Sentence examples for vanishing lemma from inspiring English sources
The following famous Lions vanishing lemma is useful for the proofs of Theorems 3.3 3.4.
If not, from the vanishing lemma, it follows that there exist two positive constants δ, ρ and ({x_{n}} in{mathbb{R}^{3}}) such that int_{{B_{rho}}(x_{n})} {{{({u_{n}} - bar{u})}^{2}}} ge delta.
Let x be an element in a p-group P with | x P | = p b. Then there exist at least b ( p − 1 ) non-linear irreducible characters vanishing at x. Lemma 2.3 ([8], Proposition 3.2.2]).
The closed-loop stability analysis is presented based on a technical lemma developed for nonlinear cascaded systems with vanishing disturbances.
According to Lemma 1 this solution is oscillatory, bounded, and not vanishing at infinity.
By Lemma 3.2, we need to verify that is a vanishing Carleson measure.
Since | G / G ′ | ≥ 4, there are at least two of these irreducible characters φ i, satisfying | U φ i / G ′ | ≥ 2. By Lemma 3.1, there are at least four irreducible characters of G, vanishing at an element, a contradiction.
In the proofs of the lemmas we often use the following inequalities valid for the function f vanishing at x = 0 and x = 1 or for the function with the first derivative vanishing at some point x ∈ [ 0, 1 ] : | f | 2 ≤ 2 ∥ f ∥ ∥ ∂ f ∂ x ∥, ∥ f ∥ ≤ 2 ∥ ∂ f ∂ x ∥, (39).
Finally, we use Lemma 3.5 to show that there exists a critical μ ∗ such that spreading occurs when μ > μ ∗ and vanishing happens when μ ∈ ( 0, μ ∗ ].
