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Using the method of approximating the Lipschitz functions by the convex functions, we give a sufficient condition for the set where Lipschitz functions on a Hilbert space are Frechet differentiable is a residual set.

我们利用Lipschitz函数空间中凸函数的逼近性质来分析Lipschitz函数的微分性质，得到Hilbert空间上Lipschitz函数的Frechet可微点集是剩余集的一个充分条件，以标题"Lipschitz函数与凸函数之间的一些联系"在中国科学杂志上发表。

In noncommutative scalar field theories, the appearing of nonplanar diagrams is related to permutation symmetry of field functions in the interaction Lagrangian, while in NCQED we can not arbitrarily change the positions of field functions in the interaction Lagrangian because different positions of the field functions correspond to different representations of the gauge group.

这里需要特别指出的是NCQED的非平面图产生的原因与非对易标量场根本不同，非对易标量场非平面图的产生是和相互作用拉氏量密度中场函数间位置的置换对称性有关，而在NCQED中我们不能随便交换相互作用拉氏量密度中场函数的位置，因为场函数的不同位置对应着规范群的不同表示。

Furthermor, we consider their nondiferentiable situation, we define nonsmooth univex functions for Lipschitz functions by using Clarke generalized directional derivative and study nonsmooth multiobjective fractional programming with the new convexity. We establish generalized Karush-Kuhn-Tucker necessary and sufficient optimality condition and prove weak, strong and strict converse duality theorems for nonsmooth multiobjective fractional programming problems containing univex functions.

而且，本文利用Clarke广义方向导数针对Lipschitz函数在原来一致凸函数概念的基础上定义了不可微的一致凸函数，并利用这类新凸性，我们研究了非光滑多目标分式规划，获得了广义Karush-Kuhn-Tucher最优性条件；弱对偶定理、强对偶定理和严格逆对偶定理。

This paper is committed to expatiate on the attribute of the functions profoundly from several aspects such as the mapping, inverse functions, compound function and etc and discuss the attributes of mapping and several special functions energetically.

本文着重从几个方面如映射、函数的复合、反函数等深刻阐述了函数的性质，且重点讨论了映射、几种特殊函数的性质和应用。

This thesis mainly discuss following issues, Theory and simple expressions for array covariance matrixes are derived when angular spread functions are symmetric distribution functions, i. e. the Uniform distribution, the Gauss distribution, the Laplace distribution and the Von Mises distribution, and a non-symmetric distribution function, i. e. the Gamma distribution. And the relation between the effective signal subspace and the array number, or and the nominal angle of the distributed source, the angular spread, the distributed functions, and the Signal-to-Noise Ratio is gained. The dimension of the effective signal subspace increases with increment of the array number. And it is more obvious to the non-symmetric distribution. The dimension of the effective signal subspace decreases with increment of the nominal angle. And the distributed source is equal to a point source as θ=π/2. The dimension of the effective signal subspace increases with increment of the angular spread.

本论文针对阵列信号处理中广泛存在的分布源现象，主要讨论了以下问题：推导了角度分布函数分别为对称的均匀分布、高斯分布、拉普拉斯分布、Von Mises分布和非对称的伽马分布时，分布源阵列接收信号协方差阵的严格模型和简化模型，得到了单个分布源的有效信号子空间随阵元数、分布源中心角、分布角、角度分布函数和信噪比的变化规律：随着阵元数的增加，对所有角度分布函数的有效信号子空间维数也随着增加，且非对称分布函数的有效信号子空间充满整个空间的可能性更大；随着分布源中心角逐步增加，有效信号子空间维数逐步减小，当θ=π/2时，等价于点源情形；随着分布源分布角逐渐加大，有效信号子空间维数也随之增加，直到有效信号子空间充满整个空间；随着信噪比的增加，有效信号子空间维数有一定程度的减少。

In Chapter 4, we give a new proof of the local Lipschitz continuity of H-convex functions on the Heisenberg group. For a Carnot group of step two, we obtain a characterization of H-convex functions and prove that H-convex functions are locally Lipschitz continuous.

第四章，我们导出了Heisenberg群上H-凸函数的Lipschitz连续性的一个新证明，对于二阶Carnot群，我们给出了H-凸函数的一个特征并证明了Carnot群上H-凸函数是局部Lipschitz连续函数。

In elliptic case,the compactness of"convex polyhedrons"is destinedto be true because of the"local Lipschitz continuity of convex functions",whilein parabolic case,we cannot expect convex-monotone functions have such goodproperties as convex functions.

在广义解存在性的证明中，参考椭圆情形由"凸多面体"逼近的思想，我们采用了由一种特殊构造的"凸单调多面体"逼近的途径，但在证明这种凸单调多面体族的紧性时，我们遇到了困难。

Calculation formulae for real number order derivative and integral of logarithmic functions and power functions have been obtained which are the important references to calculate real number order calculus of any other functions.

计算得出了对数函数、幂函数的任意实数阶导数及积分的完整计算公式，它们是推导其它函数任意实数阶微积分公式的重要参照式。

Copulas are functions that join multivariate distribution functions to their one-dimensional marginal distribution functions.

Copula函数是将多维随机变量的联合分布和其边缘分布连接起来的一种函数。

Copulas are functions that joint multivariate distribution functions to their one–dimensional marginal distribution functions.

Copula用来描述多元随机变量的一维边缘分布与其联合分布之间的函数关系。

We stopped by the old fashion ice cream parlor Manhattan Beach Creamery for some refreshment. The "No Shoes, No Shirt, No Problem" sign on the wall epitomizes the beach lifestyle.

我们去商店区的一家老式冰淇淋店坐坐，墙上挂著「没鞋，没上衣，没问题」的牌子完全捕捉了海滩生活的随性及风格。

"If therefore", argues Bellarmine we can offer our prayers and our satisfactions in behalf of those detained in purgatory, because we are members of the great body of Christ, why may not the Vicar of Christ apply to the same souls the superabundant satisfaction of Christ and his saints--of which he is the dispenser?

" "因此，如果"，认为贝拉明（德indulgentiis ，十四）"我们可以提供我们的祈祷和我们满意的代表那些被拘留在炼狱，因为我们是成员的伟大基督的奥体，为什麼可能不是牧师基督适用於同样的灵魂满意的过剩是耶稣和他的圣人-他是在配药？