Sentence examples for by using the continuity from inspiring English sources

By using the continuity of f, we obtain that the operator A is completely continuous.

By using the continuity of ∗ and [[22], Lemma 1], we get the following result.

As n → ∞, we can get d ( F − 1 x ∗, x ∗ ) = 0 by using the continuity of ξ.

By using the continuity of the function g, we deduce that the operator (mathcal{A}_{1}) is continuous.

The transfer matrix between two consecutive unit cells in the structures is obtained by using the continuity conditions.

Then, by using the continuity of ψ and lower semi-continuity of ϕ, we find ψ ( s ) ≤ ψ ( s ) − ϕ ( s ), (13).

Therefore, by using the completely continuity of and, respectively, and letting, we have (3.20).

By using the uniform continuity of f̃, we have (|tilde{f}(hat{t} -tilde{f}(t)|

Then, the shape functions can be determined by using the stress continuity conditions between interfaces and the stress boundaries on surfaces.

Then by using the interface continuity conditions between the piezoelectric fiber layer and the matrix layer, and the boundary condition at the external surface of the piezoelectric fiber reinforced laminated composite the unknown constants are determined.

The locally Lipschitz case is treated in the same manner as done in the proof of Theorem 2 (1) by stopping the process at time τ U, (2) by using the Lipschitz continuity of f and g in the ball of radius U and (3) by a truncation argument and using the almost sure boundedness of the solutions extending the convergence to the locally Lipschitz case.