对偶的

In this paper, based on the conjugate function and the dual programming of convex programming, with the dual cone, to conic programming, the dual programming is fully discussed, the expression of the dual programming is educed, the main dualities are proved, and the dualities of the familiar conic programming are studied.

以共轭函数和凸规划的对偶规划为基础，利用对偶锥的概念，全面讨论了一般锥规划的对偶问题，严格推导出锥规划对偶规划的表示形式，给出了锥规划的主要对偶性质，并用这些结果研究了常见锥规划的对偶性。

In dual Brunn-Minkowski theory, we study the properties of the dual harmonic quer-massintegrals systematically and establish some inequalities for the dual harmonic quer-massintegrals, such as the Minkowski inequality, the Brunn-Minkowski inequality, the Blaschke-Santalo inequality and the Bieberbach inequality. We establish the dual Brunn-Minkowski inequality for dual affine quermassintegrals. Recently we learned that Gardner have independently proved it by a different method. The polar body of a convex body is an important object in the context of convex geometry. Hence, after we studied the intersection bodies, it is natural to consider the inequalities for their polar bodies.

在对偶Brunn-Minkowski理论中，我们引入了对偶调和均质积分概念，系统的研究了它的性质，并建立对偶调和均质积分的Brunn-Minkowski不等式，Blaschke-Santalo型不等式和Bieberbach不等式；接着我们建立了对偶仿射均质积分的对偶Brunn-Minkowski不等式，最近我们得知这个不等式被Gardner用另外的方式证明；凸体的极体是凸几何中一个重要概(来源：2525ABf8C论文网www.abclunwen.com)念，既然相交体和投影体有对偶关系，因此在研究完投影体的极体之后自然要研究相交体的极体。

In the first part, convex duality method linearizes convex wealth equation, and the introduction of Lagrange multiplier converts original problem to an unconstrained static optimum problem. At last, we obtain another equivalent optimum problem with convex control domain.

文章前一部分运用凸对偶的方法先将财富方程线性化，Lagrange乘子的引入使最优问题等价与一个无约束的静态最优问题，最后转化为一个凸控制域的最优问题。

Bonikowskis model, the covering upper approximation operation and covering lower approximation operation are not dual to each other. William Zhu proved them determining each other, but he cannot give the quantity relation of them.

Bonikowski定义的广义覆盖粗糙集模型中，上近似算子与下近似算子不满足对偶的性质，虽然William Zhu等证明了上、下近似算子是相互确定的，但是没有给出它们之间具体的数量关系。

Wanka studied two kinds of dualproblem in finite dimensional space, namely, an extended Fenchel type and Fenchel-Lagrange dual based on the conjugate theorem in convex optimization problem.

Wanka利用有限维空间中凸优化问题的共轭理论，研究了两类对偶问题，即广义Fenchel对偶问题和Fenchel-Lagrange对偶问题，后者是经典Fenchel和Lagrange对偶问题的组合，二者都是在扰动理论基础上产生的，还提出了一个约束条件保证其凸优化问题中强对偶成立。

W anka studied two kinds of dualproblem in finite dimensional space,namely,an extended Fenchel type and FenchelˉLagrange dual based on the conjugate theorem in convex optimization problem.The latter is a"combination"of the classical Fenchel dual and Lagrange dual,which are based on perturbation theory.Then they introduced a constraint qualification whose fulfillment is sufficient in order to guarantee strong duality.

W anka利用有限维空间中凸优化问题的共轭理论，研究了两类对偶问题，即广义Fenchel对偶问题和FenchelˉLagrange对偶问题，后者是经典Fenchel和Lagrange对偶问题的组合，二者都是在扰动理论基础上产生的，还提出了一个约束条件保证其凸优化问题中强对偶成立。

Based on analysing the mechanism of substitution and compensation of ICT for labor in service sector, this dissertation constructed the cost function of translog form which was dual with the production function of the service sector, estimated the parameter of the share requirement equation affected by the factors of labor force, ICT and non-ICT, so that to calculate the Allen partial elasticity of substitution and the MES elasticity of substitution.

本文在分析 ICT 对服务业劳动的替代和补偿机制的基础上，建立与服务业生产函数对偶的超越对数形式的成本函数，估计出劳动力、 ICT 和非 ICT 等要素的需求份额方程的参数，以此计算 Allen 偏替代弹性和 MES 替代弹性。

Accordingly, we know the connection of the universal duality and boundness of a primal-dual pair feasible set.We apply these theories to practical applications in chapter four.In chapter five we make many conclusions and bring forward research expectation for the future.conic optimization; dual conic optimization; duality gap; universal duality

研究主要有：利用Nesterov和Todd的齐次模型判断锥规划与其对偶锥规划解的存在性；类似于线性规划推导出锥规划的KKT-条件；从锥规划的泛对偶性得到锥规划与对偶锥规划的零对偶间隙存在的条件，从而了解泛对偶性与原—对偶

These cases may indicate that the separatrix frequency region of bidirectional electron beams are within a large braodband from 250 to 2900 MHz and the originating ranges are in a very small (4 to 100 MHz) and different bands.

介绍云南和北京天文台射电频谱仪观测到的3个对偶的米波--微波Ⅲ型爆发，估计了双向电子束起源的频率和高度，3个事件分别揭示了在正向和反向漂移爆发之间的分界频率(约为250，1300和2900 MHz)，它们指出了一个致密的电子加速源，在这个源中产生了向上和向下两个方向注入的电子束，从这些事例可以表明不同事件的双向电子束的分界频率有一个相当大的范围(250-2900 MHz)，而它们的起源范围却是在一个很小的(大约4-100 MHz)和不同的频段范围内。

Hence, when taking the equivalent duals of all former statements, one ends up with a number of theorems that equally apply to Boolean algebras, but where every occurrence of '' ideal '' is replaced by '' filter ''.

因此，当采用所有前者陈述的等价对偶的时候，你可能终结于同等的适用于布尔代数的定理，但是这里所有''理想''的出现都被替代为''滤子''。

With Death guitarist Schuldiner adopting vocal duties, the band made a major impact on the scene.

随着死亡的吉他手Schuldiner接受主唱的职务，乐队在现实中树立了重要的影响。

But he could still end up breakfasting on Swiss-government issue muesli because all six are accused of nicking around 45 million pounds they should have paid to FIFA.

不过他最后仍有可能沦为瑞士政府&议事餐桌&上的一道早餐，因为这所有六个人都被指控把本应支付给国际足联的大约4500万英镑骗了个精光。