代数曲线

As the center or the Algebra subalgebra of a algebra plays an important role in reducing the number of state variables, the cartan subalgebra is a maximum Abelian subalgebra of an algebra, so it can be a powerful tool for the structure simplification of control systems.

由于代数的中心，或交换子代数在系统的变量降低方面具有重要作用，而Cartan子代数是李代数的极大交换子代数，因而它能成为简化系统结构的有力工具，第二部分利用向量场的积分曲线所定义的映射，构造出一个S—坐标，在该坐标下任一向量场都有一个标准的表示方法，一种无穷级数表示方法。

For a general parametric curve/surface, we usually cannot compute its exact implicit form. Even though its exact implicit form can be computed, the curve/surface implicitization involves relatively complicated computation and the degree is higher. Moreover, it may have unexpected components and self-intersections. All these unsatisfied properties limit the applications of the exact implicitization. So finding curve/surface approximate implicitization has become a practical problem. In this paper, we present an algorithm to solve the approximate implicitization of a given parametric curve by using a quadratic algebraic spline curve.

由于精确隐式化过程不一定可以实现，即使可以实现隐式曲线曲面的阶数高计算复杂，并且具有不希望的自交点和奇异分支，从而限制了隐式化的运用，所以寻求参数曲线曲面的近似隐式化问题成为很实际又重要的问题，提出利用二次代数样条曲线来实现一般平面参数曲线近似隐式化的一种算法。

Discusses algebraically and geometrically the properties of the ruled surfaces in the Euclidean three-space, especially, the principal normal surfaces on which there are some special curves, such as Mannheim curves, Bertrand curves and general helices.

在三维欧氏空间中，作为特殊曲线，Mannheim曲线、Bertrand曲线以及一般螺线具有良好的几何和代数性质。讨论了三维欧氏空间中特殊曲线的主法线曲面。

This article takes the teaching of conic sections as an example. By designing worksheets, teachers can introduce the historical material about conic sections to students. By way of using Apollonius' definition of parabola, ellipse and hyperbola, teachers can introduce the geometric aspect of "conic section" to students. By using the concept of " latus rectum " in Conics , we can connect "conic sections"-- representation of geometrical aspect, with "the equation of conic sections"-- representation of algebraic aspect to improving insufficiency of text books.

同时本文也试著从历史文本中寻找材料，简单举例说明数学教师可以如何应用这些史料在几何单元教学上，例如三角函数的正余弦定理，最后再以圆锥曲线的正焦弦为例，说明如何利用数学史料於此单元的教学，尤其是阿波罗尼斯的《锥线论》中对圆锥曲线的3个命题，将此3个命题的内容与意涵，尤其是正焦弦在圆锥曲线的几何意义上所扮演的角色，将其适当地融入教学中，将可使学生真正学习圆锥曲线的几何知识，而不再只是代数形式的几何知识。

Chapter 1 briefs the relation between invariance and computer vision and summarizes the research and application of invariance in computer vision. Chapter 2 first derives the transformations of three camera models, then makes the correpondences between the models and three typical geometrical transformation groups by analysing the transformations respectively. The correspondences supply the theoretical basis for applying geometrical invariants to resolve the problems of computer vision. In Chapter 3, we describe the geometrical invariant theory and prove some geometrical invariants of coplanar points, lines or conics by algebraic method. In order to use the invariants of conic pairs to describe general 2D shapes, we discuss the perspectively invariant representation of planar curves using conies in detail. A system consisted of two TMS320C25 and based on moment invariants is introduced in Chapter 5. The system can recognize more than 30 different shapes of object model or more than 10 plane models with similar shape in real time.

第一章简述了不变性与计算机视觉的关系，以及计算机视觉中的不变性研究和应用概况；第二章推导了计算机视觉中常用三种投影模型的变换关系，通过对这三种变换关系的分析，分别建立了这三种投影模型和几何学中的三种变换群之间的一一对应关系，为几何不变性在计算机视觉中的应用提供了理论基础；在第三章中，我们介绍了几何不变性的理论，并且用代数方法证明了共面点、直线、二次曲线的几何不变量和射影不变量；为了把二次曲线的不变量用于一般二维形状描述，在第四章中我们详细地讨论了用二次曲线实现一般平面曲线的透视不变性表示的方法；第五章介绍了用两片TMS320C25构成的、基于不变矩形特征的运动目标实时识别系统。

Proceed from general conclusion of algebra on the finite field, study this special algebra system of Elliptic Curve on the finite field, Discuss that sets up several theories of the Public Key Cryptosystems and the problems of implementation on it.

从有限域上的代数的一般结论出发，研究有限域上的椭圆曲线这一特殊的代数系统，讨论在其上建立公钥密码体制的若干理论和实现问题。

The Weil Descent's algebraic method for DLP in the Jacobian of hyperelliptic curves which generalized by Galbraith is analyzed, and the problem of whether or not the Weil Descent'method can be used to attack the HCDLP with the form of y〓+xy=f defined over GF is discussed in detail.

研究了Galbraith提出的对超椭圆曲线的离散对数问题的WeilDescent代数攻击方法，对定义在GF上的形如y〓+xy=f的HCDLP能否用Weil Descent代数方法攻击作了详细讨论。

In this paper,a rational approximation method of the norm of parametric speed is given,which is more effective than the polynomial approximation.

文章根据等距曲线逼近的关键在于对其参数速度模的逼近，提出了对参数速度模有理逼近的2种形式，在此基础上给出了平面Bézier曲线等距曲线的逼近函数和误差估计函数，并导出了保持法矢平移方向的代数方式的等距有理逼近算法。

For the regular curves, we find two Killing fields for the purpose of integrating the structural equations of the p-elastic curves and express the p-elastica by quadratures in a system of cy...

对于正则曲线的情形，我们发现了两个用于求解p-弹性曲线的结构方程的Killing向量场并用积分将p-弹性曲线在一个柱面坐标系中表示出来，而对仿射星形曲线的情形，我们用积分方法解出了欧拉-拉格朗日方程，利用Killing向量场及线性李代数s1(2，R)、s1(3，R)和s1(4，R)的分类将高阶结构方程降为一阶线性方程，因此我们用积分完全解出了中心仿射p-弹性曲线。

For the regular curves, we find two Killing fields for the purpose of integrating the structural equations of the p-elastic curves and express the p-elastica by quadratures in a syste...

对于正则曲线的情形，我们发现了两个用于求解p-弹性曲线的结构方程的Killing向量场并用积分将p-弹性曲线在一个柱面坐标系中表示出来，而对仿射星形曲线的情形，我们用积分方法解出了欧拉-拉格朗日方程，利用Killing向量场及线性李代数s1(2，R)、s1(3，R)和s1(4，R)的分类将高阶结构方程降为一阶线性方程，因此我们用积分完全解出了中心仿射p-弹性曲线。

I'm strongly against the death penalty — it's an eye for an eye.

我不赞成死刑——这是以牙还牙的报复行为。

And to get you the support you need, we're enlisting all elements of our national power: our diplomacy and development, our economic might and our moral suasion, so that you and the rest of our military do not bear the burden of our security alone.

并给你们所须的支援，我们正徵召国家所有各种的力量：我们的外交及发展，我们的经济力量与道德劝说，所以你们与其他军人不须要孤独地负起国家安全的责任。