Sentence examples for with respect to the functions from inspiring English sources

Fixing ϕ, we minimize the functional E ϕ, h 1, h 2) with respect to the functions h 1 x) and h 2 x).

The results are discussed with respect to the functions of the mutated genes.

With respect to the functions V and K, we assume that.

For, we define the truncation operator with respect to the functions and given by (3.19).

In the previous related results on boundary value problems for p-Laplacian differential equations by means of the monotone iterative method, the monotone-type conditions for nonlinear terms f with respect to the functions u or their derivatives are usually required.

end{aligned} (3.5) From this system one can derive numerous nonlinear equations with respect to the functions (Q t,x)), (R t,x)) that may have the properties of Korteweg-de Vries equation (independence of the spectral parameter on time, infinite number of conservation laws, integrability) by taking the spectral expansions (N=sum_{k=-m}^{m} N_{k}z^{k}).

In [14], Berinde introduced the concepts of comparison function and ( c ) -comparison function with respect to the function φ : R + → R +.

Let us refer by the sequence to all the orthonormal polynomials which arise from the Gram-Schmidt procedure with respect to the function.

K is said to be an invex set with respect to the function η iff ∀ x, y ∈ K, ∀ λ ∈ [ 0, 1 ] ⇒ y + λ η ( x, y ) ∈ K. Definition 2.2 ([7]).

By convoluting an arbitrary function χ (t) with Eq. (17), we now obtain the solution of the velocity potential with respect to the function generally given by Eq. (4) as follows: (18).

It is easy to see that (2.36) is linear with respect to the function M, so that we can easily find the following recursive formulae for the calculation of c αβ and ω m (s, μ).