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The sequences were aligned by using the CLUSTAL W program (Thompson et al. 1997).
The spacer sequences were compiled and aligned by using the multisequence alignment program ClustalX (1.8).
Total sequences were aligned by using the MAFFT aligner (25 ) implemented in Geneious version 5.5 (26 ).
Full-envelope protein sequences for each serotype were aligned by using the multiple alignment tool of MEGA5 (www.megasoftware.net).net
Sequencing products were purified and underwent electrophoresis with the 3100 Genetic Analyzer (Applied Biosystems) and aligned by using the multisequence alignment CLUSTALX version 1.8 (13 ).
Open reading frames were translated into amino acid sequences and aligned by using the ClustalW alignment option in Geneious version 6.0.5 (www.geneious.com/).
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Both nucleotide and predicted amino acid sequences were aligned by using the ClustalW 1.4 method (http://www.ebi.ac.uk/clustalw/).ac.uk/clustalw/
end{aligned} By using the boundedness of (z(t)), we see that (3.1) holds.
end{aligned} By using the lemma on logarithmic derivative again, we get (mu(f geq mu(A_{s})).
end{aligned} By using the fixed point index theorem, sufficient conditions for the existence of at least two and at least three positive solutions were obtained.
end{aligned} By using the uniqueness of the asymptotic center, (Tx = x), so x is a fixed point of T. Hence, (F(T)) is nonempty.
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