Concept Of Homotopy Research Articles

On Structure Fuzzy Fibration

In this paper, we introduced concept of fuzzy homotopy lifting property, fuzzy fiber structure is introduced. Also, the definition of the fuzzy lifting function and some theorems in a fuzzy topological space are discussed.

A (Discrete) Homotopy Theory for Geometric Spaces

We define the concepts of homotopy and fundamental group for geometric spaces as a generalization of metric spaces, digital spaces, and graphs; then, we compare them with corresponding concepts in these spaces. Also, we state some properties of the fundamental group of geometric spaces and some theorems to calculate them.

A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

AbstractThe large fluctuation of uncertain parameters introduces a great challenge in the stability analysis of structures. To address this problem, a novel stochastic residual error based homotopy method is proposed in this article. This new method used the concept of homotopy to reconstruct a new governing equation for stochastic elastic buckling analysis, and the closed‐form solutions of the isolated buckling eigenvalues and eigenvectors are obtained by the stochastic homotopy analysis method. On this basis, apth order origin moment of the stochastic residual error with respect to the elastic buckling equation is defined. Then, the optimal form of the homotopy series can be determined automatically by minimizing thepth order origin moment, which overcomes the disadvantage of highly relying on sample values of the existing homotopy stochastic finite element method. Moreover, the proposed method is developed to deal with the stochastic closely spaced buckling eigenvalue problem. Three mathematical examples and three buckling eigenvalue examples, including a variable cross‐section column, a 7‐story frame, and a Kiewitt single‐layer latticed spherical shell, are performed to illustrate the accuracy and effectiveness of the proposed method by comparing with the existing methods when dealing with large fluctuation of random parameters.

An Analytic Solution for Riccati Matrix Delay Differential Equation using Coupled Homotopy-Adomian Approach

An efficient modification and a novel technique combining the homotopy concept with Adomian decomposition method (ADM) to obtain an accurate analytical solution for Riccati matrix delay differential equation (RMDDE) is introduced in this paper . Both methods are very efficient and effective. The whole integral part of ADM is used instead of the integral part of homotopy technique. The major feature in current technique gives us a large convergence region of iterative approximate solutions .The results acquired by this technique give better approximations for a larger region as well as previously. Finally, the results conducted via suggesting an efficient and easy technique, and may be addressed to other non-linear problems.

Winding number and homotopy for quaternionic curves

In this paper, following a recent approach to quaternionic curves, we defined the quaternionic polar angle that enabled us to define global properties of quaternionic curves, namely, the winding number and the homotopy concept. The results admit various applications, including further analogies to plane curves, and physical applications.

A drift homotopy implicit particle filter method for nonlinear filtering problems

<p style='text-indent:20px;'>In this paper, we develop a drift homotopy implicit particle filter method. The methodology of our approach is to adopt the concept of drift homotopy in the resampling procedure of the particle filter method for solving the nonlinear filtering problem, and we introduce an implicit particle filter method to improve the efficiency of the drift homotopy resampling procedure. Numerical experiments are carried out to demonstrate the effectiveness and efficiency of our drift homotopy implicit particle filter.</p>

A Reliable Treatment for Nonlinear Differential Equations

In this paper, we use the concept of homotopy, Laplace transform, and He’s polynomials, to propose the auxiliary Laplace homotopy parameter method (ALHPM). We construct a homotopy equation consisting on two auxiliary parameters for solving nonlinear differential equations, which switch nonlinear terms with He’s polynomials. The existence of two auxiliary parameters in the homotopy equation allows us to guarantee the convergence of the obtained series. Compared with numerical techniques, the method solves nonlinear problems without any discretization and is capable to reduce computational work. We use the method for different types of singular Emden–Fowler equations. The solutions, constructed in the form of a convergent series, are in excellent agreement with the existing solutions.

A new homotopy approach for stochastic static model updating with large uncertain measurement errors

Large measurement errors are a major challenge in structural model updating. Based on the concept of homotopy, a new stochastic static model updating method is proposed to update structural models using uncertain static data with large measurement errors. First, considering the uncertainty of the static measurement, a stochastic model updating equation for element update factors is set up. To solve the stochastic model updating equation, a series of homotopy deformation equations are presented to establish the relationship between the deterministic update factors and the random update factors. Furthermore, the homotopy series expansions of the random update factors can be determined by solving the homotopy deformation equations. Since the measured degrees of freedom of updated structures are usually limited or unavailable, a static condensation technique is used for stochastic model updating. To address the ill-posed problems caused by incomplete measurement information and static measurement errors, the Tikhonov regularization method is used in the process of solving the homotopy deformation equations. Three numerical examples are given to demonstrate the validity of the proposed stochastic model updating method. The numerical results clearly show that unlike the second-order perturbation method, this new method can produce better accuracy in cases with large measurement errors. Compared with the Bayesian method with the Delayed Rejection Adaptive Metropolis sampling technique and even a fast Bayesian method, the proposed method utilizes much less computational time and provides an equivalent accuracy. Finally, static loading experiments on a two-span continuous concrete beam are implemented to validate the proposed model updating method.

A non-linear mathematical analysis of thermally radiative stratified nanoliquid featuring the aspects of magnetic field, Robin conditions and thermal radiation

The heat transportation features of current liquids are improved exceptionally via suspension of nano-sized solid elements having diameter (<100nm) and are regarded as potential working liquids for the utilizations like automobile radiators, electronic chilling systems, solar collectors, heat pipes, nuclear reactors etc. Here in present investigation, nanoliquid model encompassing diffusions (thermophoresis and Brownian) is utilized to address Jeffrey material convected flow under viscous dissipation, thermo-solutal (stratifications, Robin conditions), suction/injection and Joule heating. Hydromagnetic flow persuaded by a permeable cylinder's movement is formulated. Boundary-layer approach is opted to simplify the mathematical systems. Non-dimensionalization process is done through similarity variables. Non-linear analysis is reported by utilizing homotopy concept. Besides the results are provided on physical contextual of involved constraints.

Approximate Solutions of Coupled Nonlinear Oscillations: Stability Analysis

The current article is concerned with a comprehensive investigation in achieving approximate solutions of coupled nonlinear oscillations with high nonlinearity. These equations are highly nonlinear second-order ordinary differential equations. Via a coupling of the Homotopy perturbation method and Laplace transforms, which is so-called the He-Laplace method, traditional approximate solutions involving the secular terms are accomplished. On the other hand, in order to cancel the secular terms, an expanded frequency technique is adapted to accomplish periodic approximate solutions. Therefore, a nonlinear frequency, for each differential equation, is achieved. Furthermore, for more convenience, these solutions are pictured to indicate their behavior. The multiple time-scales with the aid of the Homotopy concept are utilized to judge the stability criteria. The analyses reveal the resonance as well as the non-resonant cases. Additionally, numerical calculations are carried out, graphically, to address the regions that guaranteed the bounded solutions. It is found that the latter method, is the most powerful mathematical tool in extracting the stability analysis of the considered system.

Rheology of hydromagnetic viscoelastic fluid subjected to dissipation aspect

PurposeThe purpose of this paper is to address the impact of gravity induced stretching flow of second grade liquid subject to thermal radiation. The flow is generated by the stretching of an impermeable cylinder. Stagnation point flow is considered. Convective type boundary conditions are applied on temperature and concentration. The present investigation further includes the aspects of magnetohydrodynamics, Joule heating, chemical reaction and viscous dissipation and heat generation/absorption.Design/methodology/approachThe ordinary differential expressions are formed using suitable similarity transformations from the governing partial differential expressions. The subsequent nonlinear ordinary differential expressions are solved analytically using homotopy concept to report the consequences of different dimensionless physical parameters in graphical and tabular forms.FindingsThe results witnessed that increasing values of curvature parameter corresponds to higher temperature and concentration. Besides this, the impacts of destructive and constructive chemical processes on the concentration distribution are noted opposite.Originality/No such analysis has yet been reported.

Comparison for the Approximate Solution of the Second-Order Fuzzy Nonlinear Differential Equation with Fuzzy Initial Conditions

This research focuses on the approximate solutions of second-order fuzzy differential equations with fuzzy initial condition with two different methods depending on the properties of the fuzzy set theory. The methods in this research based on the Optimum homotopy asymptotic method (OHAM) and homotopy analysis method (HAM) are used implemented and analyzed to obtain the approximate solution of second-order nonlinear fuzzy differential equation. The concept of topology homotopy is used in both methods to produce a convergent series solution for the propped problem. Nevertheless, in contrast to other destructive approaches, these methods do not rely upon tiny or large parameters. This way we can easily monitor the convergence of approximation series. Furthermore, these techniques do not require any discretization and linearization relative with numerical methods and thus decrease calculations more that can solve high order problems without reducing it into a first-order system of equations. The obtained results of the proposed problem are presented, followed by a comparative study of the two implemented methods. The use of the methods investigated and the validity and applicability of the methods in the fuzzy domain are illustrated by a numerical example. Finally, the convergence and accuracy of the proposed methods of the provided example are presented through the error estimates between the exact solutions displayed in the form of tables and figures.

Fundamental Group of Rough Topological Spaces

In this paper we study and define the concept of the fundamental group of rough topological spaces (RTSs), which deeply depends on the concepts of rough sets (RSs) and rough topology (RT). Working towards this stated objective, we define the concept of rough path (RPt) which gives room for the introduction of rough loop (RL). We also define the concepts of rough homotopy (RH) and later shows that it is indeed an equivalence relation. We introduce the fundamental group of rough topological spaces by showing that all the group axioms satisfied. Also, this paper establish the fact that most of the results in fundamental group of ordinary topological spaces are also hold for the fundamental group of rough topological spaces.

Heat generation in mixed convected Williamson liquid stretching flow under generalized Fourier concept

We addressed mixed convected stretching flow considering Williamson liquid subjected to variable conductivity. Transportation of heat is investigated employing generalized Fourier law and heat generation. Governing non-linear expressions are achieved through well-known boundary-layer concept. Transformation procedure converts PDEs into ODEs. Homotopy concept is used for non-linear analysis. Novel attributes of non-dimensional variables are scrutinized via graphical outcomes. We found higher temperature subjected to generalized Fourier law in comparison to well-known Fourier law.

Synchronous Maneuver Searching and Trajectory Planning for Autonomous Vehicles in Dynamic Traffic Environments

In the real-time decision-making and local planning process of autonomous vehicles in dynamic environments, the autonomous driving system may fail to find a reasonable policy or even gets trapped in some situation due to the complexity of global tasks and the incompatibility between upper-level maneuver decisions with the low-level lower level trajectory planning. To solve this problem, this paper presents a synchronous maneuver searching and trajectory planning (SMSTP) algorithm based on the topological concept of homotopy. Firstly, a set of alternative maneuvers with boundary limits are enumerated on a multi-lane road. Instead of sampling numerous paths in the whole spatio-temporal space, we, for the first time, propose using Trajectory Profiles (TPs) to quickly construct the topological maneuvers represented by different routes, and put forward a corridor generation algorithm based on graph-search. The bounded corridor further constrains the maneuver's space in the spatial space. A step-wise heuristic optimization algorithm is then proposed to synchronously generate a feasible trajectory for each maneuver. To achieve real-time performance, we initialize the states to be optimized with the boundary constraints of maneuvers, and we set some heuristic states as terminal targets in the quadratic cost function. The solution of a feasible trajectory is always guaranteed only if a specific maneuver is given. The simulation and realistic driving-test experiments verified that the proposed SMSTP algorithm has a short computation time which is less than 37ms, and the experimental results showed the validity and effectiveness of the SMSTP algorithm.

Sutterby fluid flow subject to homogeneous–heterogeneous reactions and nonlinear radiation

This paper addresses MHD Sutterby fluid flow by a rotating disk of variable thickness. Thermal radiation is nonlinear. Impact of homogeneous–heterogeneous reactions is also examined. For the development of convergent solution the homotopy concept is utilized. Skin friction coefficient, temperature, velocity and Nusselt number are addressed. Radial, azimuthal and tangential velocities are enhanced by increasing the disk thickness. For larger disk thickness coefficient there is decay in skin friction. Nusselt number is enhanced for larger Prandtl number and radiation parameter.

A method for fault diagnosis of nonlinear circuits

PurposeThe purpose of this paper is to develop a method for multiple soft fault diagnosis of nonlinear circuits including fault detection, identification of faulty elements and estimation of their values in real circumstances.Design/methodology/approachThe method for fault diagnosis proposed here uses a measurement test leading to a system of nonlinear equations expressing the measured quantities in terms of the circuit parameters. Nonlinear functions, which appear in these equations are not given in explicit analytical form. The equations are solved using a homotopy concept. A key problem of the solvability of the equations is considered locally while tracing the solution path. Actual faults are selected on the basis of the observation that the probability of faults in fewer number of elements is greater than in a larger number of elements.FindingsThe results indicate that the method is an effective tool for testing nonlinear circuits including bipolar junction transistors and junction field effect transistors.Originality/valueThe homotopy method is generalized and associated with a restart procedure and a numerical algorithm for solving differential equations. Testable sets of elements are found using the singular value decomposition. The procedure for selecting faulty elements, based on the minimal fault number rule, is developed. The method comprises both theoretical and practical aspects of fault diagnosis.

Application of Homotopy to the Ageing Process of Human Body Within the Framework of Algebraic Topology

This paper presents the ageing process of the human body from the homotopy concept in the algebraic topology. The compact connected human body with boundary is assumed to be topologically equivalent to a cylinder. The complex connected cylindrical shape of the body x=S^1*I, described by the functions f,h: x&amp;rarr;x is a Cartesian product of the vertical interval I=[0,&amp;beta;] and a circle S^1. The initial state of the body x=S^1*I is the topological shape of the infant. The ageing process called the homotopy is the family of continuous functions f_t(x) on the interval I=[0,&amp;alpha;] where &amp;alpha;&amp;gt;&amp;beta; and t&amp;ge;&amp;beta;. It is an increasing sequence of the function f_t(x) of the body x=S^1*I. The homotopy relates the topological shape of the infant to the topological shape of the adult. The homology theory assigns to the human body (x) a sequence of abelian groups H_n(x) for n=0,1,2&amp;hellip;and to the continuous function a sequence of homomorphism. The homology group characterizes the number and continuity of the compact surface of human body. The study excludes genetic, hormonal, environmental and other factors in the ageing process and recognizes the fact that ageing continues throughout life, through childhood, and adolescence into adulthood. Topologically the infant is equal to the adult since the infant continuously grows into the adult. The study found an algebraic way of relating homotopy to the process of ageing of human body. It was also established that this could offer other topologists useful topological tips in the application of homotopy to other physical continuous processes.&#x0D; &#x0D; &amp;nbsp;&#x0D; &#x0D; &amp;nbsp;

Physical aspects of irreversibility in radiative flow of viscous material with cubic autocatalysis chemical reaction

.Analysis of irreversibility in flow by a stretchable surface has gained much consideration in recent years. Entropy optimization properly computes the second law thermodynamic irreversibilities. Therefore, deterioration of entropy proficiency results in a more useful energy transport process. In this article, a physical aspect of irreversibility in radiative flow of viscous material with quartic autocatalysis chemical reaction is addressed. The flow is discussed between two stretchable rotating disks. Heat transfer occurring in this physical problem is modelled through thermal radiation, Joule heating and viscous dissipation. This is the first time the concept of homogeneous-heterogeneous reactions has been studied with entropy generation. The nonlinear flow expressions are made dimensionless. The obtained equations are then tackled through the homotopy concept. The analysis discloses that the radiation parameter and Eckert number play a vital role in the enhancement of temperature field. The tangential velocity decreases versus the magnetic parameter. The radial component of velocity boosts close to lower disks and it decreases near the upper disks versus the Reynolds number. The variations in the Nusselt number and skin friction are presented graphically with various emerging variables. It is noticed that entropy rate can be controlled by minimizing the impact of Brinkman and Reynolds numbers.

Higher homotopies between quasigroups

The concept of homotopy has been a staple of quasigroup theory since Albert's work on isotopies in the early 1940s. In this paper, we introduce and study the dual concept of a higher homotopy.