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1·Since this is a concave function, it's not just the expected value.

由于这是一个凹函数，它不仅仅是期望值。

2·The graph of a concave function is always below its tangent.

一个凹函数的图象总在它的切线的下方。

3·A branch and bound methods is proposed for minimizing concave function over a convex.

本文介绍了求解凸集上凹函数最优解的一种分支定界方法。

4·In this paper, concave function and its properties are discussed. And concave function's application in optimization are shown as well.

研究与凸函数对应的凹函数及其性质，并指出了凹函数在最优化中的应用。

5·There are convex function and concave function, Which are of expansive application in many aspects, especially in inequality proof and error estimate.

凸(凹)函数有很多特性，这些性质可广泛应用于不等式的证明及误差估计等方面。

6·On the effect boundary, risk is return's rigid monotony increase by degrees convex function , return is risk's rigid monotony increase by degrees concave function.

在有效前沿上，风险是收益的严格单调递增凸函数，收益是风险的严格单调递增凹函数。

7·There are natural cotton crystal, mineral, concave surface, the natural normal function of any minor damage, is not a quality problem.

天然水晶内有棉絮、矿质、冰裂、表面凹点、天然轻微破损都属正常现象，不属品质问题。

8·To gain sevrial decision method of convex (concave) function by the definition of convex(concave) function and Taylor formula.

利用凸（凹）函数的定义和泰勒公式给出了凸（凹）函数的几个判别法。

9·Show: If the firm's production function is concave, it's profit function is concave.

证明：如果一个公司的生产函数是凹函数，那么它的利润函数也是凹函数。

10·The production function is concave with decreasing returns.

生产函数是凹性的，边际报酬递减。