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Let (g(x)) be an odd function and (G_< G^).
Lemma 2.1 Let n ∈ N. Let f : Z → C be an odd function.
end{aligned} For any given positive integer n, let (g_{n}(s)) be an odd function.
For any given positive integer m, let (g_{m}(s)) be an odd function.
Let and be real vector spaces and be an odd function satisfying (1.2).
To apply the global bifurcation theorem, we extend to be an odd function by (3.4).
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So, is an odd function.
because is an odd function.
Assume now f is an odd function.
Then, we define on such that is an odd function.
For the restoring force being an odd function, three numerical examples are presented.
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