极小性

The maximum principle of the optimal control for the stochastic systems described by Zakaj stochastic partial sifferential equationis proved by approximately minimum point theorem of E. Ekeland. The convexity and compactness of the set of control values is not assumed, and it is not necessary for the maximum principle about differentiability in control variables included in drift term of the stoch astic system and the integrand in index functional, and costate process satisfies the stochastic partial ...

在不假便定控制变量取值的集合是凸的和紧的，不要求随机系统的漂移项和指标泛函的被积函数关于控制变量具有可微性的情况下，用E，Ekeland的近似极小点定理证明了Zakai随机偏微分方程描述的随机系统的最优控制的最大值原理，和用很简洁的方法证明了协态过程满足一个随机偏微分方程。

The definitions of generalized directional derivative and generalized gradient of Lipschitz functions defined on Riemannian manifold are presented. Some properties of the directional derivative and gradient are proved by using tangent and cotangent mapping. The minimization necessary condition of nonsmooth Lipschitz functions is given. Moreover, Fritz John necessary optimality condition in mathematical programming is provided on Riemannian manifold.

在黎曼流形上给出了Lipschitz函数的广义方向导数和广义梯度的概念，利用黎曼流形局部上与欧氏空间开集微分同胚的性质以及切映射和余切映射导出了广义梯度的性质和运算法则，证明了定义在黎曼流形上的函数取得极小值的必要条件是广义梯度包含零元素，并利用这些性质给出了黎曼流形上数学规划问题的Fritz John型最优性条件。

Based on a Hopfield neural network mode for combinational circuit test generation, a test generation algorithm with global searching ability is proposed. This algorithm is the combination of chaotic searching with gradient algorithm. By means of the inherent stochastic and ergodic property of chaotic searching, the problem of being trapped in local minimum of gradient algorithm can be solved.

基于组合电路测试生成的离散Hopfield神经网络模型，将混沌搜索与Hopfield网络的梯度算法相结合，利用混沌搜索的内随机性及遍历性来克服梯度算法易于陷于局部极小的缺点，形成一种具有全局搜索能力的测试生成有效算法。

By making use of the ergodicity specific to the chaotic theory, one can improve the chaotic-searching optimization method based on Logistic mapping.

在优化设计领域中，混沌现象的遍历性特点可以作为搜索过程中避免陷入局部极小的一种优化机制。

Within the fibroid two cystic areas are seen with minimal blood flow consistent with degenerative changes.

纤维瘤内两处囊性区被发现有符合退行性病变的极小的血流量。

As a application of the above results, some minimax inequalities and saddle point theorems can be obtained. In addition, fixed point theorems and section theorems, maximal element theorems and the existence theorems of equilibrium for abstract economies can also be discussed.

作为应用，可以得到L-凸空间中极大极小不等式和鞍点定理；进一步，还可以探讨不动点定理和截口定理，极大元定理和抽象经济的平衡存在性等问题。

Since the stability of the neural network don't require the convexity of the original maximin problem, the obtained results can be applied to solve a class of nonconvex optimization problem.

由于稳定性分析中不需要原极大极小问题的凸性，该网络可以用来求解一类非凸优化问题。

There is mutual contradiction between direction estimation of user's signal and power allocation among all users in the beamforming algorithm. A twice game beamforming algorithm based on game theory is proposed to deal with it. Beamforming game algorithm mathematics model is constructed. During the first game, direction of signal and power allocation are mapped the game theory as "player", which are modeled as the problem of maximin function and obtain direction estimation first. Then during the second game, power allocations of different users are described as a multi-user game. Power updated algorithm is designed. The existence and uniqueness of the Nash equilibrium in the twice game beamforming algorithm based on game theory are proved by mathematics derivation.

摘要针对波束成形算法中，用户的信号方向估计值和用户之间的功率分配存在着相互矛盾，本文提出了一种基于博弈论的二次博弈波束成形算法，构建了波束成形博弈算法数学模型，首先在第一次博弈的时候，将波束成形算法中的信号方向和功率分配映射为博弈论数学模型中的"局中人"，将其建模为函数的极大极小值求解问题，先求解出信号方向；然后在第二次博弈的时候，将不同用户的功率分配过程描述为一个多用户的博弈过程，设计了功率分配更新算法，通过数学推导论证了纳什平衡点的存在性和唯一性。

A complete novel assumption is presented. Then, we prove the existence of two nontrivial solutions for a class of second order superlinear difference systems with the help of Minimax principle and Linking theorem in critical point theory,.6. Two applications in physics are given. Firstly, a coupled differential system coming from second harmonic waves is discussed.

利用Ricceri建立的三临界点定理，证明了一类含参数二阶差分方程在某些新的条件下至少存在三个解；随后，利用临界点理论中的极小极大原理和环绕定理，讨论了一类二阶超线性差分方程组多重解的存在性。

A minimax theorem generally involves three assumption conditions: space structures on sets X and Y, the continuity of the functions and the concavity and convexity of functions.

一个极大极小定理一般涉及三个假设条件：集合X和Y的空间结构，函数的连续性和函数的凹凸性。

It has been put forward that there exists single Ball point and double Ball points on the symmetrical connecting-rod curves of equilateral mechanisms.

从鲍尔点的形成原理出发，分析对称连杆曲线上鲍尔点的产生条件，提出等边机构的对称连杆曲线上有单鲍尔点和双鲍尔点。

The factory affiliated to the Group primarily manufactures multiple-purpose pincers, baking kits, knives, scissors, kitchenware, gardening tools and beauty care kits as well as other hardware tools, the annual production value of which reaches US$ 30 million dollars.

集团所属工厂主要生产多用钳、烤具、刀具、剪刀、厨具、花园工具、美容套等五金产品，年生产总值3000万美元，产品价廉物美、选料上乘、质量保证，深受国内外客户的青睐