Sentence examples for under the integral from inspiring English sources

Since we have the delta function depending on ω under the integral, we obtain (32)In order to integrate this we use the residue theorem.

Define ℓ = d - 1 2. It then follows that f ∈ C ℓ ( 0, ∞ ) and we can write f ( t ) = ∫ 0 ∞ r d - 1 + ℓ Y ( r ) σ ^ ( r t ) d r. (8)In other words, we can differentiate under the integral sign ℓ times since the successive derivatives are integrable, due to (7) and (5).

We need to prove that the functions under the integral signs are measurable.

In addition, the integral can be differentiated with respect to x ∈ R under the integral sign.

If we simply ignore the scaling factor under the integral (10), we obtain the modified/simplified cost function.

That (iii) implies (i) follows from differentiating under the integral sign and then applying Theorem 1.12 in [5].

Proof We must state that the functions under the integrals of equation (3.1) are Bochner integrable on Ω, since they are strongly measurable and we can state the following obvious results.

Moreover, due to inequality (5.11), we may omit the Steklov averages if the factors under the integrals are separated by Hölder's inequality.

Under hypotheses (H1)–(H5), the integral equation (12) has at least one solution (x_) in X. Define T by begin{aligned} Tx t)= int _{0}^{t}, F t,s,x s)), ds + lambda (t),quad tin [0,tau ]. end{aligned}Let (x,yin X) satisfying (xi left( gx s),gy(s)right) ge 0) for all (tin [0,tau ]).

Thus, the advantage of the relative conservation method is the capability to discriminate SLiMs conserved under constraints of the integral protein from those conserved to serve as functional motifs.

The area-under-the-curve (AUC) scores were calculated by approximation of the integral under each plotted APoA line.