Sentence examples for by routine calculations we from inspiring English sources

Remark 2.2 It is easy to see that (H2) implies that if σ ∈ [ 1, ∞ ), then f ( t, σ u ) ≤ σ λ f ( t, u ), (2.6). and (H2′) implies that if σ ∈ [ 1, ∞ ), then f ( t, σ u ) ≥ σ − λ f ( t, u ). (2.7). Now, we present several lemmas that will be used in the proof of our results. By routine calculations we have the following results.

Substituting (2.14) into (2.12), by routine calculation, we can get (2.3).

By some routine calculations, we have begin{aligned}[b]&left.

By Lemma 2.1 and routine calculations, we have the following lemma.

By the definition of Ramanujan's theta function above and routine calculations, we can rewrite Ramanujan's circular summation (1.23) as follows (see, for details [[5], p.338]).

By a routine calculation, we obtain [ T x ] α ( x ) = { t : ( T x ) ( t ) ≥ α ( x ) } = G x. Now condition (7) becomes condition (5).

The proof follows by routine calculations, so we omit it here.

end{aligned} (4.24) Then by routine calculations in (4.18), as in the proof of Theorem 3.1, we derive the inequality biglVert phi'_{n} (t) bigrVert ^{2} + biglVert nablaphi'_{n}(t) bigrVert ^{2} + biglVert Deltaphi_{n} (t) bigrVert ^{2} le C k, T) Bigglvert Phi^{0} + sum ^{5}_{i=1} Phi^{i}_{n} Biggrvert.

The proof follows by routine calculations.

This is easily done by routine calculation under Assumptions 1 and 2.  .

By routine calculation, it is easy to show that (4.3) is satisfied with a = 1 2, b = 1 2 and 0 ≤ c ≤ 7 15.