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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。

Plus neutrons, in the nucleus cloud chamber photograph, an unknown nucleus collides with a 44He nucleus, and after the collision tHe nucleus, and after the collision the two he two particles nuclei nuclei travel in perpendicular directions relative to each other.travel in perpendicular directions relative

如右图所示，在如右图所示，在44HeHe（左上角的数字为核子数，即在核子中的质子和中子数之和）云室照片中，一个未知核子与（左上角的数位为核子数，即在核子中的质子和中子数之和）云室照片中，一个未知核子与44HeHe核子碰撞，碰撞后两个核子以彼此垂直的方向运动。

And a viviparous quadruped bends his limbs in opposite directions to a man's, and in opposite directions to one another; for he has his forelegs bent convexly, his hind legs concavely.

一胎生四足动物和相反的方向弯曲，四肢男人的，而在相反的方向彼此，因为他有他的前肢弯曲convexly，他的后腿concavely。

Based on wave propagating in three directions of elastic foundation, vibrating displacement could be assumed to be in three forms along three directions. Hooke's law and theory of waves were adopted to deduce the formula of coupled viscous boundary. The coupled viscous boundary could be applied in finite element method.

根据波在地基的3个方向都可以进行传播的特点，假定了地基中质点沿3个方向的振动表达式，通过胡克定律和波动理论，推导了弹性地基中三维动力问题的耦合粘性边界的公式。

To determinate volumes and directions of the principle stresses, constantly three 1/4-bridges should be applied to measure normal strains along three sensitive grids. Based on the analysis of the plane stress state and generalized Hooke's law, volumes and directions of the principle stresses at one point will be obtained.

在该实验中，为了测量构件表面一点的主应力大小及方向，通常采用3个1/4桥路，测量该点沿3个敏感栅方向的线应变，根据平面应变状态分析和广义胡克定律求得该点的主应力大小及方向。

The Carrier, Master and ship shall have liberty to comply with any orders or directions as to loading, departure, arrival, routes, ports of call, stoppage, discharge, destination, delivery or otherwise howsoever given by the government of any nation or department thereof or any person acting or purporting to act with the authority of such government or any department thereof, or by any committee or person having, under the terms of war risk insurance on the ship, the right to give such orders or directions.

五、承运人、船长和船舶有权遵守任何国家政府或政府部门或代表或旨在代表政府或政府部门行使职权的任何个人所作出的有关装货、离港、抵达、航线、停靠港口、停航、卸货、目的地、交货或其他事项的命令或指令，或任何委员会或个人根据船舶战争风险保险条款规定而有权作出的此种命令或指令。

The Carrier, Master and .ship shall have liberty to comply with any orders or directions as to loading, departure, arrival, routes, ports of call, stoppage, discharge, destination, delivery or otherwise howsoever given by the government of any nation or department thereof or any person actingor purporting to act with the authority of such government or any department thereof, or by any .committee or person having, under the terms of war risk insurance on the ship, the right to give such orders or directions.

五、承运人、船长和船舶有权遵守任何国家政府或政府部门或代表或旨在代表政府或政府部门行使职权的任何个人所作出的有关装货、离港、抵达；航线、停靠港口、停航、卸货、目的地、交货或其他事项的命令或指令，或任何委员会或个人根据船舶战争风险保险条款规定而有权作出的此种命令或指令。

In designing process, it will waste time on determining parting directions. For this reason, we generate a method for checking partibility by using the relations of the normal vectors of shapes. If the mold was decided as partible object, the process would evaluating the range of the parting directions. Also, the result of parting directions will be demostrated by one cone named parting cone. The total procedures of determining partibililty require O time.

基於这样的原因，本文提出一个判断可脱模性的方法，利用自由曲面上三角化之后所得到的法向量的分布情形，进行运算，若判断为可脱模模型，并计算出可脱模模型的脱模范围，使用圆锥表示之，此圆锥既称为脱模角，过程所需的时间复杂度为O；若判断为不可脱模模型则所需要的时间复杂度为O。

Studies on spatial heteroplasmy of Saperda Papulnea L. population from different directions indicated that the model of semivariogram of eggs, larvae and adults in four directions are the same, but spatial dependent range are different, belonging to geometrical anisotropy. They had stronger continuity in the direction of 0° and 45° than the other two directions.

对不同方向上青杨天牛种群空间异质性研究表明：天牛卵、幼虫和成虫在四个方向上的半变异函数拟合的模型相同，但空间依赖范围却不同，属几何性的各向异性。

Eight-direction Line Diagram is formed by The Thirteen Postures, including eight directions, four normal directions of martial arts: east, west, south, north, and four corners directions: northeast, northwest, southeast, southwest, combining with five steps: advancing, retreating, look to the left, look to the right, central equilibrium, The circular Eight-direction Line Diagram is from a square, forming from the rim of a circle through geometry principle.

八方线图是从武术方向方位的四正位：东、西、南、北，四偶位：东北、西北、东南、西南、及前进、后退、左顾、右盼、中定等八门五步十三势组成，用几何原理以外接园从正方变化为圆形八方线图。太极拳十分重视方向方位及中正安舒，有了八方线，在练拳时，有一个准确的中心点，方向方位准确无误，避免脚步下出现&差之毫厘，谬以千里&的错误。

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失