Search Results (68)

Numerous systems in electrical engineering, biology, and mechanical structures can be modeled using dynamical systems theory. This paper examines the behavior of a 3D dynamical system with six parameters, specifically its damped or periodic oscillations and asymptotic properties as functions of six physical parameters. The system is integrated explicitly through a smooth solution of a third order nonlinear differential equation, yielding exact parametric expressions that describe a heteroclinic orbit. To analyze parameter influence, we apply the Multistage Optimal Homotopy Perturbation Method (MOHPM). Its main advantage is the small number of iterations required, due to the effective choice of auxiliary convergence control functions. The MOHPM solutions agree closely with numerical results, demonstrated qualitatively through figures and quantitatively through tables. Accuracy is further assessed by comparison with the Optimal Homotopy Perturbation Method (OHPM). A qualitative analysis of errors is also provided. Full article

The present study deals with the nonlinear forced vibration of an axially functionally graded beam subjected to an electromagnetic actuator, moving load, and Casimir force, considering the curvature of the beam and it resting on a nonlinear elastic Winkler–Pasternak foundation. The presence of an electromagnetic actuator and Casimir force besides the presence of mechanical impact (moving load) and nonlinear elastic foundation is a characteristic of a real system, but this has not been studied in this form until now, currently representing a remaining gap. The governing differential equations of motion in the considered system are based on Euler–Bernoulli beam theory and von Kármán geometric nonlinearity. The material properties are expressed according to a power law function through the thickness direction. We point out that the present study is the first to consider the curvature in combination with electromagnetic actuation, Casimir force, an elastic foundation, and moving load. Unlike in other works, axial inertia is not assumed to be negligible in our investigation. The Optimal Homotopy Asymptotic Method is employed to obtain an approximate analytical expression for the nonlinear dynamic response and the nonlinear frequency. The solutions obtained are very accurate in comparison with numerical solutions, and our procedure is simple and easy to implement for nonlinear problems. The local stability near the primary resonance and internal resonance is analyzed by means of the variable expansion method, the homotopy perturbation method, equilibrium points, the Jacobian matrix, and the Routh–Hurwitz criterion. Full article

The responses of a very large floating structure (VLFS), which is modeled as a thin viscoelastic plate floating on a fluid of finite depth, are analytically studied within the framework of the nonlinear potential flow theory. We use the Laplace equation with the dynamical boundary condition to express a balance among the hydrodynamic, inertial, and viscoelastic forces. For the case of steady-state incident waves, we obtain convergent series solutions for plate deflection and velocity potential by choosing the optimal convergence-control parameter $C_0$ and proper auxiliary linear operators in the homotopy analysis method (HAM). The strain relaxation time for the viscoelastic plate is studied, and the result shows that the plate deflection decreases when the retardation time increases. The influences of other physical parameters on the viscoelastic plate are also discussed. The nonlinearity of dispersion relation and the retardation time of the plate have important and non-negligible effects on the responses of the VLFS. The results obtained here may be helpful in understanding the different physical parameters to model hydroelastic responses of a VLFS in the real ocean. Full article

In this paper, we study the homotopy theory of single intersection graphs arising from acting spaces over topological semigroups. An acting space $(\mathcal{S},\mathcal{B})$ is defined as a topological space $\mathcal{B}$ equipped with a continuous action of a topological semigroup $\mathcal{S}$, generalizing the notion of algebraic actions in a topological setting. To connect this structure with graph theory, we associate to each acting space a single intersection graph ${\mathcal{G}}_{\mathcal{S}\mathcal{B}}$, whose vertices are proper $\mathcal{S}\mathcal{B}$-subacting spaces, and two vertices are adjacent if their intersection is a singleton set. This graph construction encodes both algebraic and topological interactions between subacting spaces and provides a framework to study connectivity and homotopical properties via combinatorial methods. We then work within a categorical framework, where objects are graphical acting semigroups and morphisms are $\mathcal{S}$-acting maps, allowing us to systematically study structural properties and their invariance under morphisms. In this setting, we introduce the notion of acting fibrations and formulate the corresponding lifting problem. Our main result establishes that an $\mathcal{S}$-acting map is an acting fibration if and only if it admits an $\mathcal{A}$-lifting function, providing a characterization analogous to classical fibration theory. Furthermore, we introduce $\mathcal{A}$-regular lifting functions and analyze their role in preserving homotopical structures, including a natural homotopy extension property. 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 $\Delta$-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

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

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 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

This paper develops extension theory for Brown functors in directed graph homotopy theory. We establish a systematic method for extending Brown functors from finite directed graphs to arbitrary directed graphs using inverse limits over finite subdigraphs. We prove that this extension is well-defined and preserves essential functorial properties. Additionally, we provide an alternative characterization of this extension through the Yoneda lemma, demonstrating how extended Brown functors can be naturally identified with sets of natural transformations from representable functors. This categorical perspective offers deeper theoretical insight into the structure of extended Brown functors and establishes important connections with classical representability theory, providing the technical foundation for Brown representability in directed graph theory. Full article

Let Y be a smooth compact n-manifold. We studied smooth embeddings and immersions $\beta :M\to \mathbb{R}\times Y$ of compact n-manifolds M such that $\beta \left(M\right)$ avoids some priory chosen closed poset $\mathrm{\Theta }$ of tangent patterns to the fibers of the obvious projection $\pi :\mathbb{R}\times Y\to Y$. Then, for a fixed Y, we introduced an equivalence relation between such $\beta$’s; creating a crossover between pseudo-isotopies and bordisms. We called this relation quasitopy. In the presented study of quasitopies, the spaces ${\mathcal{P}}_d^{\mathbf{c}\mathrm{\Theta }}$ of real univariate polynomials of degree d with real divisors, whose combinatorial patterns avoid a given closed poset $\mathrm{\Theta }$, play the classical role of Grassmanians. We computed the quasitopy classes ${\mathcal{Q}}_d^{\mathsf{emb}}(Y,\mathbf{c}\mathrm{\Theta })$ of $\mathrm{\Theta }$-constrained embeddings $\beta$ in terms of homotopy/homology theory of spaces Y and ${\mathcal{P}}_d^{\mathbf{c}\mathrm{\Theta }}$. We proved also that the quasitopies of embeddings stabilize, as $d\to \infty$. Full article

The conventional power distributions methods face many challenges, such as high switching frequency, multiple carrier cycles, and single power distribution. To address the above issues, a novel power distribution based on the selective harmonic elimination (PD–SHE) strategy is proposed to achieve arbitrary power distribution and selective harmonic elimination but with low switching frequency and single carrier cycle. Firstly, the novel PD–SHE model is established based on the principles of SHE and power calculation theory, where distribution ratio is introduced to adjust power distribution arbitrarily and its constraints have been deduced. In addition, the issue of redundancy of solution is also analyzed and solved by adding valid constraints. Finally, polynomial homotopy continuation (PHC) algorithm is applied to solve the novel PD–SHE model. Then, all the physically realizable solutions can be found without choosing the initial value in the full range of modulation index. The results of simulation analysis show the effectiveness of PD–SHE strategy and the superiority of PHC algorithm in solving the global optimal solution. Moreover, the reliability of the global optimal solution for PD–SHE strategy is verified via physical experiments in terms of harmonic elimination and active power distribution. Full article

This work explores modern mathematical avenues as part of fractional calculus research. We apply fractional dispersion relations to the fractional wave equation to numerically examine various formulations of the generalized fractional wave equation. The research explores Drinfeld–Sokolov–Wilson and shallow water equations as fundamental differential equations forming the basis of wave theory studies. This work presents effective methods to obtain the numerical solution of the fractional-order FDSW and FSW coupled system equations. The analysis employs Caputo fractional derivatives during studies of fractional orders. This study develops the new iterative transform technique (NITM) and homotopy perturbation transform method (HPTM) using Elzaki transform (ET) with a new iteration method and a homotopy perturbation method. The proposed techniques generate approximation solutions that adopt an infinite fractional series with fractional order solutions converging towards analytic integer solutions. The proposed method demonstrates its precision through tabular simulations of computed approximations and their absolute error values while representing results with 2D and 3D graphics. The paper presents the physical analysis of solution dynamics across diverse $ϵ$ ranges during a suitable time frame. The developed computational techniques yield numerical and graphical output, which are compared to analytic results to verify the solution convergence. The computational algorithms have proven their high accuracy, flexibility, effectiveness, and simplicity in evaluating fractional-order mathematical models. Full article

Causal discovery involves searching intractably large spaces. Decomposing the search space into classes of observationally equivalent causal models is a well-studied avenue to making discovery tractable. This paper studies the topological structure underlying causal equivalence to develop a categorical formulation of Chickering’s transformational characterization of Bayesian networks. A homotopic generalization of the Meek–Chickering theorem on the connectivity structure within causal equivalence classes and a topological representation of Greedy Equivalence Search (GES) that moves from one equivalence class of models to the next are described. Specifically, this work defines causal models as propable symmetric monoidal categories (cPROPs), which define a functor category ${\mathcal{C}}^P$ from a coalgebraic PROP P to a symmetric monoidal category $\mathcal{C}$. Such functor categories were first studied by Fox, who showed that they define the right adjoint of the inclusion of Cartesian categories in the larger category of all symmetric monoidal categories. cPROPs are an algebraic theory in the sense of Lawvere. cPROPs are related to previous categorical causal models, such as Markov categories and affine CDU categories, which can be viewed as defined by cPROP maps specifying the semantics of comonoidal structures corresponding to the “copy-delete” mechanisms. This work characterizes Pearl’s structural causal models (SCMs) in terms of Cartesian cPROPs, where the morphisms that define the endogenous variables are purely deterministic. A higher algebraic K-theory of causality is developed by studying the classifying spaces of observationally equivalent causal cPROP models by constructing their simplicial realization through the nerve functor. It is shown that Meek–Chickering causal DAG equivalence generalizes to induce a homotopic equivalence across observationally equivalent cPROP functors. A homotopic generalization of the Meek–Chickering theorem is presented, where covered edge reversals connecting equivalent DAGs induce natural transformations between homotopically equivalent cPROP functors and correspond to an equivalence structure on the corresponding string diagrams. The Grothendieck group completion of cPROP causal models is defined using the Grayson–Quillen construction and relate the classifying space of cPROP causal equivalence classes to classifying spaces of an induced groupoid. A real-world domain modeling genetic mutations in cancer is used to illustrate the framework in this paper. Full article

In this work, we first develop the modified time Caputo fractional Kawahara Equations (MTCFKEs) in the usual Hilbert spaces and extend them to analogous structures within the theory of Hilbert algebras. Next, we employ the residual power series method, combined with the Laplace transform, to introduce a new effective technique called the Laplace Residual Power Series Method (LRPSM). This method is applied to derive the coefficients of the series solution for MTCFKEs in the context of Hilbert algebras. In real Hilbert algebras, we obtain approximate solutions for MTCFKEs under both exact and approximate initial conditions. We present both graphical and numerical results of the approximate analytical solutions to demonstrate the capability, efficiency, and reliability of the LRPSM. Furthermore, we compare our results with solutions obtained using the homotopy analysis method and the natural transform decomposition method. Full article