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).
Since the first component of ℱ is nonsingular, the reduction property of the Brouwer degree implies deg ( F, Ω # ) = ( − 1 ) m sign det M ⋅ deg ( g ( 0, ⋅ ), Ω # ∩ ( { 0 } × R s ) ). Observe now that since ∂ 2 g is never singular, | deg ( g ( 0, ⋅ ), Ω # ∩ ( { 0 } × R s ) ) | = # ( Ω # ∩ ( { 0 } × [ g ( 0, ⋅ ) ] − 1 ( 0 ) ) ), which is finite and nonzero.
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).
The invariance under homotopy of the Brouwer degree implies that deg ( I − A ( 0, ⋅ ), B ρ, 0 ) = deg ( I − A ( 1, ⋅ ), B ρ, 0 ).
The comparative degree implies a comparison and when it doesn't come, we get sore.
A decreased degree implies that the connections in the network are relatively sparse.
In this way, a better optimal solution can be obtained with a smaller possibility degree of the constraint, However, a smaller possibility degree implies a smaller reliability of the uncertain constraint.
(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.
The negative correlation between expression variation and interaction degree implies that protein with high interaction degrees do not tolerate extensive expression variation and such protein need more precise control on gene expression for an organism to function normally.
