解空间

In chapter two, under the assumption that the set of decision variables is convex and the objective functions are concave, it is shown that the set of noninferior solutions in valued space of the objective functions turns out to be a strictly decreasing, concave and no chinky block of surface, and it can be constructed recursively.

在第2章凸性假设下非劣解集的构造及其几何性质中，在假设决策变量集为凸集，目标函数向量是凹的情况下，证明了目标函数向量值空间中的非劣解集是一严格单调递减，无缝隙且凹的曲面块，并给出了m维目标函数向量值空间中非劣解集的一种递归构造方式。

Thesolutions compose self-contained inner product space (weight function is constant 1),i.e., they are basic series of Hilbert space; hence any internal waves could beexpanded into generalized Fourier series. With difference method, Sturm-Liouvilleequation is transformed to matrix eigenvalue problem. Each eigenvalue is related to ahorizontal wave number, and relevant solution is a certain mode of internal wave.

其解系构成完备的加权(权函数为1)内积空间，即Hilbert空间，从而内波现象可以展成该解系的广义Fourier级数；采用差分方法，将 Sturm-Liouville 方程转化为矩阵特征值问题，每一个特征值对应一个内波波数，该特征值对应的解则是内波的某一模态。

In this paper, we consider neutral functional differential equations and study the global existence of their solutions, stable subspaces and unstable subspaces of linear autonomous dynamical systems, local stable manifolds and local unstable manifolds of near hyperbolic equilibrium point of nonlinear autonomous dynamical systems, periodic solutions and oscillation of the neutral equations.

本文主要研究中立型泛函微分方程解的整体存在性，自治线性动力系统的稳定子空间与不稳定子空间、自治非线性动力系统在双曲平衡点的局部稳定流与局部不稳定流，周期解与解的振动性。

There is the Cauchy problem of hyperbolic system with weak damping in Chapter4. In the space of dimension N=1,3, exploiting the semigroup method, the global existence and the decay behavior of the solution with small initial data to problem (about problem with small initial data) is put forward.

当空间维数N=1，3和初值充分小时，利用半群理论得到问题整体解的存在性，并对于整体解给出一致衰减估计；当空间维数N=1 时，利用试验函数方法讨论解在有限时刻爆破的条件。

Character of solution of stochastic differential equation in dual of nuclear space was studied.

研究核空间的对偶空间的随机微分方程解的性质，得到微分方程组解 L 2收敛的条件，及在一定条件下解过程导出的测度{ Pn}为胎

The rate of spatial variability of available N, available K, total N, total P, total K content and pH value in 2007 were all less than 20%, and their area rate of relative uniformity were 100%. The rate of spatial variability of available P and organic matter were large than 20%, and their area rate of relative uniformity were individually 91.34% and 95.70%, which large than 80%.

结果表明，2006年试验地中碱解氮、全氮、全磷、全钾、有机质含量和pH值的空间变异程度都小于20%，相对均匀面积比都为100%，速效磷和速效钾的空间变异程度大于20%，但相对均匀面积比分别为82.18%和94.99%，都大于80%。2007年试验地中碱解氮、速效钾、全氮、全磷、全钾含量和pH值的空间变异程度都小于20%，相对均匀面积比都为100%，速效磷和有机质的空间变异程度大于20%，但相对均匀面积比分别为91.34%和95.70%，都大于80%。

In Chapter two, it is given of the three-dimensional boundary element and the boundary element space, Sobolev space for the contact boundary element problem with friction; it is proved of existence and uniqueness of the solution by variation inequality for the contact Boundary Element Method matrix equation under two kinds of condition that the contact zone is fixed and the contact zone is increasing along with the load; it is given of the expressionfor the error of the exact solution and the boundary element solution.

第2章，给出三维边界单元及边界元空间、有摩擦接触边界元问题的Sobolev空间，利用变分不等式证明了不变接触区和随载荷可变接触区两种情况下的接触边界元法矩阵方程解的存在唯一性，给出了准确解与边界元解的误差表达式，还证明了三维弹塑性摩擦接触问题凝缩矩阵解的存在唯一性，为三维弹塑性摩擦接触边界元法奠定数学基础。

To obtain a complete picture about shock formation in non-viscous flow, in the succeeding chapter we investigate the isothermal shocks in adiabatic flows. We find that in this case the shock solution still exists, but the structure and the properties of the solution are greatly different from above even qualitatively. In some region of the parameter space there only exists unstable shock solution and in some region of the parameter space there may exist two stable shock solutions.

为得到完整的无粘流体中激波形成的图象，在接下来的第四章，我们研究绝热流体中等温作为激波条件时的情况，发现此时激波解仍存在，但解的结构和性质甚至在定性上都与前两章有了显著的不同：某些参数空间只存在非稳定的激波解，某些参数空间则会同时出现两个稳定的激波解。

The results showed that the spatial distribution range of soil fertility in Qinling forest area on south slope was soil pH volue 5.97～6.89,organic matter 6.6～50.1 g·kg~-1, total N 0.5～3.8 g·kg~-1 ,available N 33.6～257.3 mg·kg~-1,available P 0.81～5.8 mg·kg~-1,available K 38.9～262.4 mg·kg~-1,clay particle content 11.3%～18.1%,capacity of exchangeable base 68.5～310.2 mmol·kg~-1 respectively; The spatial variation degree of available P, organic matter, and total N was big,whereas clay particle content and active acid pH value was small;the spatial variation degree of soil pH value, organic matter, available P, clay particle content, and capacity of exchangeable base decreased with the deep of soil,whereas soil total N、available N, and available K increased.

结果表明，秦岭南坡土壤肥力指标在整个土壤剖面空间分异范围分别为：土壤pH值5.97～6.89，有机质含量6.6～50.1 g·kg-1，土壤全N含量0.5～3.8 g·kg-1，碱解N含量33.6～257.3 mg·kg-1，速效P含量0.81～5.8 mg·kg-1，速效K含量38.9～262.4 mg·kg-1，粘粒含量11.3%～18.1%，阳离子代换量68.5～310.2 mmol·kg-1；其空间分异程度以速效P、有机质和土壤全N含量较大，土壤粘粒含量和土壤pH值较小；随着土层的加深，土壤pH值、有机质、速效P、土壤粘粒含量和阳离子代换量空间分异程度减小，而土壤全N、碱解N和速效K含量空间分异程度增大。

In the third chapter,We study some geometric properties and spectral propertiesof the Jacobi operator on a solvable extension G of Heisenberg groups H.In the firstsection,we give the definition of G and some fundamental geometric properties on G.Inthe second section,we discuss the Integrability of certain subbundles and the geometricstructure of the induced foliations in case of integrability.In the third section,we studythe spectral properties of the Jacobi operator of G.

可解李群与类对称空间亦密切相关，Damcek-Ricci空间即是广义Heisenberg的一可解扩张李群，在第三章中，我们构造且研究了Heisenberg群的一可解扩张李群G的几何性质，第一节讨论了G的曲率，李指数映射，整体坐标等基本几何性质；第二章研究了TG的某些子丛的可积性及可积时诱导叶片的几何结构；第三节给出了G的Jacobi算子的特征值和相应的特征子空间。

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失