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7 Matrix Exponentials and Linear Systems The solution vectors of a homogeneous linear system Can be used to construct a matrix = that satisfies the matrix differential equation associated with Eq.

5.7 指数矩阵和线性系统借助等式（1），齐次线性系统的解向量可以被用来构造一个满足矩阵微分方程的矩阵。

In this thesis, we take the delay r as a bifurcation parameter to investigate the Hopf bifurcation phenomenon in system (1). By analyzing the associated characteristic transcendental equation of system (2), and using the Hopf bifurcation theorem, we obtain one condition for the existence of Hopf bifurcation in system (1). Furthermore, based on the center manifold theorem and the method of normal form, some interesting results about the properties of Hopf bifurcation are obtained, including the direction of Hopf bifurcation and stability of Hopf bifurcating periodic solutions.

我们以时滞T作为分支参数，研究了系统(1)的Hopf分支现象：通过分析系统(2)的特征超越方程，结合利用Hopf分支定理获得了系统(1)的Hopf分支存在的一个条件；利用中心流形定理和正规形方法分析了系统(1)的Hopf分支的性质，包括分支的方向和分支周期解的稳定性。

The bilinear form of BBM equation has known, so we can similiarly get the bilinear form of associated DGH equation, then we can get the bilinear form of DGH equation and 1-soliton by a approciate method.

BBM方程的双线性已知，通过恰当的设法可以类似得到关联DGH方程的双线性形式，即得到了DGH方程的双线性形式，并由此得到了DGH方程的1—孤子解。

The implicit TDMA algorithm and first kind of firstorder upwind difference were employed to solve the conservation equation associated with appropriate boundary conditions.

通过分析离心力场作用下铝熔液在Al2O3短纤维多孔介质内的渗流传热现象，考虑离心惯性力对渗流传热过程的影响，建立了多孔介质渗流传热模型。

Choosing time delay as the control parameter, the stability of trivial equilibrium is discussed by analyzing distribution of the roots of the associated characteristic equation. It is found that Hopf bifurcation occurs from trivial equilibrium when the delay passes through critical values, then the critical values and their relations with system control parameters are obtained.

以时滞量为参数，通过分析对应线性系统特征方程的根的分布，讨论了平凡解的稳定性，得到了平凡解发生Hopf分岔而失稳的临界时滞量以及临界时滞量与系统控制参数的关系。

This text studies the obstacle problems associated with nonhomogeneous elliptic equation, gives the definition of solutions of second order degenerate nonhomogeneous obstacle problems, and making use of the Poincaré inequality and others,acquire some properties of these solutions and their lead number,filling up the blank of an nonhomogeneous obstacle problem research.

本文研究了非齐次椭圆方程的障碍问题，给出了二阶非齐次障碍问题解的定义，利用Poincar啨不等式，获得非齐次障碍问题的解及其导数的一些性质，填补了对非齐次障碍问题研究的空白。

Then the delay is used in circadian model. Using the delay as a argument, the distribution of the roots of the characteristic equation associated with the model under light is analyzed. Then the stable and Hopf bifurcation condition of the circadian model with delay is obtained.

然后，将时滞引入到昼夜节律系统模型后，以滞量为参数，讨论光照昼夜节律模型线性部分特征方程根的分布情况，得到该系统的稳定性和Hopf分支产生的条件。

The control architecture and associated error equation are built for adaptive output feedback control. Then, a stable weight adjust rule for the no-line NN is derived by Lyapunov theory, thus assuring the stability of the closed-loop system.

NN的在线补偿规律由Lyapunov稳定性方法得到，保证了系统的稳定性，可以有效补偿模型参数不确定和未建模动态引起的控制偏差。

By analyzing the distribution of the zeros to the transcendental characteristic equation associated with the trivial.solution,we obtain some- sufficient conditions on the stability and instability of the trivial solution.These results are generalized to n-unit neural networks by means of space decomposition in this Chapter.

在第二章中，我们讨论了不带自反馈与带自反馈的3元环状神经网络系统平凡解所对应超越特征方程的根的分布情况，确定了系统平凡解稳定与不稳定的充分条件，并利用空间分解的方法把相应的结果推广到n元环状神经网络系统。

In particular, are associated with the verb form of oral expression in Chinese has irreplaceable status.

尤其是一些与动词相关的口语格式在汉语表达中有着不可替代的地位。

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。