证明

Like most university graduates in mechanical engineering, I do not have the sufficient mathematical knowledge to judge major mathematical problems, i.e. Fermat's Last Theorem, Riemann's Hypothesis, Jiang Chun-xuan's proof of FLT, Wiles's in question proof of FLT, or Jiang Chun-xuan's disproof RH, etc.

像大部分机械工程大学毕业生那样，我没有足够的数学知识判断重大数学问题，如费马大定理〔FLT〕、黎曼假设〔RH〕等，以及蒋春暄对于FLT的证明、怀尔斯〔Wiles〕对FLT有问题的证明、蒋春暄对于RH的否定。

Justify the correctness of the recursive division algorithm given in page 15, and show that it takes time O(n^2) on n-bit inputs.

证明15页递归除法算法的正确性，然后证明对于n位输入，他需要O(n^2)的时间。

Chapter 6, consider a coupled generalized KdV-Burgers equation. In section 6.2, we study the initial-boundary value problem in the semi-unbounded domain, the existence of global solutions and global attractors is proved by means of a uniform priori estimate for time. In section 6.3, the Cauchy problem by using the weighted space, the existence of the global attractors for a coupled generalized KdV-Burgers in an semi-unbounded domain is proved.

第六章，考虑了一类广义耦合的KdV-Burgers方程，在第二节中讨论了半无界区域上的初边值问题，证明了整体光滑解和整体吸引子的存在性；在第三节中讨论了Cauchy问题，利用加权函数和加权空间上的插值8不等式，证明了半无界区域上整体吸引子的存在性。

Furthermore, the proof of correctness of CalcCycleTime is also the proof of the existence of cycle time, the existence of generalized eigenvectors, and the duality theorem.

本论文中给出的CalcCycleTime算法的正确性证明同时是周时存在性、广义特征向量存在性和对偶定理的证明。

It is proved that there is no limit cycle around the positive equilibra by constructing Dulac function, thus global asymptotic stability of the positive equilibria is proved in the first quadrant.

分析了该系统的平衡点性态，利用Dulac函数证明了系统在正平衡点外围不存在极限环，从而证明了正平衡点在第一象限内是全局渐近稳定的。

A predator model with refuge and toxicity is studied. By constructing a suitable Dulac function, it is shown that there is no limit cycle in the first quadrant, thus the globally stability of the equilibrium is proved. Further more, the paper analyzes the influence of refuge, toxicity and harvesting. It is amazing that ① if the death rate of predator is big or the harvesting effort is big, the existence of the predator is unrelated with refuge; if the death rate of predator and the harvesting effort are small, and the refuge is small enough, the population of both species is increased as the increase of refuge;② if d2=0, the increase of d1 has no influence on the prey population, while the population of predator is decreased.

研究一类有避难所、毒素作用和外界捕获影响的捕食系统，通过构造恰当的Dulac函数，证明系统在第一象限没有极限环，从而证明了平衡点的全局稳定性；分析了避难所、毒素、捕获作用对该系统最终种群数量的影响，发现：①当捕食者的死亡率大或其被捕获得多时，两种群的绝灭与避难所的存在无关；当捕食者的死亡率小、被捕获得多且避难所足够小时，随着食饵避难所的增加，捕食者种群数量也随之增加；②当毒素d2=0时，毒素d1的增加对食饵种群最终数量没有影响，而使捕食者种群数量下降。

Firstly, we consider SIQR epidemical models with continuous and pulse vaccinations; the reproduction numbers have been obtained for those models; by using Dulac function, we prove the global stabilities of equilibrium of SIQR model with continuous vaccination. In the SIQR epidemical models with pulse vaccinations, we have also discussed the existence of the infection-free periodic solution, the global stabilities of the infection-free periodic solutions, and the effectiveness of pulse vaccination and continuous vaccination policies has been compared for the two models.

文中首先考虑具有连续免疫接种和脉冲免疫接种的SIQR传染病模型，分别获得了它们各自的基本再生数；利用Dulac函数方法证明了具有连续免疫接种的SIQR模型无病平衡点的全局渐近稳定性；在脉冲免疫接种下SIQR传染病模型里，讨论了无病周期解的存在性，证明了该周期解的全局渐近稳定性，并且对连续和脉冲接种下的SIQR传染病模型的接种效率进行了比较。

The singular distributed parameter system can be converted to a singular system by using the eigenvector function method.

首次研究了广义分布参数系统的不变性条件，利用特征向量函数法将广义分布参数系统化为一广义系统，证明了它们的不变性条件是等价的，同时也指正了某些文献个别结论和证明

The artists create artworks that prove the existence and value of mankind, using their sensitive minds and skillful hands. When looking at their works, people share common feeling and enlightenment. The embracement of art is the pride owned only by mankind and is the very evidence of our civilization.

艺术家用他们敏感的心、才情的手为人类创造出证明人类存在和价值的艺术作品，使人们在观看这些作品时，产生通感，得到启迪，人们拥有热爱和欣赏艺术的高级情感乃是人类的尊严和骄傲，也是这个时代存在与进步的证明。

The existence and the uniqueness of the difference solution are proved. Based on the priori estimates and an inequality about norms, the stability and the convergence of difference solutions with the second-order are proved in the energy norm.

利用矩阵理论证明了差分解的存在惟一性，并利用一个重要的不等式在先验估计的基础上，运用能量估计的方法证明了该格式按无穷范数以二阶精度收敛到真实解。

She gently rebuff ed him, but agreed that they could be friends

她婉言拒绝了，但同意作为朋友相处。

If in the penal farm, you were sure to be criticized.

要是在劳改农场，你等着挨绳子吧！