Sentence examples for follows by using from inspiring English sources

For functions in (A^{p}(sigma)), when the weight σ is in (C_{p_{0}}), we have the following estimate, which easily follows by using a standard procedure, namely, from the Cauchy inequality applied to function (f^{ k)}), the subharmonicity of (|f|^{p}), (p>0), the integral Hölder inequality and the definition of class (C_{p_{0}}).

(n) The statement immediately follows by using the condition (L_{2}=1) in (23).

(r) The statement immediately follows by using the condition (L_{3}=1) in (24).

(v) The statement immediately follows by using the condition (L_{4}=1) in (25).

(j) The statement immediately follows by using the condition (L_{1}=1) in (22).

Finally, the proof of (iii) follows by using similar arguments as in the proof of (iii), Theorem 2.1.

The results of (c) and (d) follow by using similar arguments as above, and so we omit the proofs here.

The degradation and mineralization of BB3 were followed by using HPLC and TOC analysis, respectively.

The progress of the reaction was followed by using epoxide titration and high-performance liquid chromatography.

The oxidation is followed by using cyclic voltammetry, chronoamperometry and polarization measurements.

This was followed by using distilled water to dissolve the AAEM species.