Sentence examples for using the linearity of from inspiring English sources

Using the linearity of the trace, for any operator one can write, where.

Then using the linearity of φ-function, we have.

(75), (76) and (77) by (hat {sigma }_{j}), taking the trace and using the linearity of the trace operation.

Indeed, if we take, where the first coordinate function is given by, and such that t 1 satisfies (2.33), then, using the linearity of Riemann-Liouville derivative, we have.

To show the existence of Ω, let us define it on the polynomials (2.11) by (2.12), (2.13) and then we expand Ω to over the whole space (mathbb{C}_{2N} [ lambda ] ) by using the linearity of Ω.

Finally, using the linearity of W, tilde f= sumlimits_{i=1}^{N} m_{i}^{r_{i}} + W^{-1} text{Lap}_{lambda}), (33). is shoW^{-1} text{erturbed function of the smoothed consumption sum.

When dose conversion algorithm was used, the linearity of sensitivity was better about 38%.

(4.2) To use the linearity of the transform (g to u [ g ] - u [ 0 ]), we add and subtract the term (u ( x,T 0 )) to the functional (J_{alpha} ( g )).

We then use the linearity of botharguments of the covariance bracket to express the variance of the quadratureoperator as S ϕ = ∫ − ∞ ∞ [ e i ϕ e − i ϕ ] G [ 〈 a �� ˆ i , a ˜ ˆ i 〉 〈 a ˜ ˆ i , a ˜ ˆ i † 〉 〈 a ˜ ˆ i † , a ˜ ˆ i 〉 〈 a ˜ ˆ i † , a ˜ ˆ i † 〉 ] × G T [ e i ϕ e − i ϕ ] d ω ′. (67).

By using the linearity property of the operator (3.13).

As a consequence from Theorems 10 and 11, we obtain using the linearity property of Equations (15 - 16 15 - 16llowing.