Sentence examples for using the following equality from inspiring English sources

Using the following equality: (27).

Compute F γ T Open image in new window by using the following equality for all sets of feasible values s l K | k ∈ S γ Open image in new window.

which we have used the following equality: ∫ Z p k d ∗ μ q : = ∫ Z p ⋯ ∫ Z p ⏟ k -times d μ q ( ξ 1 ) d μ q ( ξ 2 ) ⋯ d μ q ( ξ k ).

For simplicity of notation, in what follows, we use the following equalities.

end{aligned} (6.8) We can obtain the following equality using the adjoint and difference problems: begin{aligned}& int_{0}^{l} 2 bigl[ u ( x,T g ) - y ( x ) bigr] Delta u ( x,T ), dx & quad = - int_{0}^{T} keta ( l,t ) Delta g ( t ), dt.

Firstly, we consider the following equality using (6): ∫ Z p f x + d d μ - q - 1 x + q d ∫ Z p f x d μ - q - 1 x = 2 q ∑ 0 ≤ l ≤ d - 1 - 1 l q d - l + 1 f l. (11 Let χ be a Dirichlet character of conductor d, which is any multiple of p (= o d d ).

Using the definition of pseudo-inverse, the following equality is established for the augmented matrix A: (12).

The following equality is used frequently in the paper: langle x-y,x-zrangle=frac{1}{2} bigl Vert x-yVert ^{2}+Vert x-zVert ^{2}-Vert y-zVert ^{2} bigr),quad forall x,y,zinmathcal{R}^{n}.

The following equality will be used frequently without reference: (2).

The following equality is true: (320).

We now consider the following equality.