We examine the “homotopy coniveau tower” for a general cohomology theory on smooth k-schemes, satisfying some natural axioms, and give a new proof that the layers of this tower for K-theory agree with motivic cohomology. We show how these constructions lead to a tower of functors on the Morel-Voevodsky stable homotopy category, and identify this stable homotopy coniveau tower with Voevodsky’s s...

Probability-one homotopy algorithms have strong convergence characteristics under mild assumptions. Such algorithms for mixed complementarity problems (MCPs) have potentially wide impact because MCPs are pervasive in science and engineering. A probability-one homotopy algorithm for MCPs was developed earlier by Billups and Watson based on the default homotopy mapping. This algorithm had guarant...

In his book “Homotopy Theory and Duality,” Peter Hilton described the concepts of relative homotopy theory in module theory. We study in this paper the possibility of parallel concepts of fibration and cofibration in module theory, analogous to the existing theorems in algebraic topology. First, we discover that one can study relative homotopy groups, of modules, from a viewpoint which is close...

This paper presents the solution of the nonlinear equation that governs the flow of a viscous, incompressible fluid between two converging-diverging rigid walls using an improved homotopy analysis method. The results obtained by this new technique show that the improved homotopy analysis method converges much faster than both the homotopy analysis method and the optimal homotopy asymptotic meth...

We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory. This has two main applications. First, by considering inverse diagrams in Voevodsky...

Integral symplectic 4–manifolds may be described in terms of Lefschetz fibrations. In this note we give a formula for the signature of any Lefschetz fibration in terms of the second cohomology of the moduli space of stable curves. As a consequence we see that the sphere in moduli space defined by any (not necessarily holomorphic) Lefschetz fibration has positive “symplectic volume”; it evaluate...

We explain how the notion of homotopy colimits gives rise to that of mapping spaces, even in categories which are not simplicial. We apply the technique of model approximations and use elementary properties of the category of spaces to be able to construct resolutions. We prove that the homotopy category of any monoidal model category is always a central algebra over the homotopy category of Sp...

This study addresses a nonlinear fractional Drinfeld–Sokolov–Wilson problem in dispersive water waves, which requires appropriate numerical techniques to obtain an approximative solution. The Adomian decomposition approach, the homotopy perturbation method, and Sumudu transform are combined tackle problem. Caputo manner is used describe derivative, He’s polynomials employed address nonlinearity...

Simple-homotopy for simplicial and CW complexes is a special kind of topological homotopy constructed by elementary collapses and expansions. In this paper we introduce graph homotopy for graphs and Graham homotopy for hypergraphs, and study the relation between these homotopies and the simplehomotopy for simplicial complexes. The graph homotopy is useful to describe topological properties of d...