Sentence examples for manipulations we get from inspiring English sources

Equivalently, By using some simple matricial manipulations, we get, where (39).

Applying the same fact and after some manipulations, we get (26).

After some straightforward yet tedious algebraic manipulations, we get ε 1 ∗ = L + 1 A 1 A 2 1 + L + 1 A 1 A 2, (41).

end{aligned} Now, based on the result of Theorem 4, and following [18], and after some manipulations, we get the desired result.

After some manipulations, we get c_{1}=1, qquad c_{2}=frac{1}{3}, and, consequently, (f_{1}(t)=1+t), which is the exact solution.

end{aligned} (4.10) If we make use of the relation Gamma xi Gamma 1-xi =frac{pi}{sin xi pi)}, then after performinGamma xi Gamma 1-xi =frac{pi}{sin xi we get the desired equation (4.8).

From algebraic manipulation, we get.

To facilitate the analysis, we consider the relative coordinates ρ12, α12, and α21 between two robots 1 and 2. After some algebraic manipulation, we get the following equations: ρ ̇ 12 = − v 1 cos α 12 − v 2 cos α 21, (4a).

With simple algebraic manipulation, we get: {widehat{p}}_i=frac{ exp left({displaystyle {sum}_{k=0}^K{widehat{beta}}_k{x}_{ik}}right)}{1+ exp left({displaystyle {sum}_{k=0}^K{widehat{beta}}_k{x}_{ik}}right)} The formula was used to compute predicted probabilities of coresidence under different combinations of parents' and adult children's socioeconomic statuses.

In conclusion, after some manipulations, we can get the following two general formulas.

where x denotes the instantaneous received SNR at SD. Substituting (10) into (18) and after some necessary but tedious manipulations, we can get a closed form expressions of the average capacity R S as.