The strict mathematical proofs for these conditions are carried out in this paper.
本论文还对这些必要性条件进行了严格的数学证明。
Then the properties of these geometric measures were summarized and the mathematical proofs were finished.
然后总结了这些几何度量的性质，并进行了数学证明；
Such material might include detailed descriptions of equipment, relevant mathematical proofs, research questionnaires or a detailed description of a cited author's intent.
这些材料可能包括设备的详细描述，相关的数学证明，研究问卷或一个引用作者的意图的详细说明。
SCC-MPC scheme's nominal stability and robust stability were analyzed. Mathematical proofs proved the nominal stability and robust stability of SCC-MPC.
对SCC-MPC的名义稳定性、鲁棒稳定性分别进行了分析，并用数学方法证明了SCC-MPC算法的名义稳定和鲁棒稳定。
SCC-MPC scheme's nominal stability and robust stability were analyzed. Mathematical proofs proved the nominal stability and robust stability of SCC-MPC.
对SCC-MPC的名义稳定性、鲁棒稳定性分别进行了分析，并用数学方法证明了SCC-MPC算法的名义稳定和鲁棒稳定。
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