Sentence examples for manipulation we get from inspiring English sources

From algebraic manipulation, we get.

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.

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).

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

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

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

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} (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).

Proof Differentiating equation (10) with respect to x, after some mathematical manipulations, we get ψ ′ ( L i ( x ) ) = ξ i ψ ′ ( x ) + ∑ j = 1 5 A j, i θ j − 1 ( 1 − θ ) 5 − j ( F i 2, (33).

By equating Eq. (57) to zero and after some manipulations, we get the closed form expression of the optimized normalized global threshold as λ 0 n ∗ = K 1 + 1 γ ln ( 1 + γ ).