光滑的

In this paper we use pointwise modulus of smoothness to study approximation direct theorem and equivalent theorem for some linear operators and quasi-interpolant operators; Using pointwise modulus we discuss the strong converse inequality on K-functional; and using a modified weighted K-functional and weighted modulus of smoothness we study approximation with Jacobi weight on operator with non-zero first order moments.

本文利用点态光滑模ω_~(2r)来研究某些线性算子及逆中插式逼近正定理和等价定理；利用点态光滑模讨论其关于K-泛函的强逆不等式；同时利用一种改变的带权K-泛函和带权光滑模研究一阶矩不为零的算子的点态带Jacobi权逼近。

It takesthe weighted average of the L2 norm of the difference of the observation and thesolution of the system and the L2 norm of the difference of conormal derivativeat the different sides of the interface for every subdomain as cost functional andthe smooth coefficients of the subproblem and the value of solution of the originalproblem at interface as identification parameters;Using the property of continu-ous functional defined on compact set,the existence of the optimal solution of theidentification problem is proved;The necessary conditions of optimality charac-terized by the system equation,the adjoit equation and the variational inequalitysimultaneously are given by introducing the conception ofdifferential andadjoit variable;An algorithm is devised and its flow graph is given.

其次，针对分片光滑动力系统的特征，结合正演过程的区域分解算法，建立了分片光滑系统的分解区域参数辨识模型，该模型以子区域上解的实测值与计算值之差的L2范数和界面两侧的通量差的L2范数的加权平均作目标泛函，各子问题的光滑系数及界面上真解的值为待辨识参量；利用紧致集上连续泛函的性质，证明了子区域上参数辨识问题最优辨识参量的存在性；引入微分的概念，借助伴随变量，给出了由系统方程，伴随方程和变分不等式共同表征的最优性必要条件；根据此必要条件设计了算法，给出了算法的程序框图。

In exterior differentiation, one studies tensors, exterior algebra, tangent bundle, tangent vector fields, one-parameter groups action on a manifold, Frobenius's theorem, tensors fields, exterior differential form, integration on manifolds, Stokes's theorem etc.

外微分部分主要涉及外代数、切丛，光滑切向量场，单参数变换群，光滑张量场，外微分形式，Frobenius定理，外微分形式的积分，Stokes定理及其应用等。

Therefore,a series of experiments on polishing quality of an ultra-smooth surface and polishing time were designed,which was judged according to the size of a scattering facula and luminance under the condition of total reflection.

为此，对所加工的全反射棱镜的超光滑表面抛光不同的时间，并在全反射条件下，根据全反射棱镜背向散射光斑的大小和亮暗的程度来判断所加工超光滑表面加工质量的检测方法，最终给出超光滑表面的抛光质量随抛光时间的变化。

The reconstruction problem is ill-posed, so two optimal criterions, the least module and the smoothness criterion base on Tikhonov regularization technique, are introduced into reconstruction algorithm. Many regularization parameters choice strategies are investigated, and the TPA(Two—Parameter Algorithm) strategy which is based on the Morozov discrepancy principles, is implemented in two regularization reconstruction algorithms.Numerical experiment results show that the nonnegative and smoothness constraint condition can overcome the difficulty of iteration semiconvergent, preconditioned technique can improve convergence rate and reconstruction accuracy, smoothness regularization criterion can meliorate ill-posed problem of reconstruction and enhance iteration stability, and the TPA is an effective strategy of regularization parameters choice.

数值试验表明：在共轭梯度法中引入非负约束和光滑约束改善了迭代的"半收敛"性，非负约束保证了解的非负性，光滑约束抑制了重建解的振荡现象，约束算法的重建精度与无约束算法相比大幅度提高；在约束共轭梯度重建算法中引入预优技术，可以加快算法的收敛速度，提高迭代的稳定性和重建精度；引入光滑准则的正则化技术可以有效改善图像重建问题的不适定性，加快迭代的收敛速度，提高迭代的稳定性和图像重建质量，计算正则参数的TPA算法在闪光照相图像重建中是有效的。

In the POPF model, wind farm is modeled by the probabilistic wind farm model considering the reactive power-slip characteristic, and the inequality constraints include not only the unit output constraints, the ratio constraints, the voltage constraints and the line current constraints but also the reactive compensation capacity constraints in wind farm and the system climbing capacity constraints per minute. By introducing the NCP function, the KKT conditions of POPF system are transformed equivalently. Based on the transformed nonsmooth nonlinear algebraic equations, the FOSMM is used to determine the POPF model expressed by the numerical characteristic of variables. The model includes nonsmooth functions, so it can be solved by a semismooth Newton-type method based on the subdifferential.

概率最优潮流模型中，风电场采用考虑无功功率—滑加热器差特性的风电概率模型，不等式约束中除了机组出力约束、有载调压变压器变比约束、电压约束和支路电流约束，也考虑了风电场无功补偿容量约束、系统的分钟级爬坡能力约束；使用非线性互补函数将概率最优潮流的KKT条件转化为一组包含有不光滑函数的非线性代数方程组，然后基于一次二阶矩法确定了以待求量的数字特征表示的POPF模型，由于该模型包含不光滑函数，因此采用基于次微分的半光滑牛顿型方加热器法求解。

In order to understand the weak solutions like piece-wise smooth solution, we must investigate the structures and properties of global smooth solution.

一般情况下，要研究清楚分块光滑解等常见的弱解，我们必须研究光滑解尤其是全局光滑解的结构和性质，这是研究高维守恒律方程方程全局解的基础。但用以往的处理方法一般来说是很困难的。

For example ,the non－ polar materials like hydrocarbons are heavy and draggy as compared to polar emollients,isoparaffins are light and smooth,the polar esters were significantly lighter and smoother,the branched chain and unsaturated esters were exceptionally light and smooth,in the case of straight chain esters,as the emollients become smoother and lighter ,the oiliness increases,in the case of unsaturated and branched chain esters,the emollient becomes lighter ,as well as less oily,when the alcohol chain length is decreased.

研究结果表明，物理化学特性明显影响润肤剂的轻柔感和油亮感，例如类似碳氢化合物的非极性物质相对于极性物质要粘稠些，异构烷烃则轻柔和光滑，带分支链的不饱和酯特别具有轻柔感和光滑感，带分支的不饱和酯相对直链饱和酯在油亮感方面有着特别的不同，直链酯随着产品光滑感和轻柔感的增加，油亮感也增加，对于带分支的不饱和酯而言，产品变得更轻柔，油亮感下降。

The reconstruction problem is ill-posed, so two optimal criterions, the least module and the smoothness criterion base on Tikhonov regularization technique, are introduced into reconstruction algorithm. Many regularization parameters choice strategies are investigated, and the TPA(Two—Parameter Algorithm) strategy which is based on the Morozov discrepancy principles, is implemented in two regularization reconstruction algorithms.Numerical experiment results show that the nonnegative and smoothness constraint condition can overcome the difficulty of iteration semiconvergent, preconditioned technique can improve convergence rate and reconstruction accuracy, smoothness regularization criterion can meliorate ill-posed problem of reconstruction and enhance iteration stability, and the TPA is an effective strategy of regularization parameters choice.

数值试验表明：在共轭梯度法中引入非负约束和光滑约束改善了迭代的&半收敛&性，非负约束保证了解的非负性，光滑约束抑制了重建解的振荡现象，约束算法的重建精度与无约束算法相比大幅度提高；在约束共轭梯度重建算法中引入预优技术，可以加快算法的收敛速度，提高迭代的稳定性和重建精度；引入光滑准则的正则化技术可以有效改善图像重建问题的不适定性，加快迭代的收敛速度，提高迭代的稳定性和图像重建质量，计算正则参数的TPA算法在闪光照相图像重建中是有效的。

The results show that the concave structure ameliorates the surface stress distribution of the nonsmooth sample,makes the local stress between concaves less apparently than that of the smooth sample,that means the local low stress areas are formed on the nonsmooth surface.

结果表明：在受力状态下，仿生非光滑凹坑改善了试样表面的应力分布，使非光滑试样表面凹坑间的区域局部应力明显小于光滑试样表面的应力，即产生了局部低应力区。

And Pharaoh spoke to Joseph, saying, Your father and your brothers have come to you.

47：5 法老对约瑟说，你父亲和你弟兄们到你这里来了。

Additionally, the approximate flattening of surface strip using lines linking midpoints on perpendicular lines between geodesic curves and the unconditional extreme value method are discussed.

提出了用测地线方程、曲面上两点间短程线来计算膜结构曲面测地线的方法，同时，采用测地线间垂线的中点连线和用无约束极值法进行空间条状曲面近似展开的分析。