Sentence examples for proof by using the from inspiring English sources

Second, we prove that the up down equilibrium point is a saddle via an elementary proof by using the Routh–Hurwitz criterion to show that the Jacobian matrix valued at the point has two and two eigenvalues in the open left- and right-half planes, respectively.

Proof By using the Frenet formulas and the definitions of harmonic curvatures, we get the result.

from which we complete the proof by using the Gronwall inequality.

Then we conclude the proof by using the Trudinger-Moser inequality [23, 24].

One completes the proof by using the Lebesgue's dominated convergence theorem.

Next we start the main part of the proof by using the expression (3.29).

Then ⪯ η is a partial order on X. Proof The proof follows by using the cone metric axioms, properties (2) and (3) for the cone, and (3.1).

The proof follows by using the convexity of Orlicz functions.

(iii) The proof follows by using the convexity of Orlicz functions.

Proof The proof is easy by using the Bernstein inequality and the complex mean value theorem, the proof is omitted here.

It holds