Sentence examples for proved that the functional from inspiring English sources

If m = n = 1, Tonelli [1] proved that the functional F is lower semicontinuity in W1,∞ (a, b) if and only if the function f is convex in the last variable.

We proved that the functional interactions between miR-101 and DNA methylation determine lung cancer cell fate in vitro and in vivo via CDH1 re-expression.

Next, we will prove that the functional J satisfies the geometry of Theorem 3.

We will prove that the functional φ satisfies the (PS -condition.

We prove that the functional equations (1.1) and (1.2) are equivalent if, are nonzero rational numbers.

In Section 4, we prove that the functional can arrive at a stationary global minimum.

Next, we will prove that the functional (mathcal{J}) defined by (4.1) satisfies ((mathit{PS})).

Proof of Theorem 1.1 Firstly, we prove that the functional φ satisfies the (PS -condition.

To prove that the functional (I_{lambda}) satisfies ((mathrm{PS})_{c_{lambda }}) condition for a.e.

First consider the case p > 2 for which we prove that the functional ϕ satisfies the Palais-Smale condition.

Now, in order to prove that the functional I satisfies the Palais-Smale condition, we will introduce an auxiliary notion.