The orthogonality basis of geometric algebra is used to convert the quantum master equation into state space model ; the equation is divided into several parts matrix is analyzed to show its compact on the evolution and the dynamics of the system 本文从量子主方程模型出发，选取一组合适的完备正交基将主方程模型转化为实向量空间上的利于设计控制方案的状态空间模型，并基于此模型针对一个典型的二能级量子系统设计最优控制方案，仿真结果验证了方案的有效性。
( 2 ) in the study of multivariate phenomena , each time variable should possess its own scaling parameter in order to allow maximal flexibility in time - frequency . the notion of multifrequency multifunction wavelets , to be introduced , is based on this point of view . multifrequency wavelets , via directional multiresolution analysis , generated by a single function is extended to multifrequency multifunction wavelets generated by a finite number of functions ( 2 )在研究多变量问题时，为了使时频分析具有最大的灵活性，要求每个时间变量都有它自己的尺度参数，鉴于此，本文从尺度函数构成正交基或riesz基出发，把一维多函数小波推广到二维多频率多函数小波，解决了构造正交或双正交多频率多函数小波所需要的理论依据。
Expanding the system ' s hamiltonian to a real symmetric matrix in an appropriate orthogonal basis vector and then diagonalizing it , we get the energy spectrum of the system and calculate the energy level spacing distribution function and the spectral rigidity . then we study the statistical character of the energy spectrum under the changing of magnetic field intension and find that the system ' s motion transfers from regular to chaos gradually 将系统的哈密顿量在一个适当的正交基矢下展开并对角化，得到系统的能谱，分别计算系统能谱的能级间距分布函数和谱刚度，研究了该系统量子能谱的统计特征随磁感应强度大小的改变而表现出的系统运动由规则到混沌的渐进变化。
In detail , a class of cardinal orthogonal scaling functions , which are abbreviated as cosf , are constructed in the third section , and are proved that they have the properties of decay ; in the fourth section , the regularities of these cosf are studied in the generalized condition ; under the two special cases , the regularities and the decays of these cosf are discussed in the fifth and sixth sections , respectively 具体为：第三节构造了一类正交基插值尺度函数(简称cosf ) ，并证明它们具有指数衰减性；第四节调查了一般情况下这类cosf的光滑性；第五节、第六节分别讨论了两种情况下这类cosf的光滑性和衰减性。
Where several mother wavelet functions were used to expand a function , it also can be seen as vector - valued wavelets that satisfy conditions in which matrics are involved . the main works of this paper are as follows : firstly , although haar function has bad vanish property in frequency domain , it is the only normal orthonormal basis with symmetry and real short - support property 首先，讨论了对于尺度函数，相应的母小波构成空间的标准正交基的充要条件，提出了构造a尺度母小波的算法，从理论上研究a尺度haar小波基的构造，提出了分解与重构公式，并对如何构造具有对称性的a尺度haar小波基进行了探讨。
