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This course offers advanced topics for students who have learned ordinary differential equations. The course includes linear algebra which has topics as matrices, linear systems of equations, eigenvalue problems, vector differential and integral calculus. Fourier series, orthogonal functions, Fourier transforms and partial differential equations will also be introduced.

本课程为提供已有常微分方程式基础的同学修习，内容包括线性代数，矩阵运算，特徵值问题，向量的微分与积分，一阶线性微分方程组，傅力叶级数和正交函数，傅力叶转换，并将对椭圆，抛物线，双曲线型式的偏微分方程式作概略介绍。

Initial boundary value problem of two dimensional unsteady diffusion equations is solved in this paper. By using time-dependent fundamental solution of two dimensional diffusion equations and the extension of double layer potential, we establish virtual boundary integral expression of diffusion equations. Then virtual boundary element method is used to implement the numerical computation.

本文在求解二维非定常扩散方程的初边值问题时，利用二维扩散方程与时间有关的基本解，基于双层位势的延拓，建立二维扩散方程的虚边界积分表达式，然后利用虚边界元法进行数值计算。

Due to equivalent transformation during the simplification procedure of equations, the solution set of equations can neither expand and nor contract, and also the transformed equations for four displacement functions are compact in the form and easy to find their solutions.

由于方程化简过程是等价变形，从而可以保证方程的解集既不会扩大也不会缩小，而位移势函数便给出了一维六方准晶中弹性解的一个通解公式。

For the long composite cylindrical panels, including the influences of the transverse shear deformation, rotatory inertia and initial geometric imperfections, the nonlinear dynamic equations of the long composite cylindrical panels were derived from Hamilton's philosophy. The nonlinear partial differential equations were transformed to ordinary differential equations by expansion in series and solved numerically by fourthorder Runge-Kutta method.

针对含初始几何缺陷的复合材料圆柱曲板，考虑横向剪切变形、转动惯量、初始几何缺陷以及非扁壳等诸多因素，利用Hamilton原理导出复合材料长圆柱曲板的非线性动力方程；用级数展开把非线性偏微分方程组转化为二阶常微分方程组，并由四阶Runge-Kutta方法数值求解。

This paper is the continuance of that of Lewalle (2001). Lewalle has made use of Gaussian wavelet to obtain the N-S equations with flexion as fundamental quantity in the infinite domain. In that work, Lapalace operator has been degraded to one-order operator. Moreover the resulted N-S equations take the form of diffusion equations with local bilinear mutual interaction kernel.

本文是Lewalle工作的继续，Lewalle利用高斯子波，得到了以弯曲度为基本量的无穷域中N-S方程，此时Lapalace算子降低为一阶导数，并且N-S方程变形为具有局部化双线性相互作用的扩散方程的形式。

The nonlinear dynamic equations including the initial geometric imperfections of laminated composite cylindrical shells were derived from Hamilton's philosophy. Assuming the transverse displacements to be composed of axially symmetrical part and axially asymmetrical part, the governing equations expressed as displacement forms of dynamic buckling were obtained by Galerkin method. The equations were solved by finite difference method. The effects of impact velocity, lamination angle, initial geometric imperfection on the axial dynamic buckling of laminated composite cylindrical shells were discussed.

由Hamilton原理导出包含初始几何缺陷的复合材料层合圆柱壳的非线性动力方程；将横向位移设为对称及非对称两部分，由Galerkin方法得到以位移形式表达的动力屈曲控制方程，通过有限差分方法求解；讨论了冲击速度、铺层角度、初始几何缺陷等各种因素对复合材料圆柱壳轴向冲击动力屈曲可能产生的影响。

This paper mainly consists of three parts:(1) Based on the Reddy's higher-order shear deformation plate theory and general von Kármán-type equations, put forward the governing equations of pre- and post-buckled vibration of several hybrid FGM plate forms which have not been studied adequately, triumphantly applying the perturbation technique combined with Galerkin method into equations solving, and get the comparatively simple result forms with complicated deduction at last;(2) As a new type of materials, piezoelectric fiber reinforced composite actuators were applied to optimal vibration control, interrelated informatons be depicted in this paper;(3) During the investigative cource, some factors not been considered adequently in other researches were discussed fully in the postbuckled vibration analysis, such as the temperature-dependent material coefficients, heat conduction and nonlinear vibration characteristic etc.

本文的研究工作主要包括：(1)基于Reddy高阶剪切变形板理论和广义Kármán型方程，给出了几类现有研究成果较少或没有彻底解决的FGM混合板结构的屈曲前和屈曲后振动特性研究的控制方程，成功将摄动-Galerkin混合法应用于求解过程，并给出了各控制方程相对直观且易于计算的解析解形式；(2)单向压电纤维增强复合材料(Piezoelectric Fiber Reinforced Composite，简称PFRC)是一种新型压电材料，本文将其应用到了结构振动控制研究当中，发现通过压电纤维体积含量的选择可以实现混合板结构的优化振动控制；(3)围绕几种有代表性的FGM混合板结构，讨论其屈曲后的振动特性，在分析中同时考虑热传导，材料物性参数的温度相关性，以及机/热/电荷载耦合效应对非线性振动特性的影响，最后给出了大量数值计算结果。

By integral transformation of basic equations, the stress and displacement expressions with unknown coefficients of elastic and viscoelastic materials were obtained in Laplace domain respectively, and introducing dislocation density functions, the singular integral equations were got according to the boundary conditions and interface connection conditions, further adopting Gauss integration and Gauss-Jacobi integration formula, the problem was reduced to algebraic equations, then it can be solved with the method of collocation dots in Laplace domain. Finally, the time response of dynamic stress intensity factor was calculated with the inverse Laplace integral transformation.

采用积分变换方法，得到Laplace域内弹性和粘弹性材料的应力和位移的含未知系数的表达式；引入位错密度函数，并通过边界条件和界面连接条件，导出反映裂纹尖端奇异性的奇异积分方程组，采用Gauss积分，并运用Gauss-Jacobi求积公式化奇异积分方程组为代数方程组，利用配点法进行求解；最后经过Laplace逆变换，求得动态应力强度因子的时间响应。

The influence of initial imperfections on the stability ofcomposite laminated plates was studied by solving the nonlinear governing equations for laminated plates with elastical restraints subject to in-plane loading using the energy variational principle and nonlinear geometric equations incorporating the initial imperfection. The governing equations were solved using generalized Fourier series to obtain the effect of the non-uniform order term.

为深入研究初始缺陷对复合材料层板稳定性的影响，利用能量变分原理和考虑初始缺陷的非线性几何方程，建立了具有弹性约束的层合板在面内载荷作用下的非线性稳定性控制方程组，运用广义傅立叶级数法对其进行求解，并进一步获得了影响非线性稳定性控制方程组的性质的非齐次项。

In this thesis the process of constructing the non-perturbative Hamiltonian theory is de-scribed and is applied to estimate the vacuum condensate. It contains the following contents:At the very beginning, by using the path integral method and eliminating the gluon freedom, aGCM action 〓 of current quarks including lower order current-current coupling was derivedfrom the QCD Lagrangian and the effective Hamiltonian operator that could hardly be doneby the normal methods was derived. After doing this, the broken vacuum is introduced whichincludes quark-antiquark condensate through the generalized Bogoliubov-Valatin transformation,the effective Hamiltonian of constituent quark was derived. The detailed formulas containingthe spatial current-current coupling term for the effective Hamiltonian and gap equations wasworked out by parameterizing the correlation kernel as a quadratic potential. And then, the gapequation was solved and the quark-antiquark condensate of vacuum was studied both in the casesof instantaneous interaction and retarded interaction. In the end, the effective Hamiltionian withtwo-body quark-quark interaction was derived with one-body approximation, and with the helpof the functional integral method the coupling non-linear dynamic equations for systems withnuclear matter was derived. Finally, these equations were solved by selfconsistent method andthe effect of nuclear matter on vacuum condensate was studied. The spatial current-current coupling term is too difficult to handle, hence the correlationkernel is assumed to be not important and usually omitted in the pure vacuum condensate, andthe instantaneous interaction generally is adopted. Retaining the spatial current-current termand partial retardation effect, the quark pairs condensate in pure vacuum was studied, and theeffect of quark mass was also studied. At present, little study is focused in the case with nuclearmatter and spatial current-current term also omitted. Under the approximation with partialspatial current-current term, the effect of nuclear matter on vacuum condensate was studied.

本论文描述了量子色动力学整体色对称模型哈密顿量方法的构建过程，得到了反映正反夸克对凝聚真空结构的关于组分夸克的有效哈密顿量算符，它隐含了胶子作用，并且准确至流-流耦合项；接着，通过参数化哈密顿量中的夸克作用关联核，导出平方禁闭势参数化选择的哈密顿量的具体公式和能隙方程；随后，应用公式，编程求解，考察了瞬时作用下和部分延迟作用下真空的正反夸克对凝聚，在计算中保留了空间流-流耦合作用；之后，导出瞬时势和延迟势下包含二体作用项的哈密顿量公式，并采用单体化近似，通过泛函变分方法得到核物质存在时耦合的非线性动力学方程；在保留部分空间双流耦合作用的近似下，求解核物质的动力学方程，考察核物质密度对真空凝聚的影响，以往考察真空凝聚，对关联核的选用，由于空间流-流耦合项不易处理，也认为作用不大，常忽略该项，并且常采用瞬时作用；本文保留空间双流项和部分延迟作用，考察了真空情形的夸克对凝聚，还考察了夸克质量对纯真空凝聚的影响，以往对核物质存在情形的真空凝聚考察很少，也都忽略空间流-流项，本文在考虑部分空间流-流项近似下，考察了核物质存在对真空凝聚的影响。

What are your goals and strategies for growth?

你的成长目标和策略是什么？

And unto the angel of the church in Sardis write; These things saith he that hath the seven irits of God, and the seven star I know thy works, that thou hast a name that thou livest, and art dead.

3：1 你要写信给撒狄教会的使者，说，那有神的七灵和七星的，说，我知道你的行为，按名你是活的，其实是死的。