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By the same arguments as above we can get the necessity parts of Theorem 1.1(i) and (ii) for (mathcal {I}_{Omega,alpha}).
By the same arguments as above we get the necessity part and the sufficiency part follows from the conclusion of (T_{Omega,alpha}) and the following lemma.
By the same arguments as above we get the necessity part and the sufficiency part follows from the conclusion of (T_{Omega,alpha}) and Lemma [9].
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com