristic class and its order reveal subtle arithmetic properties. Full article

The primary objective of this study is to provide a more precise and beneficial mathematical model for assessing corruption dynamics by utilizing non-local derivatives. This research aims to provide solutions that accurately capture the complexities and practical behaviors of corruption. To illustrate how corruption levels within a community change over time, a non-linear deterministic mathematical model has been developed. The authors present a non-integer order model that divides the population into five subgroups: susceptible, exposed, corrupted, recovered, and honest individuals. To study these corruption dynamics, we employ a new method for solving a time-fractional corruption model, which we term the q-homotopy analysis transform approach. This approach produces an effective approximation solution for the investigated equations, and data is shown as 3D plots and graphs, which give a clear physical representation. The stability and existence of the equilibrium points in the considered model are mathematically proven, and we examine the stability of the model and the equilibrium points, clarifying the conditions required for a stable solution. The resulting solutions, given in series form, show rapid convergence and accurately describe the model’s behaviour with minimal error. Furthermore, the solution’s uniqueness and convergence have been demonstrated using fixed-point theory. The proposed technique is better than a numerical approach, as it does not require much computational work, with minimal time consumed, and it removes the requirement for linearization, perturbations, and discretization. In comparison to previous approaches, the proposed technique is a competent tool for examining an analytical outcomes from the projected model, and the methodology used herein for the considered model is proved to be both efficient and reliable, indicating substantial progress in the field. Full article