椭圆空间

Thirdly, on the base of the existent error estimate of elliptical Galerkin projection operator, obtain semi-discrete and complete-discrete error estimates betwent EFG solution and exact solution of parabolic partial differential equation. Semi-discrete error estimate show that the rank of r is accordance with the approximation rank of subspace S_h. Complete-discrete error estimates show that it not only have relation with radius r of domain of influence, but also have relation with the step length of time variable and the way of discrete.Lastly, give typical examples and draw up procedures of MATLAB.

再次，在"椭圆Galerkin投影"算子的误差估计的基础上，对用EFG法解抛物型偏微分方程的EFG解与精确解之间作了半离散和全离散的误差估计，半离散的误差估计表明所给出的误差界限关于r的阶是与子空间S_h的逼近阶相一致的，全离散的误差估计表明所产生的误差不但与影响域半径r有关，而且与离散时间变量的步长τ及其离散方式有关。

The contents are the following:In chapter two, the existence and multiplicity results for the following equation of p-Laplacian type are obtained.For the elliptic quasilinear hemivariational inequality involving the p-Laplacian operator,in order to use the mountain pass theorem proving the existence result, the authors usually need to use the uniform convexity of the Sobolev space to prove the energy function satisfies the PS condition. But for the p-Laplacian type equation mentioned above, this method is no use. To overcome this difficulty, the potential function is assumed to be convex, then I prove the existence result and by using the extension of the Ricceri theorem, the multiplicity result for the problem is obtained.

在第二章我们首先考虑关于以下p-Laplacian型(p-Laplacian type)方程非平凡解及多解的存在性对于带有p-Laplacian算子的椭圆拟线性半边分不等式问题，为应用非光滑的山路引理证明解的存在性，在证明方程所对应的能量泛函满足非光滑的PS条件时，需利用Sobolev空间的一致凸性，但是对于具有更一般形式的算子的p-Laplacian型方程，不具备上述性质，在文中为克服这一困难，本人对位势泛函做了一致凸的假设，从而证明了解的存在性，并应用推广的Ricceri定理，证明了方程三个解的存在性。

Some geometric data of ellipse in hyperbolic space are considered,such as set inclusion,arc length,geodesic curvature,curvature,area and total curvature .

在双曲空间中，考察椭圆的包含关系，对弧长元素、测地曲率、曲率、面积及全曲率等几何量做出细致考察。

Location of the planetary nebulae in the outskirts of the giant galaxy Messier 87 and in the intergalactic space around the centre of the Virgo Cluster of galaxies.

图中标出了位于巨椭圆星系M87外晕远处的和位于室女座星系团中心的星系间空间中的行星状星云。

Function graphing and analyzing: 2D, 2.5D functiongraphs and animations, extrema, root, tangent, limit,derivative, integral, inverse; sequence of number: arithmetic progression, geometric progression; analytic geometry: vector, line, circle, ellipse,hyperbola and parabola; solid geometry: spatial line, prism, pyramid, cylinder, cone, dado and sphere; Ability to save graph as data file or bmp file; Ability to save animation as AVI file; Ability to move, zoom in, zoom out and rotate the scene.

函数图表和分析：2D， 2.5D 函数图表和动画，极值，根，正切，极限，导数；序列号：等差数列，等比级数；解析几何：向量，矢量，直线，圆，椭圆，双曲线和抛物线；立体几何：空间线，棱柱，三角锥，圆柱，圆锥体，梯形台和球体；保存图形为数据文件或者图像文件的功能；保存动画为 AVI 文件的功能；移动，缩小，放大和旋转场景的功能。

Second, transient flow for vertical well and fractural well in porous media with arealanisotropy has been researched in this paper. The analytic solution for vertical well andsemi-analytic solution for vertical fractural well is developed by using image inversion,superposition principle, Laplace transformation, Fourier transformation, as well as theirinverse transforms, and storage and skin are taken into account. For vertical fractural well,influence of pressure and productivity is studied when the orientation of artificial fractureisn't in accordance with the direction of major permeability, it is fundamental to evaluatepotential and analyze well test data.

文中研究了平面各向异性储层不定常渗流问题，建立了平面各向异性储层交叉直线边界中一口直井渗流的数学模型，利用镜像反演、Laplace变换、Fouier变换及其反变换推导了不同边界条件下直井的Laplace空间解析解，考虑井储和表皮的影响，分析了曲线特征及影响因素的敏感性曲线；并运用影响椭圆及Duhamel叠加原理，推导了平面各向异性储层中垂直裂缝井的压力分布方程，考虑了水力裂缝与主渗透率轴成一定角度对压力及产能的影响，为垂直裂缝井的产能评价及试井分析提供依据。

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。

Since the solutions of this problem are the critical points of the associated energy function. One generally needs some compactness such as PScondition or C-condition to prove the existence of critical points of the energy func-tion, but when we study the elliptic equation in R~N, the compactness condition does not always hold since the imbedding of the Sobolev space H_0~(1,2)into L~(2*) is not compact. In this chapter, based on the nonsmooth critical point theory, and by using the approximation technique with periodic function, the existence of nontrivial solution is obtained.

由于该方程的解就是其所对应的能量泛函的临界点，通常都要在一些紧型条件的基础上来证明其所对应的能量泛函临界点的存在性，但当我们在无界域上考虑该椭圆方程解的存在性时，Sobolev空间H_0~(1，2)到L~(2*)的嵌入非紧，从而导致所对应的能量泛函失去紧性，本章在非光滑临界点理论的基础上，应用周期逼近的方法证明该问题非平凡解的存在性。

Firstly basic interpolations are obtained from planar circle interpolation by the point by point comparison.

首先使用逐点比较法作平面圆弧插补获得基本插补量，然后以三维坐标变换将基本插补量变换成为空间椭圆的插补量。

It featured the vocabulary of the circle, oval, rectangle and hexagon, and design characters including plants as space as well as the order formed through the juxtaposition and copy of the pure geometry.

圆形、椭圆、方形和六边形成为丹麦几何景观的特征词汇，同时也生成了用植物塑造和组合空间、运用简单几何形的并置与重复来创造秩序的几何景观的设计特征。

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。