Sentence examples for extended formulae from inspiring English sources

The correctness of these extended formulae is verified by comparing them with available ones in the literature, and the importance of these formulae is on their direct application in the three-dimensional dislocation dynamics simulation involving interface or surface.

In this section, we shall establish three new formulae of products of the Frobenius-Euler polynomials to extend formulae (1.6), (1.7) and (1.8) by making use of the generating function methods.

At issue is the funding formula and whether to extend formulas based on the Gonski recommendations of baseline funding for individual students, loaded for any disadvantage.

For this reason, we will extend formula (3) rather than (2).

We extend formula (2.1) of [8] to (2.13) in this paper.

Observe that both corollaries extend formula (3) by choosing (f=psi_{m}) and (p=q=1) in Corollary 2.2, and (f=psi_{m}) and (j=1) in Corollary 2.3.

It was shown in [54] that under natural assumptions it is impossible to extend formula (3.9.1) to the class of all continuously differentiable functions.

(2) We remove the hypothesis T with bounded range and obtain the same result by the different method from [8].   (3) We extend formula (2.1) of [8] to (2.13) in this paper.

We propose new recurrences for a general class of sparse matrices to calculate Green's and lesser Green's function matrices which extend formulas derived by Takahashi and others.

The derivation of the generalized Goertzel algorithm is analog to the technique presented in Section 2. Compared to that, however, we extend formula (22) at the very beginning by unity in the form of e j2 π k N N ⋅ e - j 2 π k N N = 1 for  k ∈ ℝ, (23).

Synthesizing Formulas (1) and (2) and two vectors extends Formula (1) to the following form: (3) S p = C p + ∑ g ∈ G (p ) β p g Φ g.