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1·This paper establishes the duality theory of preference relations.

建立了基本二元关系以及偏好关系的对偶理论。

2·Moreover, we develop corresponding duality theory and related saddle point theory.

同时，我们还建立了对偶理论及相关的鞍点理论。

3·Some new ideas and methods for the application of the duality theory are presented, such as cross convolution, cross cyclical co...

在双重性原理的应用部分，还提出了交叉卷积、交叉循环卷积、借用变换对法等新概念与新方法。

4·An arbitrage pricing method for financial products in terms of generalized network model and duality theory of linear program is presented.

运用广义网络流模型和线性规划对偶理论，提出了一种金融产品的套利定价方法。

5·The generalized invex functions are introduced and the optimality theorems and duality theory are established for a class of variational problems.

文中引入了一类广义不变凸函数，对一类变分问题给出了最优解的充分性条件以及对偶理论。

6·Some new ideas and methods for the application of the duality theory are presented, such as cross convolution, cross cyclical convolution and borrowtransform-pair methods.

在双重性原理的应用部分，还提出了交叉卷积、交叉循环卷积、借用变换对法等新概念与新方法。

7·The papar calls scientits′ attention to that, the wave-particle duality theory established depond on the wavelength being small enough, but the demarcation line is not clear.

指出，虽然波粒二象性成立的前提是波长足够小，但在现有理论中却提不出一条清晰的界线。

8·In this paper, a more in-depth study was conducted to the scheduling problem of fuzzy Numbers, the fuzzy linear programming of the uncertain programming and its duality theory.

本文针对模糊数的排序问题以及不确定规划中的模糊线性规划及对偶理论进行了较为深入的研究。

9·Then, by use of multiple carrier system's frequency-sharing property and convex optimization, a subcarrier and power optimal allocation algorithm is proposed based on Lagrangian duality theory.

然后，运用凸优化技术分析了该资源分配问题，并基于拉格朗日对偶法给出了一种子载波和功率最优分配算法。

10·In this paper, a mathematical model for integral optimization design of pier shaft and its foundation is given, and the SLP method and duality theory are applied here to have a optimal solution.

本文给出了重力式桥墩与明挖基础整体优化设计的数学模型，选择序列线性规划法，并利用线性规划的对偶理论求最优解。