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The composite nonsmooth optimization problem is defined as follows:〓where h is a convex function defined on〓and is bounded from below,and〓〓is a mapping from〓to〓and〓are twicecontinuously differentiable function on〓.

复合NDO问题被定义如下：〓这里h是一个定义在〓上的凸函数，并且有下界，〓，〓是一个从〓到〓上的映照。

If the flow does not go to infinity, or the functional is bounded from below on the flow and satisfy the condition, then the limit of the flow maybe a critical point.

如果流线不走向无穷远处，或泛函在此流线上下方有界且满足条件，那么极限点就可能是临界点。

According to that Caristi fixed point theorem is equivalent to Ekeland variational principle, we weaken the condition that the functional is bounded from below in Ekeland variational principle, then we enlarge the area of the applicable functionals.

鉴于Caristi不动点定理和Ekeland变分原理的等价性，我们对Ekeland变分原理中泛函的下方有界的要求作了相应的削弱，从而扩大了适用泛函的范围。

In chapter four, we prove that a complete open Riemmannian manifold with Ricci curvature negatively lower bounded is of finite topological type provided that the conjugate radius is bounded from below by a positive constant and its excess is bounded by some function of its conjugate radius.

第四章，我们证明了对于Ricci曲率具负下界的完备开Riemannian流形，当其共轭半径有正下界且它的Excess被其共轭半径的某个函数所界定时，它就有有限拓扑型。

Lots of empirical studies show that the actual processesand results of investors'decision-making are very different from that described byMFT's optimal investment decision-making model because of judgement anddecision biases arising from investors'psychological biases and informationasymmetry , investors'bounded rationality below the MFT's requirement.

大量的实证研究表明，实际中由于投资者固有的一些心理偏差和信息不对称的存在，使得投资者的认知能力和判断能力以及判断的准确度往往存在偏差，远远低于经典现代金融理论对投资者的理性要求，投资者的实际决策过程和决策结果与经典现代金融理论的最优投资决策模型所描述的大相径庭。

FR conjugate gradient methods with perturbations are proposed. The global convergence property of the first method is proved under the condition of main directions' sufficient descent. Whereas, in the proof of the convergence for the other two methods, we only need main directions' descent. Importantly and quite interesting, boundedness conditions such as objective function being bounded below, boundedness of level set are not needed. Chapter 5 presents a version of Dai-Yuan conjugate gradient method with perturbations.

在主方向充分下降的条件下证明了第一个方法的全局收敛性，而后两个方法的收敛性是在主方向下降的条件下证明的，这些收敛性证明的一个共同特征就是不需要目标函数有下界或水平集有界等有界性条件，第5章采用Wolfe或Armijo步长规则提出了带扰动项的Dai-Yuanabbr。

This result can best be viewed as a quantum black hole version of the third law of thermodynamics. As a result of this area spectrum, the spectrum of M is discrete, bounded below, and can be made positive. Prom the physical point of view, the semiboundedness and positivity of the spectrum are very satisfying results: The semi-boundedness of the spectrum implies that one cannot extract an infinite amount of energy from the system, whereas the positivity of the spectrum is in agreement with the well-known positive-energy theorems of general relativity.

同时根据EMDA黑洞动力学，存在与面积谱对应的大于零且有下界的质量离散谱，从物理上来看，质量谱的半有界性意味着不可能从系统吸取无限的能量，而正定性则与广义相对论中的正能定理一致，也就说稳态轴对称EMDA时空的ADM能量总是大于零。

Suppose that f is a real-valued ω〓-lower semicontinuous function defined on a ω〓closed bounded subset A〓 of the dual E〓 of a locally convex space E, which is bounded below on A〓, and suppose every ω〓 closed convex subset of C〓≡ω〓-clcoA〓 is the ω〓closed convex hull of its w〓-β-exposed points.

假设X是一个局部凸空间，X〓是其对偶。A〓是X〓上的ω〓有界闭子集，f是A〓上的ω〓-下半连续实值函数且在A〓上下有界。

At present home appeared bourg bank, finance company and small finance company form of 3 kinds of new-style organizations, the article plans to combine the experience practice of lukewarm state city, put forward Chengdu area to organize the tentative plan that problem of financing of medium and small businesses solves below frame in these 3 kinds.

目前国内出现了村镇银行、贷款公司和小额贷款公司三种新型组织形式，本文拟结合温州市的经验做法，提出了成都地区在这三种组织框架下解决中小企业融资新问题的设想。

Below a lot of circumstances, real competition appears after your Idea generated the bow wave of the market, once your Idea earned big money, can draw large quantities of one great robber, great robber people went secretly how is your Idea met again?

在很多情况下，真正的竞争出现在你的Idea产生了市场的冲击波之后，一旦你的Idea赚了大钱，便会引来一大批大盗，大盗们偷去了你的Idea又会怎么样呢？

In contrast to the ubiquitous rising-sun-with-rays military flag of the Japanese, Chinese banners and ensigns feature a range of designs.

与遍地都是的太阳军旗不同，中国人的旗帜和徽章设计得各式各样。

From their small corner of Feng's Guangzhou headquarters -- a jumble of pink leashes, squeezable rubber steaks, and plastic doggy Santas for Fido's stocking -- Soleil's designers come up with at least five new products a month.

从Feng 设在广州总部的产品展示柜台上可以看到，Soleil的设计师每月至少设计出5件新产品。