完整的分析必须建立在隐函数定理的基础上。
A complete analysis must build on the implicit function theorem.
运用分歧理论，隐函数定理，以及渐近展开的方法，获得了非平凡周期解的存在性。
The existence of co-exist periodic solution is investigated by using the bifurcation theory, the implicit function theorem and the method of asymptotic expansion.
然后利用拉格朗日乘数法与隐函数定理，求出了使其中一不等式局部反向的临界值。
Furthermore, utilizing the Lagrange method of multipliers and the implicit theorem to work out the critical value which makes one of those inequality locally inverted.
首先以分阶段的诺西肽发酵过程非结构模型为基础，根据隐函数存在定理进行辅助变量的合理选择；
Firstly, based on the staged unstructured model of Nosiheptide fermentation process, secondary variables were selected according to the implicit function existence theorem.
首先在引理中得出了隐非齐次马尔可夫模型的一些性质，从而导出了隐非齐次马尔可夫模型的三元函数一类平均值的强极限定理。
At first some properties on nonhomogeneous Markov models are obtained in the lemma, then a limit theorem for the average of the three variables function of hidden nonhomogeneous Markov model is given.
首先在引理中得出了隐非齐次马尔可夫模型的一些性质，从而导出了隐非齐次马尔可夫模型的三元函数一类平均值的强极限定理。
At first some properties on nonhomogeneous Markov models are obtained in the lemma, then a limit theorem for the average of the three variables function of hidden nonhomogeneous Markov model is given.
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