Sentence examples for derivative proof from inspiring English sources

The Laplace transform of tf(t) is equal to -F'(s), where F s) is the Laplace transform of f(t) and F'(s) is its derivative (Proof [2]).

This derivative illustrates proof-of-concept for the preparation of other useful fluorophoric antimuscarinic agents which have potential utility in receptor occupancy studies and high throughput screens.

If additionally F is continuously differentiable, then the corresponding trace functional is Fréchet differentiable and there is an expression of its gradient in terms of the derivative of F. The proof of the differentiability of the trace functional is based upon Birman and Solomyak's theory of double Stieltjes operator integrals.

In such situations the major conceptual recasting will also produce new proofs but the explanatoriness of the new proofs is derivative on the conceptual recasting.

Now we shall do the estimates for the higher-order derivatives to finish the proof of Theorem 2. There exists a positive number ε ~ such that under the assumptions of Proposition 3.4, the solution (ϱ,v,H) of (19) satisfies | ( ϱ, v, H ) ( t ) | s ≤ C ( | ϱ 0, v 0, H 0 | L 1 + | ϱ 0, v 0, H 0 | s ) × ( 1 + t ) − 3 4 − ε 1, ∀ 0 < ε 1 < ε ~ (145) for all t≥0.

Proof Taking the derivative of the function ρ n with respect to x and using the estimates (92 - 94 92 - 94tain | ∂ ρ n ∂ x | ≤ C ( 1 + ∫ 0 t ( | v n | + | v n | | ∂ 2 r n ∂ x 2 | + | ∂ v n ∂ x | + | ∂ 2 v n ∂ x 2 | ) d τ ).

If E + I − 2 E I < 0, then δ ( g ) has a unique local minimum at a finite, positive value g = g 0. Proof Calculating the derivative of δ ( g ) with respect to g, we have d δ ( g ) d g = β ( 1 + g ) 2 f ( g ), where f ( g ) : = ln ( I + g E I + g E − 1 − g ) − g ( E − I ) ( 1 + g ) ( I + g E − 1 − g ) ( I + g E ).

Proof. is the derivative of with respect to SNR as SNR 0. The key point to prove this theorem is to show that when, the mutual information decreases as, and hence.

Proof Compute the full derivative of the functional V [ x ( t ), σ ( t ) ] defined by (4) along trajectories of the system (1), (2).

The key of the proof is to derive the suitable estimates for the higher order derivatives of the bilinear terms.

Moreover, if the initial data are smooth and compatible in sense that - ( u 0 i p i ) x ( 0 ) = u 0 i + 1 q i + 1 ( 0 ), i = 2, …, k, u 0 k + 1 ( x ) = u 0 1 ( x ), then the solution has continuous time derivatives down to t = 0. Proof.