Search Results (18)

This work reveals an advanced numerical scheme for obtaining approximate solutions to nonlinear fractional Kuramoto–Sivashinsky (K-S) equations involving Caputo derivatives. We introduce the Sumudu transform (ST), which converts the fractional derivatives into their classical counterparts to produce a nonlinear recurrence equation. By using the homotopy perturbation method (HPM), we construct a homotopy with an embedding parameter to solve this recurrence relation. Our proposed technique is known as the Sumudu homotopy transform method (SHTM), which delivers results after fewer iterations and achieves precise outcomes with minimal computational effort. The proposed technique effectively eliminates the necessity for complex discretization or linearization, making it highly suitable for nonlinear problems. We showcase two numerical cases, along with two- and three-dimensional visualizations, to validate the accuracy and effectiveness of this technique. It also produces rapidly converging series solutions that closely align with the precise results. 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

This essay’s title is justified by discussing a class of Yang–Mills-type theories of which standard Yang–Mills theories are special cases but which is broad enough to include gravity as a double field theory. We use the framework of homotopy algebras, where conventional Yang–Mills theory is the tensor product $\mathcal{K}\otimes \mathfrak{g}$ of a ‘kinematic’ algebra $\mathcal{K}$ with a color Lie algebra $\mathfrak{g}$. The larger class of Yang–Mills-type theories are given by the tensor product of $\mathcal{K}$ with more general Lie-type algebras, of which $\mathcal{K}$ itself is an example, up to anomalies that can be canceled for the tensor product with a second copy $\overline{\mathcal{K}}$. Gravity is then given by $\mathcal{K}\otimes \overline{\mathcal{K}}$. Full article

In this paper, we investigate properties concerning some recently introduced finite coarse shape invariants—the k-th finite coarse shape group of a pointed topological space and the k-th relative finite coarse shape group of a pointed topological pair. We define the notion of finite coarse shape group sequence of a pointed topological pair $\left(X,X_0,x_0\right)$ as an analogue of homotopy and (coarse) shape group sequences and show that for any pointed topological pair, the corresponding finite coarse shape group sequence is a chain. On the other hand, we construct an example of a pointed pair of metric continua whose finite coarse shape group sequence fails to be exact. Finally, using the aforementioned pair of metric continua together with a pointed dyadic solenoid, we show that finite coarse-shape groups, in general, differ from both shape and coarse-shape groups. Full article

Theoretical influence of the buoyancy and thermal radiation effects on the MHD (magnetohydrodynamics) flow across a stretchable porous sheet were analyzed in the present study. The Darcy–Forchheimer model and laminar flow were considered for the flow problem that was investigated. The flow was taken to incorporate a temperature-dependent heat source or sink. The study also incorporated the influences of Brownian motion and thermophoresis. The general form of the buoyancy term in the momentum equation for a free convection boundary layer is derived in this study. A favorable comparison with earlier published studies was achieved. Graphs were used to investigate and explain how different physical parameters affect the velocity, the temperature, and the concentration field. Additionally, tables are included in order to discuss the outcomes of the Sherwood number, the Nusselt number, and skin friction. The fundamental governing partial differential equations (PDEs), which are used in the modeling and analysis of the MHD flow problem, were transformed into a collection of ordinary differential equations (ODEs) by utilizing the similarity transformation. A semi-analytical approach homotopy analysis method (HAM) was applied for approximating the solutions of the modeled equations. The model finds several important applications, such as steel rolling, nuclear explosions, cooling of transmission lines, heating of the room by the use of a radiator, cooling the reactor core in nuclear power plants, design of fins, solar power technology, combustion chambers, astrophysical flow, electric transformers, and rectifiers. Among the various outcomes of the study, it was discovered that skin friction surges for 0.3 $\le F_1\le$ 0.6, 0.1 $\le k_1\le$ 0.4 and 0.3 $\le M\le$ 1.0, snf declines for 1.0 $\le G_r\le$ 4.0. Moreover, the Nusselt number augments for 0.5 $\le R\le$ 1.5, 0.2 $\le N_t\le$ 0.8 and 0.3 $\le N_b\le$ 0.9, and declines for 2.5 $\le Pr\le$ 5.5. The Sherwood number increases for 0.2 $\le N_t\le$ 0.8 and 0.3 $\le Sc\le$ 0.9, and decreases for 0.1 $\le N_b\le$ 0.7. Full article

In this article, the authors study the comparison of the generalization differential transform method (DTM) and fuzzy variational iteration method (VIM) applied to determining the approximate analytic solutions of fuzzy fractional KdV, K(2,2) and mKdV equations. Furthermore, we establish the approximation solution two-and three-dimensional fuzzy time-fractional telegraphic equations via the fuzzy reduced differential transform method (RDTM). Finding an exact or closed-approximation solution to a differential equation is possible via fuzzy RDTM. Finally, we present the fuzzy fractional variational homotopy perturbation iteration method (VHPIM) with a modified Riemann-Liouville derivative to solve the fuzzy fractional diffusion equation (FDE). Using this approach achieves a rapidly convergent sequence that approaches the exact solution of the equation. The proposed methods are investigated based on fuzzy fractional derivatives with some illustrative examples. The results reveal that the schemes are highly effective for obtaining the solutions to fuzzy fractional partial differential equations. Full article

Automated segmentation of brain tumors is a difficult procedure due to the variability and blurred boundary of the lesions. In this study, we propose an automated model based on Bendlet transform and improved Chan-Vese (CV) model for brain tumor segmentation. Since the Bendlet system is based on the principle of sparse approximation, Bendlet transform is applied to describe the images and map images to the feature space and, thereby, first obtain the feature set. This can help in effectively exploring the mapping relationship between brain lesions and normal tissues, and achieving multi-scale and multi-directional registration. Secondly, the SSIM region detection method is proposed to preliminarily locate the tumor region from three aspects of brightness, structure, and contrast. Finally, the CV model is solved by the Hermite-Shannon-Cosine wavelet homotopy method, and the boundary of the tumor region is more accurately delineated by the wavelet transform coefficient. We randomly selected some cross-sectional images to verify the effectiveness of the proposed algorithm and compared with CV, Ostu, K-FCM, and region growing segmentation methods. The experimental results showed that the proposed algorithm had higher segmentation accuracy and better stability. Full article

Achieving the smart motion of any autonomous or semi-autonomous robot requires an efficient algorithm to determine a feasible collision-free path. In this paper, a novel collision-free path homotopy-based path-planning algorithm applied to planar robotic arms is presented. The algorithm utilizes homotopy continuation methods (HCMs) to solve the non-linear algebraic equations system (NAES) that models the robot’s workspace. The method was validated with three case studies with robotic arms in different configurations. For the first case, a robot arm with three links must enter a narrow corridor with two obstacles. For the second case, a six-link robot arm with a gripper is required to take an object inside a narrow corridor with two obstacles. For the third case, a twenty-link arm must take an object inside a maze-like environment. These case studies validated, by simulation, the versatility and capacity of the proposed path-planning algorithm. The results show that the CPU time is dozens of milliseconds with a memory consumption less than 4.5 kB for the first two cases. For the third case, the CPU time is around 2.7 s and the memory consumption around 18 kB. Finally, the method’s performance was further validated using the industrial robot arm CRS CataLyst-5 by Thermo Electron. Full article

The researchers aimed to study the nonlinear fractional order model of malaria infection based on the Caputo-Fabrizio fractional derivative. The homotopy analysis transform method (HATM) is applied based on the floating-point arithmetic (FPA) and the discrete stochastic arithmetic (DSA). In the FPA, to show the accuracy of the method we use the absolute error which depends on the exact solution and a positive value $\epsilon$. Because in real life problems we do not have the exact solution and the optimal value of $\epsilon$, we need to introduce a new condition and arithmetic to show the efficiency of the method. Thus the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are applied. The CESTAC method is based on the DSA. Also, a new termination criterion is used which is based on two successive approximations. Using the CESTAC method we can find the optimal approximation, the optimal error and the optimal iteration of the method. The main theorem of the CESTAC method is proved to show that the number of common significant digits (NCSDs) between two successive approximations are almost equal to the NCSDs of the exact and approximate solutions. Plotting several graphs, the regions of convergence are demonstrated for different number of iterations k = 5, 10. The numerical results based on the simulated data show the advantages of the DSA in comparison with the FPA. Full article

This paper is devoted to establishing the most refined axiom for a digital covering space which remains open. The crucial step in making our approach is to simplify the notions of several types of earlier versions of local $(k_0,k_1)$-isomorphisms and use the most simplified local $(k_0,k_1)$-isomorphism. This approach is indeed a key step to make the axioms for a digital covering space very refined. In this paper, the most refined local $(k_0,k_1)$-isomorphism is proved to be a $(k_0,k_1)$-covering map, which implies that the earlier axioms for a digital covering space are significantly simplified with one axiom. This finding facilitates the calculations of digital fundamental groups of digital images using the unique lifting property and the homotopy lifting theorem. In addition, consider a simple closed $k:=k(t,n)$-curve with five elements in ${\mathbb{Z}}^n$, denoted by $S{C}_k^{n,5}$. After introducing the notion of digital topological imbedding, we investigate some properties of $S{C}_k^{n,5}$, where $k:=k(t,n),3\le t\le n$. Since $S{C}_k^{n,5}$ is the minimal and simple closed k-curve with odd elements in ${\mathbb{Z}}^n$ which is not k-contractible, we strongly study some properties of it associated with generalized digital wedges from the viewpoint of fixed point theory. Finally, after introducing the notion of generalized digital wedge, we further address some issues which remain open. The present paper only deals with k-connected digital images. Full article

The present paper investigates digital topological properties of an alignment of fixed point sets which can play an important role in fixed point theory from the viewpoints of computational or digital topology. In digital topology-based fixed point theory, for a digital image $(X,k)$ , let $F\left(X\right)$ be the set of cardinalities of the fixed point sets of all k-continuous self-maps of $(X,k)$ (see Definition 4). In this paper we call it an alignment of fixed point sets of $(X,k)$ . Then we have the following unsolved problem. How many components are there in $F\left(X\right)$ up to 2-connectedness? In particular, let $C_k^{n,l}$ be a simple closed k-curve with l elements in ${\mathbb{Z}}^n$ and $X:=C_k^{n,l_1}\vee C_k^{n,l_2}$ be a digital wedge of $C_k^{n,l_1}$ and $C_k^{n,l_2}$ in ${\mathbb{Z}}^n$ . Then we need to explore both the number of components of $F\left(X\right)$ up to digital 2-connectivity (see Definition 4) and perfectness of $F\left(X\right)$ (see Definition 5). The present paper addresses these issues and, furthermore, solves several problems related to the main issues. Indeed, it turns out that the three models $C_{2n}^{n,4}$ , $C_{3^n-1}^{n,4}$ , and $C_k^{n,6}$ play important roles in studying these topics because the digital fundamental groups of them have strong relationships with alignments of fixed point sets of them. Moreover, we correct some errors stated by Boxer et al. in their recent work and improve them (see Remark 3). This approach can facilitate the studies of pure and applied topologies, digital geometry, mathematical morphology, and image processing and image classification in computer science. The present paper only deals with k-connected spaces in DTC. Moreover, we will mainly deal with a set X such that $X^{♯}\ge 2$ . Full article

Any nilpotent CW-space can be localized at primes in a similar way to the localization of a ring at a prime number. For a collection $\mathcal{P}$ of prime numbers which may be empty and a localization $X_{\mathcal{P}}$ of a nilpotent CW-space X at $\mathcal{P}$ , we let $|C(X)|$ and $|C(X_{\mathcal{P}})|$ be the cardinalities of the sets of all homotopy comultiplications on X and $X_{\mathcal{P}}$ , respectively. In this paper, we show that if $|C(X)|$ is finite, then $|C\left(X\right)|\ge |C\left(X_{\mathcal{P}}\right)|$ , and if $|C(X)|$ is infinite, then $|C\left(X\right)|=|C\left(X_{\mathcal{P}}\right)|$ , where X is the k-fold wedge sum ${\bigvee }_{i=1}^k{\mathbb{S}}^{n_i}$ or Moore spaces $M(G,n)$ . Moreover, we provide examples to concretely determine the cardinality of homotopy comultiplications on the k-fold wedge sum of spheres, Moore spaces, and their localizations. Full article

Given a Khalimsky (for short, K-) topological space X, the present paper examines if there are some relationships between the contractibility of X and the existence of the fixed point property of X. Based on a K-homotopy for K-topological spaces, we firstly prove that a K-homeomorphism preserves a K-homotopy between two K-continuous maps. Thus, we obtain that a K-homeomorphism preserves K-contractibility. Besides, the present paper proves that every simple closed K-curve in the n-dimensional K-topological space, $S{C}_K^{n,l},n\ge 2,l\ge 4$ , is not K-contractible. This feature plays an important role in fixed point theory for K-topological spaces. In addition, given a K-topological space X, after developing the notion of K-contractibility relative to each singleton $\left\{x\right\}(\subset X)$ , we firstly compare it with the concept of K-contractibility of X. Finally, we prove that the K-contractibility does not imply the K-contractibility relative to each singleton $\left\{x_0\right\}(\subset X)$ . Furthermore, we deal with certain conjectures involving the (almost) fixed point property in the categories KTC and KAC, where KTC (see Section 3) (resp. KAC (see Section 5)) denotes the category of K-topological (resp. KA-) spaces, KA-) spaces are subgraphs of the connectedness graphs of the K-topology on ${\mathbb{Z}}^n$ . Full article

In this paper, we investigate the invariant properties of the coupled time-fractional Boussinesq-Burgers system. The coupled time-fractional Boussinesq-Burgers system is established to study the fluid flow in the power system and describe the propagation of shallow water waves. Firstly, the Lie symmetry analysis method is used to consider the Lie point symmetry, similarity transformation. Using the obtained symmetries, then the coupled time-fractional Boussinesq-Burgers system is reduced to nonlinear fractional ordinary differential equations (FODEs), with $Erd\acute{e}lyi$ - $Kober$ fractional differential operator. Secondly, we solve the reduced system of FODEs by using a power series expansion method. Meanwhile, the convergence of the power series solution is analyzed. Thirdly, by using the new conservation theorem, the conservation laws of the coupled time-fractional Boussinesq-Burgers system is constructed. In particular, the presentation of the numerical simulations of q-homotopy analysis method of coupled time fractional Boussinesq-Burgers system is dedicated. Full article