For logical thinking learners, it is important to learn problem solving formally and intuitively. Homotopy which is modern mathematics combines algebra and geometry together. Homotopy gives the most abstract level of explanation or the most basic invariant which is the most required characteristics for logical thinking. The homotopy lifting property gives good explanation of inductive reasoning...

A quasi-schemoid is a small category with a particular partition of the set of morphisms. We define a homotopy relation on the category of quasi-schemoids and study its fundamental properties. The homotopy set of self-homotopy equivalences on a quasi-schemoid is used as a homotopy invariant in the study. The main theorem enables us to deduce that the homotopy invariant for the quasi-schemoid in...

in this paper, we will compare a homotopy perturbation algorithm and taylor series expansin method for a system of second kind fredholm integral equations. an application of he’s homotopy perturbation method is applied to solve the system of fredholm integral equations. taylor series expansin method reduce the system of integral equations to a linear system of ordinary differential equation.

A functor is constructed from the category of globular CW-complexes to that of flows. It allows to compare the S-homotopy equivalences (resp. the T-homotopy equivalences) of globular complexes with the S-homotopy equivalences (resp. the T-homotopy equivalences) of flows. Moreover, one proves that this functor induces an equivalence of categories from the localization of the category of globular...

Homotopy coherence has a considerable history, albeit also by other names. For this volume highlighting symmetries, the appropriate use is homotopy of representations, at one time known as representations up to homotopy/homotopy coherent representations. We present brief semi-historical survey, providing some links that may not be common knowledge.

Although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. Some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.Here, after criticizi...

although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.here, after criticizi...

For a Koszul operad P , there are several existing approaches to the notion of a homotopy between homotopy morphisms of homotopy P-algebras. Some of those approaches are known to give rise to the same notions. We exhibit the missing links between those notions, thus putting them all into the same framework. The main nontrivial ingredient in establishing this relationship is the homotopy transfe...