Next, we consider the orthogonal projection
[mathematical expression not reproducible] is defined by
By (30), Lemma 5, condition (i), and the linearity orthogonal projection
of [P.sub.C], we have
On the other hand, for closed-loop systems subject to the input and output measurement noises,  developed a new closed-loop subspace identification algorithm (SOPIM) by adopting the EIV model structure of SIMPCA and proposed an orthogonal projection
approach to avoid identifying the parity space of the feedback controller.
Or equivalently, M is a reducing subspace for S if and only if S[P.sub.M] = [P.sub.M]S and [S.sup.*][P.sub.M] = [P.sub.M][S.sup.*], where [P.sub.M] is the orthogonal projection
from H onto M.
If v and w are two vector fields defined in a neighborhood of S, then the covariant derivative [[nabla].sup.S.sub.v]w is equal, at x [member of] S, to the orthogonal projection
of [D.sub.v]w, projected orthogonally to the tangent space [T.sub.x]S (cf.
In a radial plane, radially on the inside of the radially innermost point of the at least one layer of carcass reinforcement and axially on the outside of the geometric center of the bead wire, the radius of curvature at a point on the exterior surface of the bead is greater than the radius of curvature at the point of orthogonal projection
of the point of the exterior surface of the bead onto the at least one layer of carcass reinforcement.
In this study, an automated method is proposed to align high-throughput metabolite profiles using the orthogonal projection
technique, which is a powerful chemometric strategy .
An element [x.sup.r] [member of] X is a reflection point with respect to H of the element x [member of] X if and only if [h.sub.x] := (x + [x.sup.r])/2 is an orthogonal projection
of the point x onto H, that is, if and only if
Denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the vector bundle so induced and, correspondingly, denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the orthogonal projection
. Define the linear Poincare flow as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].