对偶性

Next, we discuss the relations between left quasi-dual bimodules and left dual-bimodules, we obtain that a left quasi-dual bimodule is a left dual bimodule if it satisfies one of the following conditions: sM is minimal injective and MR is a M-minimal injective kasch-module; MR is a M-minimal injective kasch-module and for any two ideals LI and L2 ofSS rM(L1 n L2)-rw(L1)+rM(I2); sM is minimal injective and for any two submodules A and B of MR,Lastly, we applicate the quasi-duality on smash product algebra R#H, and obtain an answer of the semiprime problem, i.e., let H be a finite-dimensional semisimple Hopf algebra and R be an H-module algebra, if R is left quasi-dual and semiprime, then R#H is semiprime.

我们得到：一个左拟对偶双边模如果满足下列条件之一，则它将成为一个左对偶双边模：_sM是单内射的并且M_R是一个M-单内射kasch-模；M_R是一个M-单内射kasch-模并且对_sS的任意两个理想，有r_M(L_1∩L_2)=r_M(L_1)+r_M(L_2)；_sM是单内射的且对M_R的任意两个子模，有l_s=l_s+l_s。2 在第2.3节中我们将拟对偶性应用于smash积代数R#H，部分解决了半素问题。

We study structure of and decomposition of constraints, soldering of chirality, and especially, duality of chiral p-form actions.

探讨了约束的结构和约束的分解、手征性链接、手征P次形式作用量的对偶性分析等。

In this paper, based on the conjugate function and the dual programming of convex programming, with the dual cone, to conic programming, the dual programming is fully discussed, the expression of the dual programming is educed, the main dualities are proved, and the dualities of the familiar conic programming are studied.

以共轭函数和凸规划的对偶规划为基础，利用对偶锥的概念，全面讨论了一般锥规划的对偶问题，严格推导出锥规划对偶规划的表示形式，给出了锥规划的主要对偶性质，并用这些结果研究了常见锥规划的对偶性。

Uncertainty theory is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.

不确定理论是基于规范性、单调性、自对偶性、可列次可加性和乘积测度公理的数学系统，是研究不确定现象的数学分支。

In the paper, the basic theory, the key technology of New GPS is investigated. The framework of GPS knowledge base is constructed based with databases based on the general matrix model and Inference Mechanisms based on the duality principle of New GPS.

本文通过对新一代GPS 的理论基础、关键技术的分析和研究，依据新一代GPS 的通用矩阵模型和对偶性原理，研究并提出了GPS 知识库的基本框架，GPS 知识库由基于通用GPS(来源：A31BC论文网www.abclunwen.com)矩阵模型搭建的数据库和基于对偶性原理设计的知识推理机构组成。

Finally, we verified that the doublet four dimensional potential electromagnetic theory has obvious SO (2) duality symmetry.

最后验证了电磁对偶的双四维势电磁场理论具有SO(2)电磁对偶性。

Chapter 4,based on strong efficiency rather than efficiency or weak effi-ciency,discusses the similar problem to chapter 3.The concepts of strongsupremum and strong infimum for the set in finite-dimensionalEuclidean space are given,s-conjugate functions and s-subdifferentials theory is developed,by means of which,the duality problem ofmulitobjectivre programming for strong efficient solutions or called absoluteoptimal solutions is analysed,and a new duality theory-s-conjugateduality theory is proposed.

第四章，基于强有效性而不是有效性和弱有效性，给出了类似于第三章的结果，提出了有限维欧氏空间中集合的强上、下确界等概念，研究了点集映象的强共轭函数和强次微分问题，建立了点集映象的s-共轭函数及其s-次微分理论，并借助于这一理论，讨论了多目标规划关于绝对最优解或称强有效解的对偶性问题，建立了基于强有效性的s-共轭对偶理论。

In chapter 4, Based on the orthogonality relations and duality of C-algebras, we study the decompositions of C-algebras, and give a Fourier inversion formula on C-algebras, which generalizes the Fourier inversion formula on the center of a finite group algebra over complex. As an application, we characterize the integral closure of C-algebras over the ring of integers, and show a necessary and sufficitient condition for elements of C-algbras being algbraic integral over integers.

第四章 我们应用C-代数的正交性及对偶性刻画了C-代数的一个分解，给出了C-代数上这个完全分解的Fourier反演公式，从而推广了Z分解的Fourier反演公式；作为Fourier反演公式的一个应用，我们刻画了Z在整数环Z上的整闭包，进一步我们研究了C-代数在整数环Z上的整闭包，给出了C-代数在整数环Z上为整元的一个充要条件。

Because the duality connects strong couplings with weak couplings, it may be possible to compute quantum effects through the weak version which is dual to the strong one.

由于对偶性建立了强耦合理论与弱耦合理论之间的联系，因此它提供了一条可能的途径使人们用强耦合理论的对偶弱形式做量子效应的计算。

Accordingly, we know the connection of the universal duality and boundness of a primal-dual pair feasible set.We apply these theories to practical applications in chapter four.In chapter five we make many conclusions and bring forward research expectation for the future.conic optimization; dual conic optimization; duality gap; universal duality

研究主要有：利用Nesterov和Todd的齐次模型判断锥规划与其对偶锥规划解的存在性；类似于线性规划推导出锥规划的KKT-条件；从锥规划的泛对偶性得到锥规划与对偶锥规划的零对偶间隙存在的条件，从而了解泛对偶性与原—对偶

On the other hand, the more important thing is because the urban housing is a kind of heterogeneity products.

另一方面，更重要的是由于城市住房是一种异质性产品。

Climate histogram is the fall that collects place measure calm value, cent serves as cross axle for a few equal interval, the area that the frequency that the value appears according to place is accumulated and becomes will be determined inside each interval, discharge the graph that rise with post, also be called histogram.

气候直方图是将所收集的降水量测定值，分为几个相等的区间作为横轴，并将各区间内所测定值依所出现的次数累积而成的面积，用柱子排起来的图形，也叫做柱状图。