Grassmann manifold

Grassmann manifold

[′gräs·mən ′man·ə‚fōld]
(mathematics)
The differentiable manifold whose points are all k-dimensional planes passing through the origin in n-dimensional euclidean space. Also known as Grassmannian.
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Since the transfer function of the reduced system is only related to the subspaces spanned by the transformation matrices, the [H.sub.2] model reduction error is seen as the cost function defined on the Grassmann manifold, and the [H.sub.2] optimal model reduction problem is treated as a minimization problem on the Grassmann manifold.
In this paper, based on the Grassmann manifold, we investigate [H.sub.2] optimal model reduction of the coupled system with unstable subsystems.
[H.sub.2] optimal model reduction for ODE systems based on the Grassmann manifold is investigated in Section 4 while two model reduction algorithms for coupled systems are presented in Section 5.
4 [H.sub.2] optimal model reduction of the ODE system on the Grassmann manifold
Grassmann manifold and step-wise forward component selection using support vector machines were adopted to perform the FNC measure and extract the functional networks' connectivity patterns (FCP).
We selected 82 TRD patients and 41 healthy controls for the data set to complete the multivariate data analysis on the Grassmann manifold and step-wise forward component selection using the support vector machines to obtain the FNs.
Discriminant analysis of functional connectivity patterns on Grassmann manifold. Neuroimage 2011;56:2058-67.
We recall that the oriented Grassmann manifold [[??].sub.n,k] [congruent to] SO(n)/SO(k) x SO (n - k) consists of oriented k-dimensional vector subspaces in Euclidean n-space [R.sup.n].
The manifold [[??].sub.n,k] is the universal double covering space for the Grassmann manifold [G.sub.n,k] [congruent to] O(n)/O(k) x O(n - k) of unoriented k-dimensional vector subspaces in [R.sup.n].
Given an oriented Grassmann manifold [[??].sub.n,k] such that 6 [less than or equal to] 2k [less than or equal to] n and (n, k) [not equal to] (6,3), let t be the largest integer such that [2.sup.t] - 4 [less than or equal to] n - k.
For the oriented Grassmann manifold [[??].sub.n,k] such that 6 [less than or equal to] 2k [less than or equal to] n We have that