Sentence examples for by replacing the function from inspiring English sources

Aljarrah et al. (2014) proposed quantile based approach to refine the T-X(W) family by replacing the function W with Q Y, where Q Y is the quantile function of a random variable Y.

The reduction is achieved by replacing the function v in (2.11), (where (u_{tau tau }) is replaced by ((u_tau )_tau ) in the proof of Lemma 2.1), with the function v in (2.57), (where ({tilde{v}}_1) is defined in (2.60), and f is any non-negative function in (C^2 {bar{Omega }})) satisfying (fge 1+ C_1|x|^2) on (partial Omega )).

In the proof of Theorem 1.1, by replacing the function (v_1) in (2.8) with the function v in (2.57), we obtain at a maximum point (x_0), using (2.3), begin{aligned} u_{11}(x_0) le C sqrt{1+ M_2}, end{aligned} (2.58 where C depends additionally on A and (K|phi |_{2 Omega }).

(1.15) In view of the alternating term ((-1)^{n-1}) in the Green's function ({mathcal{G}}_{n} t,s)), Hartman and Lyapunov type inequalities for the 2-point boundary value problem begin{aligned} x^{(2n)}(t)+q(t)x t)=0;quad aleq tleq b, end{aligned} (1.16) satisfying the boundary conditions (1.8) can be obtained by replacing the function (q^(t)) by (|q(t)|) in (1.9) and (1.10), respectively.

To accommodate the GLMs, the existing p.vector and T.fit functions of the maSigPro package that account, respectively, for first and second regression steps of the method have been modified by replacing the function lm by glm.

It takes into account the quantum character of the final state by replacing the Dirac function by an Airy function.

The proposed sliding mode control strategy eliminated the chattering phenomenon by replacing the sign function with a saturation function, and by replacing the constant coefficients in the reaching law with adaptive ones.

Can Theorem 3.1 be improved by replacing the control function θ by other functions?

Furthermore, by replacing the sigmoid function with the modified Versorial function, the modified CNFSLMS MCNFSLMSS) algorithm with low computational complexity is also presented.

The claim that the function s ↦ Φ i ( x, y ; p, g, f s ) is log-convex on I is an immediate consequence of Theorem 3.7 and Remark 3.5, and (24) can be obtained by replacing the convex function f with the convex function f ( z ) = log Φ i ( x, y ; p, g, f z ) for z = r, s, t in (21), where r, s, t ∈ I such that r < s < t.

The time-varying cross spectrum can be defined with the Cohen's class distributions by replacing the autocorrelation function in (17) with the cross-correlation function.