对偶关系

It is proved that the upper approximation of such fuzzy rough set in a fuzzy approximation space is just its image derived according to generalized extension principle and binary fuzzy similar relation.

通过对偶方式定义了模糊集的上、下近似算子，给出了模糊粗糙集在相应的模糊关系及模糊集的截集下的表示定理，证明了这种模糊粗糙集关于模糊近似空间的上近似恰为其在二元模糊相似关系下导出的广义扩张原理之下的像。

A differential equation is introduced to construct canonical dual function. The corresponding perfect duality theory is established to show the relationship between the KKT points of the primal problem and the canonical dual problem.

通过引入常微分方程和构造Canonical对偶函数的局部形式，引入了相应的对偶定理，勾勒出了原问题的KKT点和对偶问题的KKT点两者之间的关系。

After that, by making use of dual basis, the general reconstruction formula about frame is discussed, and the computation formula for the dual frame is presented, in which no operators are contained.

然后，利用对偶基概念讨论了一般框架下信号的重构公式，并给出了对偶框架的具体计算公式，该公式不包含任何算符，可以明显看出基矢的相关性与框架类型之间的关系。

On conic optimization, we work over some properties of cone and its dual cone, and solve the relations between special cones (obtuse cone, orthogonal cone, superior and inferior obtuse cones) and their dual cones.Furthermore, sufficient and necessary condition under which they exist are proved detailed .

本文围绕锥规划问题，对锥及其对偶锥的性质进行了研究，解决了一些特殊锥(钝锥、直角锥、优劣钝锥)及其对偶锥之间的关系，并对它们存在的充要条件给予了详细的证明。

A special kind of prime dual ideals are defined in a lattice implication algebra, then their structures and properties are discussed. It is proved that the implication operation on this lattice implication algebra is determined by these prime dual ideals, and all of these prime dual ideals compose a lattice implication algebra which is lattice implication isomorphic to the former lattice implication algebra.

在格蕴涵代数中定义了一类特殊的素对偶理想，讨论了它们的结构和性质，证明了该格蕴涵代数中的蕴涵运算可以由这些特殊的素对偶理想所确定，并且这些特殊的素对偶理想全体自然地构成一个格蕴涵代数，它和原格蕴涵代数具有格蕴涵代数同构关系。

By Lagrange dual and scalarization of vector optimization, characterizations of solutions for which objective mapping satisfies weak convexity conditions are shown. The strongly and weak dual theorems for generalized subconvex-like mapping are obtained.

通过适当定义对偶问题和向量优化问题的标量化研究各解之间的关系，刻画目标映射满足弱凸性条件的优化问题解的特征，在目标映射是广义锥次类凸的条件下给出有关的对偶定理。

Chapter 1 gives the background,current research process of relatedproblems and summarizes this thesis\'s work.In chapter 2,we study the Brownian motion with holding and jumping on the boundary.We use the resolvent method to obtain the infinitesimal generator because the domain of the infinitesimal generator is essentially the same as the range of the resolvent.Knowledge of this range and of the differential operator determines uniquely the infinitesimal generator.Since the semigroup generated by the DHJ is not strongly continuous,to use the nice property of strongly continuous semigroup in analytic theory,in chapter 3 we show that the dual is strongly continuous and derive ergodicity through spectral radius formulas and finally obtain the ergodic theorem by duality. In chapter 4,we discuss a class of a more general process---one dimensional Feller diffusion proposed by W.Feller in 1954.The Feller diffusion allows the possibility of jumps from boundary to boundary,not only from boundary to the interior.We give the stationary distribution of this process.

具体地，本文的结构如下：第一章给出了问题产生的背景，研究现状及本文的主要工作；第二章研究了在边界上逗留后随机跳的布朗运动，我(来源：3dABC论文网www.abclunwen.com)们用预解算子的方法得到其无穷小生成元，因为无穷小生成元的定义域本质上就是预解算子的值域，知道这个值域和微分算子形式就能唯一地决定无穷小生成元；由于DHJ过程产生的半群不是强连续的，为利用强连续半群的一些漂亮性质，在第三章中我们证明其对偶半群是强连续的，然后由谱半径公式得到遍历性并且最后由对偶得到遍历定理；第四章讨论了Feller在1954年引入的更广的一类过程----一维Feller扩散过程，Feller扩散过程允许有从边界到边界的跳发生，即不仅仅局限于从边界到内部的跳，在这一章中，我们给出了一维Feller扩散过程的平稳分布；在第五章，我们讨论了一些相关的问题，给出了DHJ过程对应的PDE问题及特征值与收敛速度的关系。

The main contributions of the second part of this dissertation are focused on the cryptographic properties of logical functions over finite field, with the help of the properties of trace functions, and that of p-polynomials, as well as the permutation theory over finite field: The new definition of Chrestenson linear spectrum is given and the relation between the new Chrestenson linear spectrum and the Chrestenson cyclic spectrum is presented, followed by the inverse formula of logical function over finite field; The distribution for linear structures of the logical functions over finite field is discussed and the complete construction of logical functions taking on all vectors as linear structures is suggested, which leads to the conception of the extended affine functions over finite field, whose cryptographic properties is similar to that of the affine functions over field GF (2) and prime field F〓; The relationship between the degeneration of logical functions and the linear structures, the degeneration of logical functions and the support of Chrestenson spectrum, as well as the relation between the nonlinearity and the linear structures are discussed; Using the relation of the logical functions over finite field and the vector logical functions over its prime field, we reveal the relationship between the perfect nonlinear functions over finite field and the vector generalized Bent functions over its prime field; The existence or not of the perfect nonlinear functions with any variables over any finite fields is offered, and some methods are proposed to construct the perfect nonlinear functions by using the balanced p-polynomials over finite field.

重新定义了有限域上逻辑函数的Chrestenson线性谱，考察了新定义的Chrestenson线性谱和原来的Chrestenson循环谱的关系，并利用一组对偶基给出了有限域上逻辑函数的反演公式；给出了有限域上随机变量联合分布的分解式，并利用随机变量联合分布的分解式对有限域上逻辑函数的密码性质进行了研究；给出了有限域上逻辑函数与相应素域上向量逻辑函数的关系，探讨了它们之间密码性质的联系，如平衡性，相关免疫性，扩散性，线性结构以及非线性度等；讨论了有限域上逻辑函数各类线性结构之间的关系，并给出了任意点都是线性结构的逻辑函数的全部构造，由此引出了有限域上的"泛仿射函数"的概念；考察了有限域上逻辑函数的退化性与线性结构的关系、退化性与Chrestenson谱支集的关系；给出了有限域逻辑函数非线性度的定义，利用有限域上逻辑函数的非线性度与相应素域上向量逻辑函数非线性度的关系，考察了有限域上逻辑函数的非线性度与线性结构的关系；利用有限域上逻辑函数与相应素域上向量逻辑函数的关系，揭示了有限域上的广义Bent函数与相应素域上的广义Bent函数的关系，以及有限域上的完全非线性函数与相应素域上向量广义Bent函数之间的关系；给出了任意有限域上任意n元完全非线性函数存在性与否的完整证明，并利用有限域上平衡的p-多项式的性质给出了有限域上完全非线性函数的一些基本构造方法。

In this paper,some approximate optimal solutions and an augmented Lagrangian function in nonlinear programming were introduced,duality map and duality problems based on the augmented Lagrangian function were established, relationship between the approximate optimal solutions of augmented Lagrangian function and primal problem was discussed.

介绍了几种近似最优解和增广拉格朗日函数，建立了基于增广拉格朗日函数的对偶映射和相应的对偶问题，讨论了增广拉格朗日函数的几种近似解和原问题的几种近似解的关系，得到的结果推广了一些已有的结论。

The relation between the nondegeneracy of their primal-dual optimal solutions and their strictly complementary optimal solutions is characterized, and then the conditions in which the solutions of a pair of primal and dual conic programmings are unique respectively are obtained.

从几何的观点描述一对原始-对偶锥规划可行域的极点，给出原始-对偶非退化的最优解与严格互补的最优解之间的关系，从而得到锥规划最优解唯一的条件。

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。