虚根

第一部分主要讨论了Kac-Moody代数中的一类基本问题，即给定一个实根或虚根，其对应的实根向量和虚根向量该如何表示？

Part I is mainly discussing an elementary problem in Kac-Moody Algebra : how to describe the real and imaginary root vectors corresponding to a given real or imaginary root ?

一个子准多项式所对应虚根的穿越方向根据这个子准多项式所含的虚根的大小顺序决定。

The crossing direction of an imaginary root can be directly determined by the magnitude order of the imaginary roots of the factored quasi-polynomial where it belongs .

并在此基础上讨论了C（2）型李代数的虚根性质。

Properties of imaginary roots of the C ( 2 ) type Lie algebras are also discussed .

不定型Kac－Moody代数IB（n（a））的虚根系

The Imaginary Root System of Indefinite Kac-Moody Algebra of IB_ ( n ( a ) Type

广义Kac-Moody代数的虚根与权

Imaginary Roots and Weights for Generalized Kac-Moody Algebras Exploratory right

给出了秩为2的广义Kac-Moody代数的虚根系具体刻画，讨论了其虚根所决定的反射与其Weyl群之间的联系。

Given the set of imaginary roots of a class of rank 2 generalized Kac Moody algebras .

同时，利用这些结论，在具体的应用中可以得到该类方程的纯虚根存在的充要条件、横截条件及Hopf分支的存在性，因此，具有重要的应用价值。

Based on these conclusions , it expounds the existence of the pure imaginary roots , the transversal condition , and the existence of Hopf bifurcation for this class equation . So it is important significance in practical application .

虚根是Kac&Moody代数的重要概念。

The imaginary roots of Kac Moody algebras are the important concept .

C~（2）型李代数虚根集的刻划

The root system of c ￣( 2 ) lie algebra

一类双曲型Kac&Moody代数的特殊虚根系

The Special Imaginary Root System for a Class of Hyperbolic Kac Moody Algebras

怎样解出虚根的值呢？

How do you get an imaginary solution ?

根据这些性质我们知道，时滞系统的虚根穿越情况可以直接由系统的结构信息得到。

According to these properties we can directly determine the crossing directions via the structure information .

但是由于用试凑的方法解决起来比较麻烦，而且用他们的方法所得出的稳定性条件也不够完全，只能解决问题的纯虚根的情形，零根的情形却未考虑。

But it has trouble to solve the problems by means of cut and try , moreover .

详细的说，这里的结构信息指哪些特征虚根属于同一个子准多项式。

More specifically , here the structure information means which imaginary roots belong to one factored quasi-polynomial .

首先研究了一般形式的时滞系统虚根穿越情况并得到一些重要的性质。

First , we study the crossing directions of the imaginary roots and some important properties are obtained .

进一步，我们可以断定一个一般形式的时不变参数时滞系统随时滞的增加将最终不稳定，如果该系统含有至少一个特征虚根。

Moreover , we can confirm that a time invariant delayed system will be ultimately unstable as delay increases if there exists at least one imaginary root .

经线性化处理后用状态空间法求出系统特征方程的根&拉普拉斯算子实根（衰减系数）和虚根（阻尼自然频率）。

After linearizing them , the roots of the characteristic equation is derived , roots in laplacian & ( attenuation coefficient ) and the imaginary-roots ( damping frequency ) .

如果第一列区间数以零为下界，则特征方程有成对虚根，系统临界稳定。

If the first row interval numbers take zero as a Lower bound , then the character equation has pairs of conjugated imaginary radixes , and the system is in critical stability .

两个复参数和它们的共轭复数为源于变形协调条件的特征方程的四个根。对于理想正交各向异性弹性材料，该特征方程只有纯虚根和复根两种形式。

The two parameters and their conjugates being four roots of a characteristic equation emanating from the compatibility condition could be purely imaginary numbers or complex numbers for any ideal orthotropic elastic material .

针对双零加一对纯虚根特征值系统高维分岔问题，用矩阵表示法研究其系统的最简规范形。

In order to analyze the high order bifurcation of system , the simplest normal form ( SNF ) for the singularity of a pure imaginary pair and double zero eigenvalues was studied by using matrix representation method .

考虑到参数变动下的重复仿真，对耦合块依赖图进行分层处理，生成对应于变动参数集的变动子图；通过施加虚根节点将变动子图转化成最小求解树结构。

Considering the efficiency of repetitious simulation after parameters varying , the altering sub-graph of the altering parameters set is built after hierarchically processing the coupling blocks graph . And the sub-graph is converted into a minimum solving tree by adding a virtual root node .