Sentence examples for a derived section from inspiring English sources

A derived section (E in {text {DSec}}((Delta X)^o,mathcal {C})) of such a stable model prefibration is homotopy special if the section (mathsf{ev}(E)) obtained by applying the evaluation functor (8.5) is special in the sense of Definition 5.10.

By definition, we have to show that for any (E in {text {DSec}}(I',gamma ^*mathcal {C})), the functor (E' mapsto {text {Hom}}(gamma ^*E',E)) from ({text {DSec}}(I,mathcal {C})) to abelian groups is represented by a derived section (gamma _*(E) in {text {DSec}}(I,mathcal {C})), and the natural map (gamma ^*gamma _*E rightarrow E) is an isomorphism.

For any good model prefibration (mathcal {C}) over a Reedy category I with the matching expansion (rho :M(I) rightarrow I), a derived section (sigma ) of (mathcal {C}) over I is a section (sigma in {text {Sec}}(M(I),rho ^*mathcal {C})) that is homotopy cartesian along all maps f in M(I) vertical with respect to (rho ).

Say that a derived section of (widetilde{mathcal {C}}) on either of these two categories is homotopy special if it is homotopy cartesian along all special maps, and let ({text {DSec}}_+ subset {text {DSec}}) stand for the full subcategory spanned by homotopy special sections.

end{aligned} (9.15 Any object (E in {text {Ho}}(mathcal {C})) defines a constant derived section (E_X in {mathcal {D}} X,mathcal {C})).

For any two derived sections (A,B in {text {DSec}}(I,mathcal {C})), we have a natural spectral sequence converging to ({text {Hom}}^cdot (A,B)) whose (E_1 -term is givE_1 -termis{aligiven E_1^{n,m} = prod _{i in overline{I}_n}{text {Hom}}^m(overline{L}_i(A),overline{M}_i(by).

Now assume given a simplicial set X, a homotopy special good stable model prefibration (langle mathcal {C},mathcal {C}' rangle ) over the category ((Delta X)^o), and two homotopy special derived sections (A,B in {text {DSec}}_+((Delta X)^o,mathcal {C})).

In order to apply Proposition 8.4, it is useful to obtain a version of the isomorphisms (7.7) and (7.8) valid for derived sections.

Another corollary of Proposition 8.4 is extended functoriality for the categories of derived sections.

Section Botrycephalae is a derived clade of section Phyllodineae.

Using the arguments above, we propose to use a robust metric derived in Section 4 to penalize the residual and address the impulsive sampling noise problem.