算子方程

A 2×2 matrix differential Lenard operator is extended to a 3×3 oneand the soliton hierarchy with their Lax pairs are deduced easily.

并使孤立子方程谱问题的伴随表示与Lenard算子对有机的连系在一起。

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。

Assuming that the transport operator is linear, the tokamak particle transport equations are systematically analyzed.

在输运算子为线性算子的条件下，对托卡马克等离于体粒子输运方程的求解进行了系统的分析。

Then in terms of dual operators, an approach to reduce nonlinear and noncommutative operator systems to a single equation is given.

其次， 利用对偶算子给出了将非线性非交换算子方程组化为单个方程求解的算法。

The standard parabolic equation method was obtained by the decompocition of the scalar wave equation and the expansion of the pseudo-differential operator.

通过对传统波动方程的分解和伪微分算子的近似展开，获得了标准抛物线方程。

In this paper, by using the notable Mountain Pass Lemma of critical point theory, some sufficient conditions for the existence and multiplicity of periodic and subharmonic solutions to a class of second order nonlinear functional difference equations with Jacobi operators are obtained.

论文研究了一类具有Jacobi算子类型的二阶非线性泛函差分方程(1)的周期解和次调和解的存在性与多重性。首先将方程（1）的解的存在性问题转化为某个泛函临界点的存在性问题，然后验证此泛函满足著名的山路引理的条件，从而得到论文的主要结论。

Based on the surjection of maximal monotone operators, we prove that some semilinear heat equations with weak Lipschitz nonlinear terms are exactly controllable by L〓 0, T: H〓(Ω control; Based on the critical point theorem of coercive convex functionals, we prove that some parabolic equations with Lipschitz nonlinear terms are globally approximately controllable and finite-dimensional exactly controllable by controls acting on mobile supports, and finally, we extend the results to parabolic equations in R〓.

首先利用极大单调算子的满射原理证明了半线性热方程在较弱Lipschitz条件下可通过L〓0.T；H〓(Ω控制实现精确能控。然后通过构造强制凸泛函并利用其临界点理论证明了带Lipschitz非线性项的抛物型方程可通过作用在在移动支集上的控制来实现整体近似能控与有限维精确能控，最后将整体近似能控与有限维精确能控结论推广到无界区域R〓上。

By choosing the special values of parameters in the hierarchy, the hierarchies of the general TD, TD, the general C-KdV and C-KdV can be obtained. In order to investigate the Hamiltonian structure of this soliton hierarchy, a new Lenard gradient sequence {Gj} and its corresponding 2×2 Lenard pair of operators are introduced.

对该方程族中的参数取不同的值，可得到广义TD族，TD族，广义C-KdV族和C-KdV族；为研究其Hamilton结构，通过定义新的Lenard递推序列{G_j}得到了该等谱方程族的2×2 Lenard算子对

The other one retrieves vertical velocity from the reflectivity using statistical algorithm and assimilates the vertical velocity taking Richardson's equation as observational operator. It is found that the vertical velo city derived by the former is usually too weak while the later gets the distinct three dimensional convective structure of meso-scale system with strong vertical velocity.

其一是采用Richardson方程通过水平风场、温度和气压来对垂直速度进行诊断，其二是利用统计方法由观测的反射率因子反演出垂直速度后，再采用Richardson方程作为观测算子对反演的垂直速度进行同化。

We study on the controllability of parabolic systems with a nonlinear term involving the state and the gradient ,the null controllability property of a 1-d parabolic equation involving a fractional power of the Laplace operator ,null controllability of some systems of two parabolic equations with one control force ,and exact controllability of the parabolic system with bilinear control.

我们研究了含有状态和梯度的非线性的项抛物系统的能控性，含有拉普拉斯算子的分数阶的1-d抛物方程的零能控性，带有一个强控制的两个抛物方程的一些系统的零能控性和具有双线性控制抛物系统的精确能控性。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。