In the rest of the paper the equivalence classes are identified by their representatives, which are vector functions
from [??] or [??] and thus, a vector function
should always be understood as an equivalence class.
(i) Due to the closedness of A, a weak solution of (3) can be equivalently defined to be a strongly continuous vector function
y : I [??] X such that, for all t [member of] I,
g(u) is strictly monotone on [bar.[OMEGA]]; namely,
where [u.sub.n] is the approximation of the desired solution that exactly satisfies the boundary conditions and corresponds to the number n of trial functions; [u.sub.[upsilon]] is the vector function
and [U.sub.n] is the matrix composed of trial functions whose components are constructed with the help of R-functions; an (t) is the vector containing the coefficients to be determined.
G(x, U) = [[g.sup.1](x, U), [g.sup.b](x, U), ..., [g.sub.n](x, U)] are given nonlinear vector functions
, and x [member of] Q is domain.
For any vector function
x(t), y(t) [member of] [R.sup.n], [bar.x](t) = [sup.sub.t-[tau][less than or equal to]s[less than or equal to]t]x(s) and [bar.y](t) = [sup.sub.t-[tau][less than or equal to]s[less than or equal to]t](s) satisfy the following condition:
then this vector function
is called a regular solution of (5), and (5) is said to be regularly solvable.
In case of eigendecomposition-based segmentation, a vector function
f is simply the truncated spectral coordinates of a vertex denoted by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] components have non-zero first derivatives in a neighborhood of the point [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined by the nominal values of parameters and initial conditions.
(7) [r.sub.g] is the vector function
for parabolic-profile cutter surfaces of segment a; Eq.
Among the different nodes based on their network performance this new routing protocol (INASDR) will built the node vector function
which will direct the communication state of the whole network.
where x = [([x.sub.1], [x.sub.2], ..., [x.sub.n]).sup.T] and y = [([y.sub.1], [y.sub.2], ..., [y.sub.n]).sup.T] [membre of] [R.sup.n] are the state vectors; f, g : [R.sup.n] [right arrow] [R.sup.n] are continuous nonlinear vector function
; u(f, x, y) [membre of] [R.sup.n] is the controller to be designed.