对偶化

Since a proposed application of Dirac structures is to the reduction of some mechanical systems, the reduction of Dirac structures is considered in detail in this paper, without using the existence of momentum mappings or the introducing of the notion of admissible function, etc..

因为引入Dirac结构的—个基本目的即应用于一些力学系统的约化，本文详细地讨论了Dirac结构的约化问题，包括其在Poisson流形以及Jacobi流形等的约化中的重要应用，重点利用了Dirac结构的特征对以及对偶特征对的工具，从而避开了讨论矩映射的存在性以及容许函数等一些复杂概念的介入。

The main theory results includes:(1) Using the properties of Hilbert transform, perfectly reconstruction and new type of lifting scheme, a new type of dual-tree binary coefficients complex wavelet with linear phase is achieved.(2) For linear systems that can be diagonalized by GFT and DST-II matrices, an efficient MGM method is proposed, convergence is proved.(3) We discuss the algebraic structure when Toeplitz matrix is transformed by multi-band wavelet,show that Toeplitz matrix is composed of generating function is transformed to a band and sparse matrix when wavelet applied to this matrix, based on the above results, an efficient solution of Toeplitz equations is obtained, and the computational complex is O,where N is the order of matrix.

理论成果主要包括：(1)对于对偶树二进制系数复数小波，利用Hilbert变换对性质、完全重构条件并结合新的提升格式构造研究了含参系数多进制小波构造方法，作为特例得到具有线性相位的对偶树二进制系数复数小波构造方法；(2)对于广义离散傅立叶变换与正弦变换对角化系统，提出了高效、快速的多重网格算法，理论上证明了算法的收敛性；(3)研究了Toeplitz矩阵在多进制小波变换下的代数结构，验证了多项式生成函数构成的Toeplitz系统在小波变换下的稀疏带宽性质，从而建立基于小波变换求解Toeplitz系统的快速求解方法，运算量级控制在O，其中N为系统的阶。

Results showed that the teaching assumption is supported by this study. Guided discovery and peer discussion are effective instruction strategies for symbolizing a proposition and finding symbolic rule of equivalence relation of dual propositions.

教学过程的观察记录显示在教师的导引与学生的小组讨论下，学生确可发现对偶命题等价并符号化，而在求等价命题的问题上，将原命题符号化再辅以符号规则是成功的关键步骤，教学假设获得支持。

This paper is composed of the algorithm design of dual-tree binary coefficients complex wavelets,MGM algorithm for linear systems that can be diagonalized by generalized discrete Fourier transform and type-II discrete sine transform matrices, wavelet and MGM algorithms for Toeplitz systems and their applications in signal and image processing.

本文以多进制小波与对偶树二进制系数复数小波算法设计、三角变换对角化系统MGM算法、Topelitz系统的小波与MGM结合算法作为理论主线，以对偶树复数小波与扩散方程方法结合的图像去噪，信号与恢复作为应用背景展开讨论。

First of all,by use of the modern differential geometry and the double complex functionmethod,the definitions of dual and self-dual operators as well as ordinarycomplex Ashtekar connections are extended into parametric forms.

首先利用现代微分几何和二重复函数方法将对偶，自对偶算符的定义以及普通复阿西特卡联络推广为参数化形式。

This thesis is divided into six parts. The first chapter is preface, the current status of research in the inverse problems for parabolic partial differential equations is reported; the second chapter is "regularization methods for numerical differentiation and their applications ", in this chapter we investigate many regularization methods from a viewpoint of regularization theory and algorithm, some applications in the inverse problems for parabolic partial differential equations are given; the third chapter is "spectral regularization methods". Based on Fourier analysis, within the framework of regularization theory, we apply the spectral methods to some ill-posed problems. Many numerical experiments are done in order to show the validity of the methods; the fourth chapter is devoted to wavelet dual least squares method and a revised wavelet method; in the fifth chapter,we combine finite difference method with method of lines and apply it to the backward heat conduction problem in time; in the sixth chapter "identification problems for unknown source ", the essence and the degree of two problems related to source identification are pointed out, at the same time, some numerical methods are reported.

本文分为六个部分，第一章前言简要分析了国内外抛物型偏微分方程反问题的研究现状；第二章数值微分的正则化及其应用从正则化理论和算法的角度出发，考察了许多正则化方法，还给出了数值微分在抛物型偏微分方程反问题的一些应用；第三章谱正则化方法是在Fourier分析的基础上，在一般正则化理论的框架下，给出了这种方法在各种不适定问题中的应用，数值实验表明谱方法是有效的；第四章研究了小波对偶最小二乘方法和改进的小波方法；第五章主要研究了有限差分方法结合线方法在时间反向热传导问题中的应用；第六章是未知源识别问题，主要指出了两类未知源问题的不适定程度和不适定本质，同时报告了一些数值方法。

We introduce the notion of characteristic pairs and give the definition of dual characteristic pairs of Dirac structures on Lie bialgebroids. Using the dual characteristic pairs, we give the if and only if conditions for which a maximally isotropic subbundle of the double of a Lie bialgebroid is a Dirac structure.

本文在李双代数胚上，引入了Dirac结构的特征对并给出对偶特征对的概念，利用对偶特征对，给出李双代数胚double的极大迷向子丛是Dirac结构的充要条件；其次，分别利用特征对与对偶特征对，将可约Dirac结构分为第一类可约与第二类可约，在此基础上，建立Poisson流形的两类对应约化定理。

Gr\ochenig and Balan, Casazza, Heil, and Landau introduced the concepts of localization.

Gr\ochenig 和 Balan， Casazza， Heil， Landau 等人提出了两种框架局部化的概念，这两种局部化在近几年被应用到了Gabor框架，小波框架和采样定理，得出了许多关于框架的超完备性、冗余度、框架界和密度的关系以及局部化框架的对偶框架的结构方面的重要结果。

While Yü Sin shows how supple he can be in spite of the cramping antithetical style of the Fu, Su succeeds in softening and thawing this rigid style, smoothing over its angularity and making the sharp points of the riming antitheses melt into one another.

如果说庾信向人们展示了如何在词赋的严苛对偶格式下体现出婉转优美的话，苏东坡则成功地柔化和融解了这种僵硬的骈偶形式，磨光了其棱角，使生硬的对偶调和无间。

The main aims of this disseration are to extend the complex self-dual grav-itational theory-the Ashtekar theory in four dimensional spacetime and este-bhsh a parametric self-dual gravitational theory in order to solve the problemof reality conditions in the Ashtekar theory.

本文的主要目的是推广四维时空的复自对偶引力理论-阿西特卡理论，建立一个参数化的自对偶引力理论，进而解决阿西特卡理论中面临的至关重要的实性条件问题。

However, as the name（read－only memory）implies, CD disks cannot be written onorchanged in any way.

然而，正如其名字所指出的那样，CD盘不能写，也不能用任何方式改变其内容。

Galvanizes steel pallet is mainly export which suits standard packing of European Union, the North America. galvanizes steel pallet is suitable to heavy rack. Pallet surface can design plate type, corrugated and the gap form, satisfies the different requirements.

镀锌钢托盘多用于出口，替代木托盘，免薰蒸，符合欧盟、北美各国对出口货物包装材料的法令要求；喷涂钢托盘适用于重载上货架之用，托盘表面根据需要制作成平板状、波纹状及间隔形式，满足不同的使用要求。