Sentence examples for the proof of example from inspiring English sources

Let be as in the proof of Example 2.4.

Similar to the proof of Example 3.6 we obtain that (4.40).

The rest of the proof is similar to the proof of Example 1.3.

As in the proof of Example 5.1, the conditions satisfy the ones of Theorem 2.1.

Repeating the arguments in the proof of Example 2.8, we can easily get the conclusion that ψ 2 ( t ) ψ a, 2 ( t ) attains its maximum at t = 1/2.

Reviewing the proof of Example 2.2, f is a γ-admissible mapping, α ( 0, 0 ) ≥ 1 and if { x n } is a sequence in X such that α ( x n, x n ) ≥ 1 for all n ∈ N ∪ { 0 } and x n → x as n → + ∞, then γ ( x, x ) ≥ 1.

Further work is needed to investigate the impact of various parameters on assemblies and to refine the process of forming two-dimensional AuNR nanoarrays as demonstrated in the proof-of-concept example.

The approach may be useful for pure structure alignment, but its performance needs to be demonstrated quantitatively in terms of accuracy and speed, rather than by the proof-of-concept examples given here.

Suppose that β = 1 3 and α ∈ [ 0, 1 ) such that α + β < 1. Clearly, T is an injective, continuous and sequentially convergent mapping on X. We shall prove that conditions of Corollary 8 hold and T has a fixed point. Proof For the proof of this example, we have the following cases.

It is similar to the proof of (3 - 6) in Example 2.9.

As in the proof of the previous example, we have w(t/u lesssim u^{varepsilon}w(t),quad u>1, therefore rho_{H}(Psi_{u} g)lesssim u^{alpha+varepsilon} rho_{H}(g),quad u>1, gin G_{a}.