Sentence examples for equivalent to the functional from inspiring English sources

So every solution of Eq. (1.5) with m = 1 is also an additive mapping (briefly, 1-function); (b) Eq. (1.5) with m = 2 is equivalent to the functional quadratic equation.

The functional equation (1.3) is equivalent to the functional equation (2.31).

In this section, we will investigate that the functional equation (1.1) is equivalent to the presented functional equation (1.4).

For m =2,4, Lee and Chung [46],[47] showed that Eq. (1.5) is equivalent to the quadratic functional equation and the quartic functional equation, respectively.

So every solution of Eq. (1.5) with m = 2 is also a quadratic mapping (briefly, 2-function); (c) Eq. (1.5) with m = 3 is equivalent to the cubic functional equation.

Then the following assertions hold: (a) Eq. (1.5) with m = 1 is equivalent to the additive functional equation.

It is well-known that the FERZ is equivalent to the second famous functional equation, namely FEJT begin{aligned} vartheta Big ({-frac{1}{z}}Big )= -iz)^{frac{1}{2}}vartheta (z), end{aligned} (1.17)satisfied by Jacobi's (vartheta )-function begin{aligned} vartheta (z)= -iz 2sum _{k=1}^{frac{1}{ (i pi k^2 z)= sum _{k=-infty }^infty end{aligned^2 z).

This can help current and future mission planning system designers who may wish to use the FAMs, or something equivalent, to design the functional architecture of their system(s).

This implies through condition (2.21) that n1 ≡ 0 for all λ ≤ λ0 with some λ0 < 0. By Proposition 2.10, the latter condition is equivalent to the positivity of the functional F0 for every λ ≤ λ0, in particular for λ = λ0.

x ˜ is a local maximum of H. (jj) x ˜ is a global maximum of H. (jjj) L ( x ˜ ) = z and sup x ∈ X 〈 L ( x ), x 〉 ≤ 0. Proof First, observe that the symmetry of L is equivalent to the fact that the functional H is Gâteaux differentiable with derivative given by H ′ ( x ) = L ( x ) − z. for all x ∈ X ([11], p.235).

This is equivalent to minimize the functional beta {displaystyle {int}_{varOmega}varphi left({left|nabla uright|}_{varepsilon}right dx} (22).