Homotopy Decomposition Research Articles

The homotopy decomposition of the suspension of a non-simply-connected five-manifold – ADDENDUM

The homotopy decomposition of the suspension of a non-simply-connected five-manifold – ADDENDUM

The homotopy decomposition of the suspension of a non-simply-connected five-manifold

In this paper we determine the homotopy types of the reduced suspension space of certain connected orientable closed smooth $five$ -manifolds. As applications, we compute the reduced $K$ -groups of $M$ and show that the suspension map between the third cohomotopy set $\pi ^3(M)$ and the fourth cohomotopy set $\pi ^4(\Sigma M)$ is a bijection.

Homotopy decomposition method to analysis fractional hepatitis B virus infection model

Mathematical models have been employed to investigate the factors that govern the progression of infectious diseases in viral infections. This article investigates a fractionalised model for hepatitis B virus infection, curing infected cells. We analysed the fractional hepatitis B virus infection model using the homotopy decomposition method (HDM). Additionally, the existence and uniqueness of the fractional model are considered. The graphical solutions depicted that when introducing a fractional-order derivative to the model, the peak of viral propagation is reduced, and cell damage is minimised by initiating treatment at an early stage; however, the disease takes a longer time to be eradicated. The outcomes may also be valuable in the fields of clinical science and medicine. This model provided insights into the natural history of HBV infection, including the different stages of infection (acute, chronic, and resolved) and the likelihood of progression from one stage to another. This information is essential for predicting disease outcomes and planning appropriate clinical management for patients.

Homotopy of manifolds stabilized by projective spaces

AbstractWe study the homotopy of the connected sum of a manifold with a projective space, viewed as a typical way to stabilize manifolds. In particular, we show a loop homotopy decomposition of a manifold after stabilization by a projective space, and provide concrete examples. To do this, we trace the effect in homotopy theory of surgery on certain product manifolds by showing a loop homotopy decomposition after localization away from the order of the image of the classical ‐homomorphism.

A fractional mathematical model with nonlinear partial differential equations for transmission dynamics of severe acute respiratory syndrome coronavirus 2 infection

This study presents a fractional mathematical model based on nonlinear Partial Differential Equations (PDEs) of fractional variable-order derivatives for the host populations experiencing transmission and evolution of the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) pandemic. Five host population groups have been considered, the Susceptible, Exposed, Infected, Recovered, and Deceased (SEIRD). The new model, not introduced before in its current formulation, is governed by nonlinear PDEs with fractional variable-order derivatives. As a result, the proposed model is not compared with other models or real scenarios. The advantage of the proposed fractional partial derivatives of variable orders is that they can model the rate of change of subpopulation for the proposed model. As an efficient tool to obtain the solution of the proposed model, a modified analytical technique based on the homotopy and Adomian decomposition methods is introduced. Then again, the present study is general and is applicable to a host population in any country.

Suspension homotopy of 6–manifolds

For a simply connected closed orientable manifold of dimension $6$, we show its homotopy decomposition after double suspension. This allows us to determine its $K$- and $KO$-groups easily. Moreover, for a special case we refine the decomposition to show the rigidity property of the manifold after double suspension.

An FTSA Trajectory Elliptical Homotopy for Unmanned Vehicles Path Planning With Multi-Objective Constraints

This article investigates a fixed-time simultaneous arrival (FTSA) problem in terms of the equilibrium of path lengths of unmanned vehicles. A novel trajectory elliptical homotopy method (TEHM) is designed to solve the FTSA problem of unmanned vehicles in a multi-objective constrained environment. Considering the constraints of obstacle avoidance and kinematics of unmanned vehicles, the trajectories elliptical homotopy is selected for path planning. The obtained trajectory homotopy guarantees obstacle avoidance and motion stability at the same time. To handle the non-cooperative and dynamic obstacle avoidance, a trajectory elliptical homotopy decomposition (TEHD) algorithm is proposed with an FTSA constraint. Based on the TEHM and TEHD, a multiple unmanned vehicle fixed-time regular-triangle formation algorithm is designed and implemented on real vehicles. Simulations and experiments validate the performance of the proposed methods and show how fixed-time arrival formation under obstacles and kinematic constraints was obtained.

Analytical Approaches for Approximate Solution of the Time-Fractional Coupled Schrödinger–KdV Equation

In this article, we use the homotopy perturbation method and the Adomian decomposition method with the Yang transformation to discover analytical solution to the time-fractional coupled Schrödinger–KdV equation. In the Caputo sense, fractional derivatives are described. A convergent series is used to calculate the solutions of fractional PDEs. Analytical results achieved applying the homotopy perturbation and decomposition techniques are numerically calculated and represented in the form of tables and figures. The simplicity, efficacy, and high degree of accuracy of the used method are then demonstrated by comparing these solutions to the actual solutions and the results. Finally, the applied approaches are the most popular and convergent methods for solving nonlinear fractional-order partial deferential problems.

The homotopy type of a once-suspended 6-manifold and its applications

Let M be a closed, oriented, simply connected 6-manifold. After localization away from 2, we give a homotopy decomposition of ΣM in terms of spheres, Moore spaces and other recognizable spaces. As applications we calculate generalized cohomology groups of M and determine the homotopy types of gauge groups of certain bundles over M.

On the existence and uniqueness analysis of fractional blood glucose-insulin minimal model

In our research work, we suggest the modified minimal model of fractional order and analyze it using the homotopy decomposition method (HDM). The minimal model is quite a useful mathematical model which describes the behavior of glucose-insulin metabolism. The original model was given in the 80s and has been updated over a different period. In this modified model, we add a one-factor diet which plays an important role in the blood-glucose analysis. We obtained the numerical results by using the homotopy decomposition method. HDM is extremely useful, significant, and very simple. We also discuss the existence and uniqueness of the fractional model.

On the Homotopy Decomposition for the Quotient of a Moment–Angle Complex and Its Applications

On the Homotopy Decomposition for the Quotient of a Moment–Angle Complex and Its Applications

Self-closeness number and weak homotopy decomposition

For any CW-complex X X , there exists a weak homotopy decomposition X ( m ) X^{(m)} and a self-closeness number of X X . In this paper, we study the self-closeness number of a weak homotopy decomposition of X X . We prove that the self-closeness number of X ( m ) X^{(m)} is dominated by the self-closeness number of X X . Moreover, we determine the set of self-homotopy classes and the group of self-homotopy equivalence classes of a weak homotopy decomposition of X X , and herein we provide some examples of the self-closeness number for homotopy m-sections.

New Exact Solutions of Burgers’ Equation Using Power Index Method

In this article, we focus on the new exact solutions of Burger’s equation by using a new technique which is known as the power index method (PIM). In this method, we choose suitable indexes of independent variables and similarity transformation so that the partial differential equation may be converted into ODE. We have obtained analytic solution of the ODE by using symbolic package Maple. We have got exact solution of Burgers’ equation by using analytic solution of ODE and similarity transformation. The proposed method has been effectively employed to find new exact solutions for the nonlinear Burgers’ equation. Finally, the proposed resulting answers are compared with the homotopy perturbation, decomposition, and variational iteration solutions.

The fibre of the degree 3 map, Anick spaces and the double suspension

AbstractLet S2n+1{p} denote the homotopy fibre of the degree p self map of S2n+1. For primes p ≥ 5, work by Selick shows that S2n+1{p} admits a non-trivial loop space decomposition if and only if n = 1 or p. Indecomposability in all but these dimensions was obtained by showing that a non-trivial decomposition of ΩS2n+1{p} implies the existence of a p-primary Kervaire invariant one element of order p in $\pi _{2n(p-1)-2}^S$. We prove the converse of this last implication and observe that the homotopy decomposition problem for ΩS2n+1{p} is equivalent to the strong p-primary Kervaire invariant problem for all odd primes. For p = 3, we use the 3-primary Kervaire invariant element θ3 to give a new decomposition of ΩS55{3} analogous to Selick's decomposition of ΩS2p+1{p} and as an application prove two new cases of a long-standing conjecture stating that the fibre of the double suspension $S^{2n-1} \longrightarrow \Omega ^2S^{2n+1}$ is homotopy equivalent to the double loop space of Anick's space.

Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

The fundamental groups and homotopy decompositions of algebraic topology have applications in systems involving symmetry breaking with topological excitations. The main aim of this paper is to analyze the properties of homotopy decompositions in quotient topological spaces depending on the connectedness of the space and the fundamental groups. This paper presents constructions and analysis of two varieties of homotopy decompositions depending on the variations in topological connectedness of decomposed subspaces. The proposed homotopy decomposition considers connected fundamental groups, where the homotopy equivalences are relaxed and the homeomorphisms between the fundamental groups are maintained. It is considered that one fundamental group is strictly homotopy equivalent to a set of 1-spheres on a plane and as a result it is homotopy rigid. The other fundamental group is topologically homeomorphic to the first one within the connected space and it is not homotopy rigid. The homotopy decompositions are analyzed in quotient topological spaces, where the base space and the quotient space are separable topological spaces. In specific cases, the decomposed quotient space symmetrically extends Sierpinski space with respect to origin. The connectedness of fundamental groups in the topological space is maintained by open curve embeddings without enforcing the conditions of homotopy classes on it. The extended decomposed quotient topological space preserves the trivial group structure of Sierpinski space.

Simplicial G–complexes and representation stability of polyhedral products

Representation stability in the sense of Church-Farb is concerned with stable properties of representations of sequences of algebraic structures, in particular of groups. We study this notion on objects arising in toric topology. With a simplicial $G$-complex $K$ and a topological pair $(X, A)$, a $G$-polyhedral product $(X, A)^K$ is associated. We show that the homotopy decomposition [2] of $\Sigma (X, A)^K$ is then $G$-equivariant after suspension. In the case of $\Sigma_m$-polyhedral products, we give criteria on simplicial $\Sigma_m$-complexes which imply representation stability of $\Sigma_m$-representations $\{H_i((X, A)^{K_m})\}$.

Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps

AbstractA free homotopy decomposition of any continuous map from a compact Riemannian manifold 𝓜 to a compact Riemannian manifold 𝓝 into a finite number maps belonging to a finite set is constructed, in such a way that the number of maps in this free homotopy decomposition and the number of elements of the set to which they belong can be estimated a priori by the critical Sobolev energy of the map inWs,p(𝓜, 𝓝), withsp=m= dim 𝓜. In particular, when the fundamental groupπ1(𝓝) acts trivially on the homotopy groupπm(𝓝), the number of homotopy classes to which a map can belong can be estimated by its Sobolev energy. The estimates are particular cases of estimates under a boundedness assumption on gap potentials of the form$$\begin{array}{} \displaystyle \iint\limits_{\substack{(x, y) \in \mathcal{M} \times \mathcal{M} \\ d_\mathcal{N} (f (x), f (y)) \ge \varepsilon}} \frac{1}{d_\mathcal{M} (y, x)^{2 m}} \, \mathrm{d} y \, \mathrm{d}x. \end{array}$$Whenm≥ 2, the estimates scale optimally asε→ 0. Whenm= 1, the total variation of the maps appearing in the decomposition can be controlled by the gap potential. Linear estimates on the Hurewicz homomorphism and the induced cohomology homomorphism are also obtained.

Application of a hybrid method for systems of fractional order partial differential equations arising in the model of the one-dimensional Keller-Segel equation

In this paper, we apply a hybrid method due to coupling the Laplace transform with the Adomian decomposition method (LADM) for solving nonlinear fractional differential equations that appear in the model of Keller-Segel equations with one dimension. We explain the adopted method is with several examples. It turns out that the reliability of LADM and the reductions in computations show that LADM is widely applicable. We also compare our results with the results of homotopy decomposition method (HDM).

On the topology of complexes of injective words

An injective word over a finite alphabet V is a sequence $$w=v_1v_2\cdots v_t$$ of distinct elements of V. The set $$\text {Inj}(V)$$ of injective words on V is partially ordered by inclusion. A complex of injective words is the order complex $$\Delta (W)$$ of a subposet $$W \subset \text {Inj}(V)$$. Complexes of injective words arose recently in applications of algebraic topology to neuroscience, and are of independent interest in topology and combinatorics. In this article we mainly study Permutation Complexes, i.e. complexes of injective words $$\Delta (W)$$, where W is the downward closed subposet of $$\text {Inj}(V)$$ generated by a set of permutations of V. In particular, we determine the homotopy type of $$\Delta (W)$$ when W is generated by two permutations, and prove that any stable homotopy type is realizable by a permutation complex. We describe a homotopy decomposition for the complex of injective words $$\Gamma (K)$$ associated with a simplicial complex K, and point out a connection to a result of Randal-Williams and Wahl. Finally, we discuss some probabilistic aspects of random permutation complexes.

A study of the fractional‐order mathematical model of diabetes and its resulting complications

Diabetes is a worldwide problem that affects one of every 11 persons nowadays. The IDF Diabetes Atlas (Eighth edition, 2017) states that approximately 415 million people in the world are living with the disease and that this number will rise to 629 million by the year 2045. It is a very serious problem of the world. A major part of the world population is affected by this disease and its resulting complications. In this paper, we propose to investigate a fractional‐order model of diabetes and its resulting complications. The mathematical model's parameters define the population of diabetic patients and those who are diabetic with complications at a given time t. We have also discussed the existence, uniqueness, and stability of the fractional‐order model, which we consider here. We make use of the homotopy decomposition method (HDM) in order to solve the problem.