Sentence examples for by using the definitions from inspiring English sources

Recently, Savaş defined ℐ-asymptotically lacunary statistical equivalent sequences by using the definitions ℐ-convergence and asymptotically lacunary statistical equivalent sequences together.

By using the definitions of ζ q (s, x) and ζq,E (s, x), we can define the q-analogues of Dirichlet's L-function.

In this section, we define K − ⪯ set-valued mappings on partially ordered metric spaces, and by using the definitions, we obtain the existence of common fixed points.

By using the definitions in the last two displays, inequalities (73) and (75), and finally Sobolev embedding theorem, we then have Open image in new window so that (76) follows.

The main purpose of this paper is by using the definitions and properties of Chebyshev polynomials to study the power sum problems involving Fibonacci polynomials and Lucas polynomials and to obtain some interesting divisible properties.

We can expand the left hand side of Eq. 34 by using the definitions M ⊺ v i = ∑ l ∈ 1 M v i ( l ) and v i ⊺ ζ k = v i ( l + ) − v i ( l − ), which yield ∑ i = 2 n ∑ j ≠ i E [ M ⊺ v i v i ⊺ ζ k ] E [ M ⊺ v j v j ⊺ ζ k ] (38).

By using the definition of positive and negative definite matrices and stability theory of fractional-order systems, sufficient conditions for robust stabilization of closed-loop perturbed fractional-order linear interconnected systems are given.

By using the definition, we have (5.55).

Indeed, by using the definition of, we have (3.4).

(ii) By using the definition, easy to prove.

But by using the definition of in (5.44) the proof of the lemma is complete.