极小条件

Soc _RR is a injective left R-module.(4) Soc _RR is a left self-injective ring.|Theorem 4.2.9 Let R be a commutative notherian ring, then the following are equivalent:(1) R is a FS-ring.(2) R is a PS-ring.(3) Every minimal ideal of R is injective.(4) Every minimal ideal of R is p-injective.

定理4.2.9若R是交换的Noethe(来源：Aae0eBe0C论文网www.abclunwen.com)r环，则以下条件等价：R是FS-环；R是PS-环；R的每个极小理想都是内射的；R的每个极小理想都是p-内射的；R是IS-环；R的每个极小理想都是投射的；R的每个极小理想都是平坦的。

Based on conditions on the conjugate quadrature filters constructing the orthogonal scaling function and wavelet, the orthogonal conditions were designed as an objective function which was minimized by the parameter optimization in the parameterized formula of complex conjugate quadrature filters. The compact support, symmetry, quasi-orthogonal filters and wavelets were obtained.

从构建正交尺度函数与小波的共轭正交滤波器的条件出发，把滤波器的正交条件设计成目标函数，利用共轭正交复滤波器的参数化算式进行参数优化使目标函数极小，得到紧支、对称、准正交的滤波器及其小波。

The numerical results indicated that the mixing time decreases about linearly with increasing current intensity. The mixing time decreases first then increases with increasing current frequency and comes up to its minimum value at 30Hz. Applying the traveling magnetic field to the uptake or downtake when the exciting current intensity is 200 A at 10 Hz, the mixing time can be shortened by 9%~17%. With the operating conditions kept unchanged, shorter mixing time is available if applying the traveling magnetic field to the uptake instead of downtake, and the time can be shortened by 18%~26% if applying the magnetic field to both the uptake and downtake. Moreover, the mixing time decreases with increasing argon blowing flowrate, and the time comes down to its minimum value if the argon blowing flowrate is 1 600 NL/min. After the moment, the mixing time increases with argon blowing flowrate.

计算结果表明：混匀时间随电流强度的增大而减小，并且近似成线性关系；混匀时间随电流频率的增加先减小后增大，并且在30Hz 时达到极小值；在励磁电流强度为200A ，频率为10Hz 的条件下，在上升管或下降管施加行波磁场混匀时间可缩短9%~17%；在相同操作条件下，在上升管处施加行波磁场混匀时间小于在下降管施加行波磁场的混匀时间；同时在上升管和下降管施加行波磁场，混匀时间可缩短18%~26%；混匀时间随吹氩量的增大而减小，在吹氩量为1600NL/min 时混匀时间达到极小值，吹氩量继续增大时，混匀时间反而增大。

In Chapter 6, based on discretization technique an implementable algorithm for nonconvex generalized semi-infinite minimax problems is presented and, utilizing properties of generalized quasi-directional derivative, its global convergence is proven under weak conditions.

对于广义极大极小问题，本文第六章在较弱的条件下，利用广义伪方向导数的性质，用离散化的技巧给出了非凸广义半无限极大极小问题的一种可实现的全局收敛算法。

The abstract result contains several concrete results in the literature and can also be used to deal with some new cases for resonant differential equations.In the introduction, we briefly introduce the development process of the variational methods. In Chapter 2, we list some basic knowledges refering to the variational methods, including the Sobobev space,—△ operator, the weak solution and the minimizing sequence methods and some minimax theorems. In Chapter 3, we introduce the research process of Hamiltonian system of second order and the semilinear elliptic problems, using the methods introduced previously. In Chapter 4, we prove the main theorem of the thesis, and apply it to the problems in the previous Chapter, and can also be applied to some new resonant cases.

在前言中，简要介绍了变分法的产生、发展过程，在第二章中我们介绍了有关变分法的一些基本知识，包括Sobolev空间，—△算子，弱解，极小化序列方法和一些极小极大定理，在第三章中我们介绍了非线性项有界或满足次线性条件，以及它满足推广的Ahmad-Lazer-Paul条件时，二阶Hamiltonian系统和半线性椭圆问题的研究历程，最后在第四章中我们证明了本论文的主要定理，并把它应用到第三章的问题中，使得前面的几种共振的情形都可以统一到这个抽象的结果中。

To obtain quadratical convergence, however, strict complementarity condition at the Danskin point was used. The condition is too strict to be satisfied in many practical problems such as discrete semi-infinite minimax problem. Another kind of Newton method for finite minimax problems was presented by E. Polak and, without strict complementarity at the Danskin point, superlinear convergence (of order 3/2) was proven.

Polak等人提出了一种直接求解极大极小问题的二阶收敛的牛顿法，但是为获得二阶收敛速度要求在Danskin点处满足严格互补条件，这个条件太强，很多实际问题尤其是半无限极大极小问题的离散化不满足该条件；他们又给出另外一种牛顿法，在不假设严格互补条件成立的情况下，证明了它的超线性（3/2阶）收敛性。

A Study on Fraunhofer Double-Slit Diffraction by Applying Simulation Technology;2. Based on Kirchhoffs theory,the approximate intensity formulas of double-slit diffraction of Gaussian Beam is derived under paraxial approximation with power series expansion methods.

依据基尔霍夫衍射理论，利用幂级数展开的方法推导出旁轴条件下高斯光束双缝衍射的近似光强计算公式，并通过数值计算对双缝衍射的基本特性进行了讨论，得出了与其他文献一致的结论，得出了高斯光束下的双缝衍射极小值点比平行光束双缝衍射的极小值点向外移，观察屏中心光能变小；光强分布可由激光斑在衍射屏处的参数调制的结论。

The generalized criterion for minimum zone in the determination of form error of surface is presented on the basis of linear minimax techniques, and the criteria of such surfaces as for cylindricity and straightness with arbitrary direction are derived from this generalized criterion.

本文讨论形状误差最小区城的判别准则和判别方法。根据线性极差极小化问题解的特征，给出形状误差最小条件的统一判别准则。推导出各种形状误差最小区域判别法(包括圆柱度误差和任意方向直线度误差)，并在几何判别的基础上得出代数判别的格式。

In the last of this dissertation, we give necessary and sufficient conditions for the existence of and the expressions of perpositive semidefinite solutions and bipositive semidefinite solutions to some systems of quaternion matrix equations.

给出了某些四元数矩阵方程组有广亚半正定解，双半正定解的充要条件及其表达公式；研究了矩阵方程组的最小二乘解，极小范数的最小二乘解，在某些约束条件下的最小二乘解和极小范数的最小二乘解，给出了其有这些解的充要条件及其表达公式。

E. thickness of the copper wire wall, length of the single conductor wire, height of the solenoid, the exciting current and the current density, the working temperature of the conductor, time of exciting and demagnetizing, the cooling water pressure and flux density, etc..

所建立的数学模型包含3个目标函数（即能耗极小化目标函数、铜耗极小化目标函数和纯铁用量极小化目标函数）、7个设计变量（即铜管规格参数a、b、t，线圈匝数N〓、N〓，铁铠磁通密度B〓和冷却水压p）和8类14个约束条件（即铜管壁厚、单饼导线长度、螺线管高度、激磁电流与电流密度、导体工作温度、激退磁时间，冷却水压和铁铠磁通密度等约束条件）。

They weren't aggressive, but I yelled and threw a rock in their direction to get them off the trail and away from me, just in case.

他们没有侵略性，但我大喊，并在他们的方向扔石头让他们过的线索，远离我，以防万一。

In slot 2 in your bag put wrapping paper, quantity does not matter in this case.

在你的书包里槽2把包装纸、数量无关紧要。