The relative homotopy theory of modules, including the (module) homotopy exact sequence, was developed by Peter Hilton (1965). Our thrust is to produce an alternative proof of the existence of the injective homotopy exact sequence with no reference to elements of sets, so that one can define the necessary homotopy concepts in arbitrary abelian categories with enough injectives and projectives, ...

We study some properties of A-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of A-homotopy theory. These concepts and results are well-suited to the study of certain quotients via geometric invariant theory. As a case study in the geometry of ...

in this paper, we conduct a comparative study between the homotopy perturbation method (hpm) and adomian’s decomposition method (adm) for analytic treatment of nonlinear volterra integral equations, and we show that the hpm with a specific convex homotopy is equivalent to the adm for these type of equations.

Spaces of maps into classifying spaces for equivariant crossed complexes, II: The general topological group case. Abstract The results of a previous paper [3] on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for cross...

We obtain some explicit calculations of crossedQ-modules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2-types and second homotopy modules of certain homotopy pushouts of maps of classifying spaces of discrete groups. Introduction A crossed module M = (μ : M → P ) has a classifying space BM (...

We demonstrate existence of waves localized at the interface of two nonlinear periodic media with different coefficients of the cubic nonlinearity via the one-dimensional Gross–Pitaevsky equation. We call these waves the surface gap solitons (SGS). In the case of smooth symmetric periodic potentials, we study analytically bifurcations of SGS’s from standard gap solitons and determine numericall...

In this study, we used two unique approaches, namely the Yang transform decomposition method (YTDM) and homotopy perturbation (HPTM), to derive approximate analytical solutions for nonlinear time-fractional Zakharov–Kuznetsov equations (ZKEs). This framework demonstrated behavior of weakly ion-acoustic waves in plasma containing cold ions hot isothermal electrons presence a uniform magnetic flu...

The importance of partial differential equations in physics, mathematics and engineering cannot be emphasized enough. Partial are used to represent physical processes, which then solved analytically or numerically examine the dynamical behaviour system. new iterative approach Homotopy perturbation method this article solve fractional order Fokker–Planck equation numerically. Caputo sense is cha...

In this paper, nonlinear Klein-Gordon equation with quadratic term is solved by means of an analytic technique, namely the Homotopy analysis method (HAM).Comparisons are made between the Adomian decomposition method (ADM), the exact solution and homotopy analysis method. The results reveal that the proposed method is very effective and simple.

1 .l Summary IN [l] QUILLEN introduced the notion of a model category (a category together with three classes of maps: weak equivalences, fibrations and cofibrations, satisfying certain axioms (1.4 (iv))) as a general framework for “doing homotopy theory”. To each model category M there is associated a homotopy category. If W C M denotes the subcategory of the weak equivalences, then this homot...