关于矩阵方程A^2=J解的分类问题

设A 是一个n×n(0—1)阵,J 是n×n 阶矩阵,J 的每个元素都为1。本文利用Jordan 标准型讨论矩阵方程A^2=J 的解的分类问题。

一维对称正则长波方程的动力学行为及行波解的分类

利用平面动力系统的分岔理论和方法,定性地分析了一维对称正则长波方程的动力学行为及其精确解的分类,获得了该方程的行波系统在不同参数条件下的轨线图.通过对所有的轨道进行分析,结合解的分类图,得到了该方程行波解的显式表达式.这些解包括有理解、孤立波解、爆破解、周期解等,展示了由参数变化引起的分岔现象,并丰富了一维对称正则长波方程精确解的类型.

关于指数Diophantine方程2~x+p^y=z^2的解的个数的上界估计

设P是一个固定的奇素数.得到了方程2~x+p^y=z^2的所有正整数解(x,y,z)的一个分类.此外,证明了:如果P≡1(mod 4)并且P≠17,那么Diophantine方程2~x+p^y=z^2的全部正整数解(x,y,z)的个数N(p)满足估计N(p)≤4.

Analysis of a stochastic model for algal bloom with nutrient recycling

In this paper, the dynamics of a stochastic model for algal bloom with nutrient recy- cling is investigated. The model incorporates a white noise term in the growth equation of algae population to describe the effects of random fluctuations in the environment, and a nutrient recycling term in the nutrient equation to account for the conversion of detritus into nutrient. The existence and uniqueness of the global positive solution of the model is first proved. Then we study the long-time behavior of the model. Conditions for the extinction and persistence in the mean of the algae population are established. By using the theory of integral Markov semigroups, we show that the model has an invari- ant and asymptotically stable density. Numerical simulations illustrate our theoretical results.