The invariance under homotopy of the Brouwer degree implies that deg ( I − A ( 0, ⋅ ), B ρ, 0 ) = deg ( I − A ( 1, ⋅ ), B ρ, 0 ).
This, together with the additivity property of the Brouwer degree, implies that deg [ I − A g, Ω, 0 ] = 1, which, together with the existence property of the Brouwer degree, implies that there exists u ∈ Ω such that A g ( u ) = u.
We infer that deg ( I − A ( 1, ⋅ ), B ρ, 0 ) ≠ 0, and the existence property of the Brouwer degree implies that A ( 1, ⋅ ) has at least one fixed point u which is also a solution of (3.1).
For every λ ∈ ( 0, Λ ], the existence property of the degree implies that T s x + C ( λ, x ) + ε J ψ x = 0 for some x ∈ D ( C ) ∩ Ω, which contradicts property ( P ).
Therefore, the homotopy invariance property of Brouwer's degree implies that (operatorname{deg}(H tau, cdot,cdot )),D,0)=mathrm{constant}) for (tauin[0,1]) and then operatorname{deg}(Psi_{0},D,0)=operatorname{deg}bigl(Hbigl(1,( cdot,cdot bigr),D,0bigr)=operatorname{deg}bigl(Hbigl 0, cdot,cdot ) bigr),D,0bigr)=operatorname{deg}(Phi,D,0).
A decreased degree implies that the connections in the network are relatively sparse.
Strikingly, mouse apoE binds to HDL3 particles to a much reduced degree, implying that mouse apoE resident apolipoprotein interactions are less favorable in this case.
Lemmas 3.9 and 3.10, (3.16), and the excision property of the degree imply that operatorname{deg}(operatorname{id}-mathcal{F}_{j},U_{j}) = operatorname{deg}(operatorname{id}-mathcal {F}_{j},Omega) = 1 quadmbox{for } j=1, 2. (3.17).
(3.11) Since A has no fixed point on (partial(pm P_{e}setminus{ theta})), (3.11) and the permanence of the topological degree imply that operatorname{deg}bigl I-A,mathbin{mathrel{pm} stackrel{circ}{ P_{e}}}cap( operatorname{deg}bigl I-A,mathbin{mathrel{pmtheta bigr)=i bigl(A,( pm P_{e})cap(Omega_{R_{4}}stackrel{circ}ne{ Omega_{P_{e} pm P_{e}}capr)=0.
Then the homotopy invariance and normalization properties of topological degrees imply that begin{aligned} operatorname{deg}(h_{lambda},B_{R},0) =&operatorname{deg} bigl( ( I- lambdaPsi ),B_{R},0 bigr)=operatorname{deg}(h_{1},B_{R},0) =&operatorname{deg}(h_{0},B_{R},0)=operatorname{deg}(I,B_{R},0)=1, end{aligned} since (0in B_{R}).
All the tested cell lines expressed both receptors albeit to varying degrees, implying that canonical activin A signaling could take place in these cells.
