线性常微分方程(组)的算子方法介绍及其研究展望

综述了线性微分方程(组)的算子方法,侧重地介绍了作者所发展的一系列方法和重要的结果与解公式.提出了算子方法研究的几点展望.     

Hecke Algebras and Automorphic Forms

The goal of this paper is to carry out some explicit calculations of the actions of Hecke operators on spaces of algebraic modular forms on certain simple groups. In order to do this, we give the coset decomposition for the supports of these operators. We present the results of our calculations along with interpretations concerning the lifting of forms. The data we have obtained is of interest both from the point of view of number theory and of representation theory. For example, our data, together with a conjecture of Gross, predicts the existence of a Galois extension of Q with Galois group G ₂(F₅) which is ramified only at the prime 5. We also provide evidence of the existence of the symmetric cube lifting from PGL₂ to PGSp₄.     

求解刚性常微分方程的并行广义Rosenbrock方法

本文构造了求解刚性常微分方程的并行广义Rosenbrock方法（PEROWs），分析了方法的收敛性和数值稳定性。通过用Powell方法优化方法的稳定域，构造了二级四阶并行格式PEROW4，并证明该方法是A－稳定的。新方法比同级的并行Rosenbrock方法MPROW3及PRM3均高一阶，因而在计算精度上处于优势。此外，PEROW4能使得各处理机上的负载基本均衡，从而达到非常理想的加速比和并行效率。     

"常微分方程"课程建设探讨

根据常微分方程课程特点及常微分方程的发展,探讨了常微分方程精品课程建设中师资队伍建设和教材改革与建设的重要性,提出了对改革有关教学内容以及编写具有本课程特色的优秀教材的具体设想.     

Coefficients of Drinfeld modular forms and Hecke operators

Consider the space of Drinfeld modular forms of fixed weight and type for Γ₀(n)⊂GL₂(Fq[T]). It has a linear form bn, given by the coefficient of tm+n(q−1) in the power series expansion of a type m modular form at the cusp infinity, with respect to the uniformizer t. It also has an action of a Hecke algebra. Our aim is to study the Hecke module spanned by b₁. We give elements in the Hecke annihilator of b₁. Some of them are expected to be nontrivial and such a phenomenon does not occur for classical modular forms. Moreover, we show that the Hecke module considered is spanned by coefficients bn, where n runs through an infinite set of integers. As a consequence, for any Drinfeld Hecke eigenform, we can compute explicitly certain coefficients in terms of the eigenvalues. We give an application to coefficients of the Drinfeld Hecke eigenform h.     

Laguerre Tau Methods for Solving Higher-Order Ordinary Differential Equations

In this paper, we present two numerical methods for solving higher-order differential equations using the Laguerre Tau method. These methods generate linear systems, which can be solved by Gauss elimination with maximal partial pivoting strategy. Results of some numerical experiments and theoretical analysis are presented.     

Oscillation and Nonoscillation Criteria for Two-Dimensional Systems of First Order Linear Ordinary Differential Equations

Sufficient conditions are established for the oscillation and nonoscillation of the system     

On the traces of Hecke operators

From Eichler's results on traces of Hecke operators and making essential use of the theory of Atkin-Lehner on newforms, we derive in a simple manner an equality between the traces of Hecke operators acting on different spaces of cusp forms.     

Using Root Functions for Eigenvalue Problems of Ordinary Differential Operators

We study eigenvalue problems for an ordinary differential operator L acting on L ₂(?)-spaces (Problem 1) and on L ₂(J)-spaces (Problem 2). Here J is a bounded but large interval. Assuming that in Problem 1 the spectral parameter s lies in the set of normal points of L, we show that the structure of eigenspaces for both problems is similar to the structure of finite complex-valued matrices. In the case of a finite matrix, the geometry of eigenspaces is described by the Jordan form. In the case of ordinary differential operators, the corresponding geometry is described by a sequence of root functions. Therefore, the main tool of our studies is root functions for complex-valued analytical matrix functions.     

Equidistribution of Hecke eigenforms on the modular surface

For the orthonormal basis of Hecke eigenforms in , one can associate with it a probability measure on the modular surface . We establish that this new measure tends weakly to the invariant measure on as tends to infinity, and obtain a sharp estimate for the rate of convergence.

     

一类非线性常微分方程的非平凡解

本文论讨了三个问题：①二阶非线性微分方程的初值问题；②运用非线性算子证明了非平凡解的存在性和唯一性；③给出了一个非负非平凡解的估计式．     

A Priori Estimates for Solutions of Ordinary Differential Equations of Emden--Fowler Type

Mathematical Notes -     

一类高阶线性变系数常微分方程的通解

利用升阶法研究了一类高阶线性变系数常微分方程,给出了齐次方程的通解公式,并讨论了非齐次方程待定的特解.     

一阶微分方程数值解的一种误差估计方法

针对于微分方程数值解,介绍了一种新的误差估计方法.方法证实了伪谱方法具有精度高速度快的优点,进而引出了修正的伪谱方法.     

几类三阶非线性常微分方程组边值问题正解的存在性

研究几类二阶非线性常微分方程组的边值问题,在合适的条件下,利用锥拉伸不动点定理获得了其正解的存在性.     

A Chebyshev-Gauss Spectral Collocation Method for Ordinary Differential Equations

In this paper, we introduce an efficient Chebyshev-Gauss spectral collocation method for initial value problems of ordinary differential equations. We first propose a single interval method and analyze its convergence. We then develop a multi-interval method. The suggested algorithms enjoy spectral accuracy and can be implemented in stable and efficient manners. Some numerical comparisons with some popular methods are given to demonstrate the effectiveness of this approach.     

Constructively Factoring Linear Partial Differential Operators in Two Variables

We study conditions under which a partial differential operator of arbitrary order n in two variables or an ordinary linear differential operator admits a factorization with a first-order factor on the left.The process of factoring consists of recursively solving systems of linear equations subject to certain differential compatibility conditions.In the general case of partial differential operators, it is not necessary to solve a differential equation. In special degenerate cases, such as an ordinary differential operator, the problem eventually reduces to solving some Riccati equation(s). We give the factorization conditions explicitly for the second and third orders and in outline form for higher orders. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 165–180, November, 2005.     

3阶齐次线性常微分方程的球面解曲线

首先给出3阶齐次线性常微分方程的球面解曲线的定义,然后讨论其球面解曲线与系数函数的关系,最后给出了球面解曲线的几何性质.     

Contact Classification of Linear Ordinary Differential Equations: I

It is known that a linear ordinary differential equation of order n3 can be transformed to the Laguerre–Forsyth form y ⁽ⁿ⁾= _i=3 ⁿ a _n–i (x)y ^(n–i) by a point transformation of variables. The classification of equations of this form in a neighborhood of a regular point up to a contact transformation is given.     

在常微分方程教学中融入数学建模思想的探索与实践

结合常微分方程理论性强和应用性强的特点,探索如何在教材、教学方法等方面融入数学建模思想的方法.