For each prime $p$, we define a $t$-structure on the category $\widehat{S^{0,0}}/\tau\text{-}\mathbf{Mod}_{harm}^b$ of harmonic $\mathbb{C}$-motivic left module spectra over $\widehat{S^{0,0}}/\tau$, whose MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent to abelian $p$-completed $BP_*BP$-comodules are concentrated in even degrees. We prove $\mathcal{D}^b({{BP}_*{B...

In this paper we elaborate a general homotopy-theoretic framework in which to study problems of descent and completion and of their duals, codescent and cocompletion. Our approach to homotopic (co)descent and to derived (co)completion can be viewed as ∞-category-theoretic, as our framework is constructed in the universe of simplicially enriched categories, which are a model for (∞, 1)-categorie...

In this paper, a new spectral-iterative method is employed to give approximate solutions of fractional logistic differential equation. This approach is based on combination of two different methods, i.e. the iterative method cite{35} and the spectral method. The method reduces the differential equation to systems of linear algebraic equations and then the resulting systems are solved by a numer...

where Xs+1 is the fiber of gs. We get an exact couple of homotopy groups and a spectral sequence with E 1 = πt−s(Ks) and dr : E s,t r → Es+r,t+r−1 r . This spectral sequence converges to π∗(X) (where X = X0) if the homotopy inverse limit lim←Xs is contractible and certain lim 1 groups vanish. When X is connective, it is a first quadrant spectral sequence. For more background, see [Rav86]. In th...

This paper is devoted to the Cradle Theorem. It is a combinatorial contraction discovered when studying a crucial point of the effective Bousfield-Kan spectral sequence, an unavoidable step to make effective the famous Adams spectral sequence. The homotopy equivalence TOP(Sp,TOP(Sq, X)) ∼ TOP(Sp+q, X) is obvious in ordinary topology, not surprising in combinatorial topology, but it happens the ...

This paper is devoted to the Cradle Theorem. It is a combinatorial contraction discovered when studying a crucial point of the effective Bousfield-Kan spectral sequence, an unavoidable step to make effective the famous Adams spectral sequence. The homotopy equivalence TOP(Sp,TOP(Sq, X)) ∼ TOP(Sp+q, X) is obvious in ordinary topology, not surprising in combinatorial topology, but it happens the ...