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Thin film equations; nonlinear cross-diffusion systems; two-dimensional nonlinear reaction-diffusion equations; invariant subspace method; invariant sets

基础科学，数学，数学分析薄膜方程；非线性交错扩散方程组；二维非线性反应扩散方程；不变子空间方法；不变集

The constitutive equation between current density vector and electric field strength vector is tranformed into s-domain by the principle of Laplace transform,and the inverse Laplace transform is adopted for the explicit equations between them in time-domain. Finally through exponential time stepping,the FDTD iteration equations are obtained,which can be easily solved in discrete-time-domain.

该方法是将电流密度矢量与电场强度矢量之间的本构方程基于拉普拉斯变换原理转到复频域，然后再逆变换到时域得到它们之间显式的方程，最后再结合指数差分，得到离散时域的显式的FDTD迭代方程，解决了本构方程中电流密度矢量的分量相互耦合而不易直接离散的困难。

It is able to perform spatial transformational movement, rotation, hoisting, tilting, planar rotation and tilted rotation. Based on configuration analysis, kinematic position equations were formulated, kinematic model of parallel fine-tuning structure was discussed as a key and inverse position analysis and velocity mapping equations were presented. Kinematic simulation of the mechanism was conducted using MATLAB software. A case study was provided, which applied the mechanism in tunnel erector assembling manipulators.

通过对该机构进行分析研究，建立其运动学位姿方程，重点探讨了并联微调机构的运动学模型，建立了其位置逆解与速度映射解析方程，运用MATLAB软件对该机构进行了运动学仿真分析；同时提供了该串并联机构应用于隧道管片拼装机械手的具体实例。

Inverse M-matrices often occur in relation to systems of linear or non-linear equations or eigenvalues problems in a wide variety of areas including finite difference methods for partial differential equations, input-output production and growth model in economics, iterative methods in numerical analysis, and Markov processes in probability and statistics.

其中包括偏微分方程的有限差分法、经济中的投入产出和增长模型、数值分析中的迭代法以及概率论和数理统计中的马尔可夫过程。

Summary: The concept of matrix and its determinant computing, matrix determinant, matrix sub-block with the elementary transformation, invertible matrix, rank of matrix; vector and its computation, the linear relationship between vector, vector group of rank; linear equations of the nature and structure of linear equations; matrix eigenvalue and eigenvector, similar to matrix and matrix diagonalization conditions, the standard quadratic form with the normal forms, quadratic and symmetric matrix There are qualitative.

内容提要：行列式矩阵的概念及其运算，方阵的行列式，矩阵的分块与初等变换，可逆矩阵，矩阵的秩；向量及其运算，向量间的线性关系，向量组的秩；线性方程组的性质与结构，线性方程组的求解；矩阵的特征值与特征向量，相似矩阵与矩阵可对角化条件，二次型的标准形与规范形，二次型和对称阵的有定性。

In the beginning of quadratic equations such as, x2-2=0,forced upon us the irrational number, and led to the study of methods of operating with radicals, and to the solution of radical equations.

而二次方程，例如x2-2=0的出现，让我们不得不接触无理数，同时也带来了对于求根方法，以及根方程的研究。

Meanwhile, we proved the unboundedness of solutions of Lienard equations with weaker damping terms when the rotation number is irrational. We still obtained the unboundedness of solutions of Lienard equations with asymmetric nonlinearities at resonance.

另外，还通过应用Birkhoff遍历定理证明了当旋转数为无理数时，弱阻尼的Lienard方程无界解的存在性；同时，证明了共振条件下具有不对称非线性项的Lienard方程无界解的存在性。

For different polynomials g, if the characteristic polynomial of a n matrix A is irreducible, then we get some theorems to determine matrix equations g =A solvable; if it is reducible, then, to see n-dimension space vectors M over a field F as F-module, we use module theory to determine these equations solvable such that it is simpler and clearer to investigate these questions.

对于不同多项式g，当n阶矩阵A的特征多项式为不可约的，我们给出了矩阵方程g=A有解的判定定理；当A的特征多项式为可约的，把域F上的n维线性空间M作为由A导出的F -模，我们利用模论知识来决定矩阵方程g=A有解性，从而使这一问题变了简单，研究思路更加清晰。

Iterative method is used for equations or equations approximate root of a common algorithm design methods.

迭代法是用于求方程或方程组近似根的一种常用的算法设计方法。

Variational equations of motion of each flexible link are derived under Jourdain variational principle, and then the mixed algebraic-differential equations of flexible multibody systems are obtained with the kinematic constraint conditions.

用Jourdain速度变分原理建立各机械臂的动力学变分方程，在此基础上根据各物体之间的运动学约束关系，建立柔性多体系统的微分代数混合动力学方程。

For every one of my New Year resolutions, I have been a backslider for years.

说到我的新年决心，我每年都做

With empathy and focus on a feeling level, we can all relate to the common ground of mutuality of emotions.

有了同情心和感觉上一级的重点，我们都可以涉及到相互的情感共同点。