Sentence examples for making use of the approximation from inspiring English sources

end{aligned} (4.69) Making use of the approximation property (2.5) of pressure, the error estimate (vert !vert !vert sigma-sigma_{h}vert !vert !vert ^{2}_{0,0}) and (vert !vert !vert D mathbf{u}-mathbf{u}_{h})vert !vert !vert ^{2}_{0,0}) in Theorem 4.3, the truncation errors (2.11) anD mathbf{u}-mathbf{u}ral discretion, we can derive the required result (4.62).

The proof is similar to that of Lemma 5.3 and hence omitted, though here we make use of the approximation (5.4) rather than (5.5).

In this section we demonstrate the use of the software iNA, which makes use of the approximation methods described in the previous section, to predict the noise characteristics of two gene regulatory networks involving post-transcriptional regulation by non-coding RNA and negative autoregulation via post-translational modification.

Making use of the Max-log approximation, the LLR for a given bit,, can be approximated by [14] (14).

We successfully demonstrate the method with a classic beam diffusion test case in 2D, making use of the Lévy area approximation for the correlated Milstein cross terms, and generating a computational saving of a factor of 100 for ε= 10−5.

Making use of the max-log approximation, the LLR for a given bit b p (belonging to an arbitrary subcarrier q and stream i) can be approximated by [21] ℒ i ML [ q ] s i [ q ], b p ≈ 1 2 max b ∈ ℬ p, + 1 − 1 σ n 2 ∥ r [ q ] − A [ q ] ℳ ( b ) ∥ 2 − 1 2 max b ∈ ℬ p, − 1 − 1 σ n 2 ∥ r [ q ] − A [ q ] ℳ ( b ) ∥ 2, (12).

Since the CDI channel model falls into this case, the ergodic capacity upper bound can be found by taking the expectation of (23) over the channel distribution and setting ρ = 0. Making use of the high SNR (P ≫ 1) approximation log(1 + xP) ≈ log xP), the ergodic capacity upper bound is then given by (26).

Throughout the section, we assume that { q n } is a sequence in ( 0, 1 ) such that q n → 1. Theorem 6 For each f ∈ C 2 ∗ [ 0, ∞ ), we have lim n → ∞ ∥ B n, q n ( f ) − f ∥ 2 = 0. Proof Making use of the Korovkin type theorem on weighted approximation [25], we see that it is sufficient to verify the following three conditions: lim n → ∞ ∥ B n, q n ( t k ; x ) − x k ∥ 2 = 0, k = 0, 1, 2. (3.14).

From Table 1, we can obtain a good approximation with the exact solution by making use of the proposed method.

The N B ^ estimator is also an approximation to the full likelihood, but makes use of the continuous approximation to simplify the calculations.

Then we make use of the numerical approximation scheme recently developed in [44].