THE POINCAR BIFURCATION OF A CUBIC HAMILTONIAN SYSTEM WITH HOMOCLINIC LOOP

In this paper, we discuss the Poincare bifurcation of a cubic Hamiltonian system with homoclinic loop. We prove that the system can generate at most seven limit cycles after a small perturbation of general cubic polynomials.     

A topological characterization of the ω-limit sets for analytic flows on the plane, the sphere and the projective plane

In this paper the ω-limit sets for analytic and polynomial differential equations on the plane are characterized up to homeomorphisms. The analogous problem is solved in full detail for analytic flows on the sphere and the projective plane. We also explain how to carry on the same program for analytic flows defined on open subsets of these surfaces.     

Hilbert模形式与一类Dirichlet级数的特殊值

在文［２］中，Ｗ．Ｋｏｈｎｎ对权为ｋ和ｌ的任意二个歧点型模形式ｆ和ｇ（其变换群是全模群ＳＬ＿２（Ｚ））定义了一类Ｄｉｒｉｃｈｌｅｔ级数Ｌ＿（ｆ，ｇ，ｎ）（ｓ），利用Ｌ＿（ｆ，ｇ；ｎ）（ｓ）（为整数），可构造一个线性映射Ｗ＿ｇ：Ｓ＿ｋ→Ｓ＿（ｋ－ｌ）．并且讨论了Ｌ＿（ｆ，ｇ；ｎ）的一些特征值．在本文中，我们将［２］中的结果推广到Ｈｉｌｂｅｒｔ模形式的情况，并得到类似的结论．     

Alternative approaches to the construction of the Poincare-invariant spin operators of the Dirac particles

At present a number of methods of constructing the Poincare-invariant spin operators for relativistic particles with half-integer spin in the one-particle theory are well known. The method of odd operator constructing, the Lorentz method of bilinear covariant form transformation, and the method with the Foldy–Wouthuysen representation belong to them. New approaches to the construction of spin operators are developed in the present work, namely, a method of separating space-like component directly from the spin matrices of bilinear covariant forms, including the method of multiplication of the covariant Hamiltonian of the Dirac equation by these matrices. By this means we succeeded in constructing the Poincare-invariant spin operators by simpler and mathematically faultless methods. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 10–15, November, 2008.     

On the Existence and Construction of Tight Fourth Order Quadrature Rules for the Sphere

We study the so-called tight quadrature rules for polynomials of degree 4 on the unit sphere S ^D-1 and present precise formulae for the first 6 components of the nodes in terms of the parameter u := . In particular, we reobtain the well-known necessary condition for the existence of such rules saying that u has to be an odd integer and we sharpen it under an additional assumption.As a constructive application, two explicit tight fourth order quadrature rules for the case D = 7 are given.     

A counterexample to the ``composition conjecture"

In this note we construct a class of counterexamples to the ``composition conjecture" concerning an infinitesimal version of the center problem for the polynomial Abel equation in the complex domain.

     

Theorems about the Existence of Solutions to Problems with Nonlocal Initial Value

Recently much work has been devoted to nonlocal problems. However, very little has been accomplished in the literature for nonlocal initial problems with nonlinear boundary conditions. It is the purpose of this paper to prove the existence results for solutions to a semilinear parabolic PDE with linear homogeneous boundary conditions, and to other ones with nonlinear boundary conditions, provided the ordered upper and lower solutions are given. Semigroup, fractional order function spaces and generalized Poincaré operators play an important role in proving the existence of solutions. Supportedpartly by Research Grant 19531060 of China NSF and Grant 97024811 of the Foundation of Doctoral Program     

有界星形域上的Poincaré不等式

本文得到了有界星形域上的Poincar啨不等式 ,证明了Poincar啨域在满足一定条件的拟共形映照下是不变的 .     

MATHEMATICAL ANALYSIS FOR QUADRILATERAL ROTATED Q1 ELEMENT Ⅱ：POINCARE INEQUALITY AND TRACE INEQUALITY

This is the second part of the paper for the mathematical study of nonconforming rotated Q1 element (NRQ1 hereafter) on arbitrary quadrilateral meshes. Some Poincare Inequalities are proved without assuming the quasi-uniformity of the mesh subdivision. A discrete trace inequality is also proved.     

选取任意庞加莱截面的新方法

针对用庞加莱截面分析非线性系统时较难选择合适截面的问题,提出了一种投影时域法.该方法可以在时域上直观地判断所选平面是否合适并实时准确地调整平面的方向与位置.文中给出了投影时域法的完整定义并从理论上详细分析了该方法的原理;同时研究了在时域上判断所选平面是否是合适的庞加莱截面的条件以及如何调整所选平面的方向与位置;最后,利用投影时域法对三种典型的三维或四维非线性系统进行了仿真实验,实验结果证明了该方法的有效性和实用性.