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Chapter 5 is devoted the study of fuzzy optimization in fuzzy convex analysis structure given in chapter 4. We consider the problems of minimizing and maximizing a convex fuzzy mapping over a convex set and develop necessary and/or sufficient optimality conditions.

在第5章中，首先，在第4章所建立模糊凸分析的结构基础上，考虑了凸模糊映射的极值问题，得到了凸模糊映射取得极值的充分/必要条件。

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。

The polar form is used as the representation for one of the composite functions, whereas the Bernstein representation is used for another function. At the last part of this chapter, we consider the problem about generalized Bezier surface, which is defined in arbitrary convex polygon.

我们利用函数复合的思想去处理这个问题，被复合的两个函数一个用Polar形式表示，另外一个用Bernstein基形式表示，这样就可以充分利用Polar形式的多元仿射性，直接生成新的控制顶点，简化了证明过程。

Besides merits of the two dimensional generation of random aggregate model, It consider the effect of the radius of the different aggregates, which were assumed to obey normal distribution. Consequently, the numerical model generated is reasonable.5. The 3-D random model of convex polyhedron aggregate was generated with 3D-RAS.

三维混凝土随机骨料投放算法是二维投放算法的延伸，除了具有二维算法的一切优点外，还考虑了骨料粒径的变化，假定骨料粒径服从正态分布，从而使该软件生成的随机骨料模型和实际结构更为接近。5。

In fact, the former only consider the strict convex programming, while the latter considers the convex programming.

支持向量机中有一类重要的变形方法，虽然很有效，但缺乏相应的统计学习理论基础。

In this paper, we consider the optimal control problem of time-delayed forward-backward stochastic system when control set is convex.

本文研究了在控制集为凸集的情况下带时滞的正倒向随机控制系统的最优控制问题。

If it is out of session and it would be a delay to wait for the opening of an ordinary session, the Council of Ministers must proceed to convoke an extraordinary session of the National Assembly in order to consider whether to approve or disapprove the Emergency Decree without delay.

如果是的会议，那将是一个毫不拖延地等待开放的一个普通会议上，部长理事会必须召集一次特别会议国民大会，以考虑是否批准或不批准的紧急法令不延迟。

If you consider so, Copenhagen is certain to alter your mind.

假如你是这么以为的话，哥本哈根确定能够改变你的想法。

Some socialists consider language as the copestone of culture---without language, no culture; in the other hand, language is influenced by culture, reflecting culture at the same time.

有些社会学家认为，语言是文化的冠石---没有语言，就没有文化；从另一方面看，语言又受文化的影响，反映文化。

Obviously, as I said before you have to consider copyright issues.

显然，我已经说过，你要考虑版权问题。

The absorption and distribution of chromium were studied in ryeusing nutrient culture technique and pot experiment.

采用不同浓度K2CrO4(0，0.4，0.8和1.2 mmol/L)的Hoagland营养液处理黑麦幼苗，测定铬在黑麦体内的亚细胞分布、铬化学形态及不同部位的积累。

By analyzing theory foundation of mathematical morphology in the digital image processing, researching morphology arithmetic of the binary Image, discussing two basic forms for the least structure element: dilation and erosion.

通过分析数学形态学在图像中的理论基础，研究二值图像的形态分析算法，探讨最小结构元素的两种基本形态：膨胀和腐蚀；分析了数学形态学复杂算法的基本原理，把数学形态学的部分并行处理理念引入到家实际应用中。