Sentence examples for manifold critical points from inspiring English sources

M ¯ is a C 1 manifold; critical points of I restricted on M ¯ are critical points of I restricted on E ˜ P ; for any x ∈ E ˜ P ∖ { 0 }, there exists a unique t x such that t x x ∈ M ¯ and I 1 ( t x x ) = min t ∈ ( 0, ∞ ) I 1 ( t x ) ; I restricted on M ¯ satisfies the (PS) condition.

For each (yin mathcal{K}^), we denote by (W_{u}^{infty} y)_{infty}) the unstable manifold of the critical points at infinity ((y)_{infty}).

In this case, the functional J 0 possesses a n-dimensional manifold Z of critical points, given by Z = { δ ˜ ( a, λ ), a ∈ S n, λ > 0 }.

By construction X k 0 ∞ is a finite cw complex, where the cells of dimension an integer r in X k 0 ∞ are given by the unstable manifolds of the critical points at infinity ( y ) ∞ such that i ( y ) ∞ = r.

At the beginning of this subsection, we give some basic definitions with will allow us to describe the unstable manifolds of the critical points at infinity in V ( 1, ε ).

Homogeneous manifolds satisfy this condition, but we exhibit examples of locally homogeneous manifolds which are not critical points in dimensions ≥ 3.

So, it is convenient to consider Φ on the Nehari manifold that contains all nontrivial critical points of Φ and on which Φ turns out to be bounded from below.

Observe that W ˜ u ( y ) correspond to W s ( y ), the stable manifold of the critical point y along the flow lines of ( − ∂ K ).

Two critical points should be made.

Notice that, just like for usual critical points, it is associated to each critical point at infinity w ∞ of J stable and unstable manifolds W s ∞ ( w ∞ ) and W u ∞ ( w ∞ ) (see [26], pp.356-357).

I highlight six critical points.