Then we construct the corresponding transfer matrix to determine the rtt integrability of dirac oscillator and gain the conserved quantities of the system according to quantum determinant 接着构造出相应的整体转移矩阵，确定dirac谐振子在rtt意义下的量子可积性的问题，并由量子行列式确定体系的守恒量族。
Continuity , integrability and differentiability of riemann function are discussed ; especially , the non - differentiable properties on [ 0 , 1 ] are proved , and dirichlet ' s function is comparated with it 摘要从黎曼函数的简单特徵入手讨论它的连续性、可积性、可导性，特别是证明了黎曼函数在区间[ 0 ， 1 ]上处处不可导，并结合狄利克雷函数加以引申和推广。
At the same time , we try to analysis the relations between the special cases of dirac structure and the poisson structure or symplectic structure , moreover the sufficiency and necessary condition in which the dirac structure is integrable is given 定义了dirac结构的可积性，给出了dirac结构的可积性的充要条件即似dirac结构l完全可积充要条件是( l ， | l ， [ . , . ] )是李代数胚。
The content of the section two is about the dirac structure for lie algebroid . in this section the maximal isotropic subbundle , the characteristic pairs and the dual characteristic pairs are introduced . and some related conclusions are quoted directly 引入了极大迷向子丛，对偶特征对和特征对的概念，直接引用了已有的相关的结论，重要的是特征对尤其是对偶特征对在讨论极大迷向子丛可积性方面起着关键作用，二者是不可分割的。
The integrability conditions and coefficient conditions for the appearance of 5 and 6 limit cycles from the neighborhood of the equator are obtained . an example of cubic system with 6 limit cycles bifurcating from the equator is given for the first time 同时计算出系统的前6个赤道环量，得到了系统在赤道邻域的可积性条件及在赤道附近分支出5个和6个极限环的系数条件，从而首次给出了一个平面三次系统在赤道附近分支出6个极限环的计算实例
Second , we try to find another kind of realization of yangian so that we can study the symmetry of this system based on a different point of view . we find that there is yangian symmetry in dirac oscillator , as a result , we are able to shift one degenerate state to anther in the same energy level . then we construct the corresponding transfer matrix to determine the rtt integrability of dirac oscillator 然后，本文寻找dirac谐振子的另外一种yangian实现形式，从另一个角度研究该体系的对称性，从而表明dirac谐振子这里这种实现下具有yangian对称性，这样，我们可以实现在同一个能级的不同简并态之间的跃迁，再构造出相应的整体转移矩阵，确定dirac谐振子在rtt意义下的量子可积性的问题。
In this dissertation , with the aid of many types of constructive transformations and symbolic computation , some topics in nonlinear waves and integrable system are studied , including exact solutions , painleve integrability , backlund transformation , darboux transformation , symmetry ( similarity reduction ) , conditional symmetry , lax integrable hierarchy , liouville integrable n - hamilton structure , constraint flow , involutive system , lax representation , r - matrix , separation of variables and integrable couplings . chapter 2 and 3 are devoted to investigating exact solutions of nonlinear wave equations : firstly , the basic theories of c - d pair and c - d integrable system are presented 本文以构造性的变换及符号计算为工具，来研究非线性波和可积系统中的一些问题：精确解(如孤子解、周期解、有理解、 dromion解及compacton解等) 、 panileve可积性、 backlund变换、 darboux变换、对称(相似约化) 、条件对称、 lax可积族、 liouville可积的n - hamilton结构、约束流、对合系统、 lax表示、 r -矩阵、变量分离及可积的耦合系统
The problem of position control is studied in this paper , and the results of my work are as follow : in first the dynamic model of underactuated manipulator is established , the non - holonomic characters of the constraint is analysed , and then the judgement method that if constraint is integrable is gived . so it is proved that the constraint of the manipulator in the paper is un - integrable , and the dynamic system is nonholonomic 本文以该类机器人为对象，进行了位置控制的研究，主要完成以下几方面工作：建立了具有非驱动关节机器人的动力学数学模型，对模型中的约束条件，即非驱动臂的动态方程的可积性进行了分析，给出可积性的判别方法，同时证明了该模型的约束条件为非完全约束。
Equation ( 4 ) is said to belong to limit circle type if all solutions of equation ( 4 ) belong to l ~ ( 2 ) ( simply denoted by l . c . ) equation ( 4 ) is said to belong to lagrange stable if all solutions of equation ( 4 ) belong to ( simply denoted by l . s . ) . in chapter 4 , we study criteria for the linear nonhomogeneous differential equation belonging to the limit circle type 方程( ’ )称为极限圆型的，若方程( ’ )的所有解都属于护[ a ， co ) (简记为l . c . ) ;方程( ’ )称为拉格拉日稳定，若方程( ’ )的所有解均属于lco [ a ， co ) (简记为l . s . ) .由于方程( ’ )解的平方可积性及有界性的研究在微分算子理论、按微分方程的特征函数展开理论以及无界区间上受控系统的最佳控制理论等方面具有重要应用
With the help of successful experiences the space with - hereditarily closure - preserving cs * pair - networks and the space with - hereditarily closure - preserving psedobases have gained , this thesis makes a research of the space with - hereditarily closure - preserving pair - networks at the aspects of the yan pengfei ' s question of space with - hereditarily closure - preserving pair - networks and discusses the equivalent depiction of this space and the connection between this space and related generalized metric space and the topological properties of this space . at the same time , it make further consideration of the similar structure of space with - weakly hereditarily closure - preserving pair - networks 本文围绕燕鹏飞关于遗传闭包保持双网络空间的问题，借助具有遗传闭包保持cs ~ *双网络空间和具有遗传闭包保持伪基空间已取得的成功经验，对具有遗传闭包保持双网络的空间进行研究，探索该空间的等价刻画及与相关广义度量空间的联系，并由此讨论这类空间的可加性、遗华南师范人学硕十学位论文传性、可积性及映射性质等拓扑运算性质
