向量的

In the second chapter, elementary theories and ideas of one step forming inverse approach were described and warped modification method, implemented by Cook, was introduced into one step forming simulation and based on idea of surface spreading; new quadrilateral element modification arithmetic was put forward. By spreading warped quadrilateral onto plane in local coordinate system, a plane quadrilateral element whose shape and area were equal to original warped quadrilateral element was gotten and internal force vectors of four nodes and stiffness matrix of elements were improved to increase precision of simulations on this plane element.

第二章从一步成形逆有限元法的基本理论和思想出发，将Cook基于弹性理论的四边形单元的翘曲修正算法引入基于塑性的一步成形，并且基于曲面展开的思想，本文提出一种新的四节点单元翘曲修正方案，通过把翘曲的单元展开的方法在局部参考系中得到一个与实际翘曲单元的形状与面积非常接近的平面单元，在该平面单元中进行单元刚度及内力向量的计算，进一步改善模拟结果的精度。

Also in the 3-D space, we assign the 20 amino acids to 20 vertices of the dodecahedron. By the symmetry of the dodecahedron we obtain 3-D representation of 20 amino acids, and 3-D graphical representation and the corresponding numerical sequence of protein sequences. And similarity and dissimilarity analysis based on the invariants of graphs and characteristics of numerical sequences are given for nine RNA secondary structures of RNA-3 of virus. We construct sequence phylogenetic tree of a group of cytochromes C protein.

在DNA三联体密码子表示的基础上，在半复平面上给出了蛋白质序列的非退化的2-D图形表示，同时利用复向量的主要特征—模和相位，给出了蛋白质序列的一种数值刻划，进一步在3-D空间里，把20种氨基酸分别分配给正12面体的20个顶点，根据正12面体的对称性得到了20种氨基酸的3-D表示，进而得到了蛋白质序列的3-D图形表示和对应的数字序列，并利用图的不变量和数字序列的特征比较了9种动物的神经元

Matrices, Vector and Vector Calculus, Newtonian Mechanics-single Particle, Oscillations, Nonlinear Oscillations and Chaos, Gravitation, Some Methods in the Calculus of Variations, Hamilton's Principles Lagrangian and Hamiltonian Dynamics, Central-Force Motion, Dynamics of a System of Particles, Motion in a Noninertia Reference Frame, Dynamics of Rigid Body.

矩阵和向量的计算、单一质点的牛顿力学、线性与非线性的振动运动、重力、微积分上的变分法介绍、哈密顿原理、拉氏及哈氏力学、连心力下的运动、质点系的运动力学、在非惯性坐标中的运动、刚体的运动。

According to the definition of an interval number complementary judgment matrix, the definition of complete consistency and satisfactory were introduced.

根据区间数判断矩阵的概念给出了区间数判断矩阵的完全一致性定义，通过对完全一致性性质的讨论演绎出区间数判断矩阵的一致性和满意一致性定义及其判别方法，并且研究了一致性和满意一致性定义的合理性，提出了求解区间数判断矩阵排序向量的递推排序方法，最后用算例验证了方法的有效性。

In the third part, we study an application of stereographic projection transformation in statistics, deriving Box-Muller formula (use two independent uniform random number to generate two independent standard normal random number) and spherical symmetric distribution (a product of a positive random variable and a random vector with uniform distribution) in multivariate statistical analysis.

第3章，研究了球极投影变换在统计中的应用，通过球极投影变换获得了利用两个相互独立的均匀随机数生成两个相互独立的标准正态随机数的Box-Muller公式以及多元统计分析中的球对称分布（一个正的随机变量和一个与之相互独立的均匀分布的随机向量的乘积）。

After illustrating the conception pseudo-symplectry ,the evolvement and extension of the conception symplectry and the formation of pseudo-symplectic space ,the nature of pseudo-symplectic space and the inner relationship and differences between pseudo-symplectic space and symplectic space are reflected here.

通过对辛空间的深入分析和研究，从空间中的度量与向量的迷向性问题入手并展开较为详细的讨论，定义"伪辛"的概念，阐述"辛"概念的演变、扩充与伪辛空间生成的事实，进而揭示伪辛空间的本质属性及其与辛空间的内在联系与区别。

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。

There are many situations where observed data are in the form of direction cosines or vectors but with an unknown positive scalar so that only the direction is known.

实际中常有这样的情形，当我们观察到的数据是以方向的形式或者以向量的形式给出但却具有未知的度量单位，因而只知道它们的方向，这时我们观察到的就是球面数据。

An intelligent fault detection and diagnosis method based on space vector is presented, fault diagnosis knowledge self-acquisition and self-tuning and symptom optimum algorithm are infered. 4、An early fault detection and diagnosis method based on ANN for process industry is presented, which has perfect self-learning and self-tuning ability and can establish diagnosis model automatically with the process of fault diagnosis in the absence of sufficient prior knowledge.

提出一种基于特征空间向量的智能故障诊断方法，推导了其故障诊断知识的自动获取算法和故障诊断的自适应算法以及征兆信号优化选取算法； 4、提出一种基于人工神经网络的生产过程早期故障检测与诊断方法，该故障诊断方法知识获取容易，具有良好的自学习和自适应能力，能够在故障先验知识不完备的情况下，通过与用户的交互，逐步自动建立起完善的故障模型知识。

The universal expression of Laue equation and two theorems on reciprocal space and reciprocal vector are obtained and proved.

讨论和研究了干涉函数对衍射的影响，提出并证明了Laue方程的一般表达式和有关倒易空间与倒易向量的2个定理，研究了作为三维晶体特例的二维晶体的衍射，讨论了二维晶体倒易点阵的物理内容和点阵点的权重。

Breath, muscle contraction of the buttocks; arch body, as far as possible to hold his head, right leg straight towards the ceiling (peg-leg knee in order to avoid muscle tension).

呼气，收缩臀部肌肉；拱起身体，尽量抬起头来，右腿伸直朝向天花板（膝微屈，以避免肌肉紧张）。

The cost of moving grain food products was unchanged from May, but year over year are up 8%.

粮食产品的运输费用与5月份相比没有变化，但却比去年同期高8％。