大代数

Group is an important concept in Modem Algebra. Viewed from a different angle it may be divided into two different categories: finite group and infinite group; and Abelian group and non-Abelian group.

群是《近世代数》的一个重要概念，从不同的角度出发，群可以分为有限群和无限群两大类，又可以分为交换群和非交换群两大类。

Methods of algebraic geometry rely heavily on sheaf theory and other parts of homological algebra .

代数几何方法在很大程度上依赖束理论和其他地区的同调代数。

We find serval conditions that are equivalent to 2-tournament matrices being singular,give the bound for the maximum algebraic multiplicity of 0 as an eigenvaulle of n order2-tournament,obtion the spectral radius of a singular 2-tournament matrix of order nis bounded above by n-2,and investigate 2-tournament matrix achieving this bound.

第五章讨论了2-竞赛矩阵的奇异性和特征值之间的关系，得到了2-竞赛矩阵奇异的充要条件，给出了2-竞赛矩阵的零化向量的刻划，得到了得分向量，3-圈数目与奇异性的关系；给出了2-竞赛矩阵的特征值0的代数重数的上下界，得到了奇异的2-竞赛矩阵谱根的最大值及谱根达到最大值的竞赛矩阵的得分向量的特征。

This paper reviews the historical development of the algebraic solutions of polynomial equations, introduces the life and scientific achievements of Cardano together with the historical background of Ars Magna. Based on this, the paper makes an elaborate study on Ars Magna.

本文首先回顾了多项式方程代数解法的发展过程，介绍了卡尔达诺的生平、科学与数学成就以及《大术》的历史背景，然后在此基础上详细研究了《大术》各章的内容。

Chapter 9: We report a large quantity of numerical experiments of 13 different algebraic multigrid algorithms for solving the Poisson equation, anisotropic equation, equation with cross-derivative terms, general matrix problems with large off-diagonal positive entries, biharmonic equation, Toeplity matrix, elasticity systems, finite element discretization of the Laplacian and even 3D problems. Particular attention is focused on asymptotic convergence factors and CPU-time consumed. Numerical results for many different types of practical problems demonstrate the efficiency and robustness of the proposed algebraic multigrid methods.

第九章：在各种代数多重网格算法的基础上，进行了大量的数值试验，具体给出了十三种不同的代数多重网格方法求解泊松方程，各向异性方程，带混合导数项的方程，带有大的非对角正元素的一般矩阵问题，重调和方程，托普利兹矩阵，弹性力学方程组，拉普拉斯算子的有限元离散，甚至三维问题的较为丰富的数值结果，重点关注它们的渐近收敛因子和所需的CPU时间，来源于不同类型问题的计算结果既为代数多重网格理论分析和算法的改进提供了很实用的资料，同时也证实了本文给出的代数多重网格算法的效绩和稳健性。

By using the solutions of a new auxiliary elliptic equation,a direct algebraic method is proposed to construct the exact solutions of some nonlinear evolution equations.The main difference between this method and previous auxiliary elliptic equation methods is that the balance order becomes smaller after using the new auxiliary elliptic equation.Therefore,the derived algebraic equations are greatly simplified.

利用一个新的辅助椭圆方程将求解非线性发展方程精确解的问题转化为一个代数方程组进行求解，与已有的辅助椭圆方程法的主要不同是，应用这一新的辅助椭圆方程后降低了平衡次数，减少了所得的代数方程组的个数和方程的项数，从而大大地简化了代数方程组的求解。

Symmetry group is a connected compact Lie group, the system has also a fine cascade decomposition.

该章正是从这一角度出发，通过参数确定了线性控制系统的对称代数，而且构造出的对称代数其维数是最大的。

Based on the Lie algebra that describes the symmetries, it is proved in this dissertation that a sufficiently large class of control Hamiltonians can be provided by the universal enveloping algebra of the Lie algebra, on which the quantum mechanical control systems can be modelled.

基于描述量子力学系统对称性的李代数结构，本文证明该李代数的泛包络代数提供了足够大的一类控制哈密顿量，因而可以在其上建立量子力学控制系统的模型。

Together with some basic knowledge of Linear Algebra, we prove that there exists a polynomial time algorithm for finding the shortest contractible cycle in an locally LEW-embedded graph. 3. By some results of Minor Theorem, we prove the crossing numbers of some circular graphs C(10,4), C(9,3), C(8,3) on the projective plane.

利用C.Thomassen在大边宽嵌入方面的工作，得到了局部大边宽嵌入图的一些性质，再利用线性代数的知识，证明了在局部大边宽条件下存在多项式时间算法可以找到一个图的最短可收缩圈。

Its main goal is to explore information in the K-theory groups of the index C*-algebras, the Roe algebras C*, by using the large-scale geometrical structure of proper metric spaces, including noncompact complete Riemannian manifolds, finitely generated groups, etc., so as to establish connections among geometry, topology and analysis of the geometric spaces, and furthermore, to solve other relating problems, say, the Novikov conjecture, the Gromov-Lawson-Rosenberg conjecture on positive scalar curvature, the idempotent problem in the theory of C*-algebras.

粗几何上的指标理论是"非交换几何"领域九十年代以来发展起来的重要研究方向，它孕育于非紧流形上的指标理论，其主要目标是通过几何空间（如非紧完备黎曼流形、有限生成群等）的大尺度几何结构探索指标代数，即 Roe代数，的K-理论群的信息，从而建立几何空间的几何、拓扑与分析之间的联系，并应用于解决其他重要问题，如Novikov猜测、Gromov-Lawson-Rosenberg正标量曲率猜测、群C*-代数幂等元问题等。

And Pharaoh spoke to Joseph, saying, Your father and your brothers have come to you.

47：5 法老对约瑟说，你父亲和你弟兄们到你这里来了。

Additionally, the approximate flattening of surface strip using lines linking midpoints on perpendicular lines between geodesic curves and the unconditional extreme value method are discussed.

提出了用测地线方程、曲面上两点间短程线来计算膜结构曲面测地线的方法，同时，采用测地线间垂线的中点连线和用无约束极值法进行空间条状曲面近似展开的分析。