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.     

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.     

中心与焦点判定定理证明之补充

中心与焦点判定问题的研究是微分方程定性理论课程教学的重要内容之一,然而我们发现在近30年出版的10多种国内外教材中对其中一个主要定理的涉及Poincare形式级数的证明均有缺陷,本文之目的就是对这一证明做了完整的补充.     

碰撞振动系统的一类余维二分岔及T2环面分岔

建立了三自由度碰撞振动系统的动力学模型及其周期运动的Poincaré映射,当Jacobi矩阵存在两对共轭复特征值同时在单位圆上时,通过中心流形-范式方法将六维映射转变为四维范式映射.理论分析了这种余维二分岔问题,给出了局部动力学行为的两参数开折.证明系统在一定的参数组合下,存在稳定的Hopf分岔和T2环面分岔.数值计算验证了理论结果.     

一类非自治动力系统超次谐周期解的不存在性

在非线性动力系统的研究中， Melnikov函数被广泛地用来作为微扰哈密顿系统是否发生次谐或超次谐分岔乃至混沌的判 据. 但是在大多数情况下，经典的Melnikov方法往往只给出存在次谐周期解的结论. 产生 该结果的原因被归之为在经典的Melnikov方法中只采取了一阶近似，因而高阶Melnikov方 法被发展用来判断超次谐周期解的存在性. 本文对一类非自治微分动力系统进行了研究，证 明了在这样一类系统中如果存在周期解则只可能是次谐周期解，超次谐周期解不可能存在， 并进一步证明了在一类平面问题中所定义的旋转(R)型超次谐周期解同样不可能存在.作为 该结论的一个应用，文中考察了几个典型的算例，结果表明现有的二阶Melnikov方法判断 平面扰动系统是否存在超次谐周期解的结论是不恰当的，并提供了一个简单的几何上的解释.     

斯托克斯空间用邦加球表示光偏振态的再研究

从斯托克斯子空间引入邦加球 ,在邦加球上表示光的任意偏振态 ,给出了表达式