Newton—Riemann时空中的动力学（Ⅳ）

讨论了Newton-Riemann时空中的Lagrange力学及其与N－R时空中的Newton力学及Hamilton力学的关系。     

分数阶微分方程的比较定理

本文给出了非线性Riemann—Liouville分数阶微分方程和Caputo分数阶微分方程与相应的非线性Volterra积分方程的等价性,并在此基础上建立了分数阶微分方程的比较定理．     

不同形式柯西—黎曼方程的比较与分析

在复变函数中Cauchy-Riemann方程是判断函数可微与解析的主要条件，同时也是复分析与偏微方程理论之间的一座桥梁。本从不同的角度导出了Cauchy-Riemann方程的几种形式，并在此基础上研究复变函数的导数。     

非正则型Hilbert边值问题

本文考虑了解析函数非正则型的Hilbert边值问题．利用对称扩张法将问题化为等价的正则型Riemann边值问题，获得了问题的通解及可解性条件，同时给出了问题可解的一个必要条件．     

一类完备Riemann流形上的有界调和函数

本文我们将对一类完备Ｒｉｅｍａｎｎ流形上的有界调和函数所组成的线性空间的维数的上界进行估计，同时给出了一个关于测地球体积的Ｂｉｓｈｏｐ－Ｇｒｏｍｏｖ型体积比较定理。     

多复变数的Schwarz导数(Ⅴ)

本文从Thurston的观点出发,用二阶逼近来定义与讨论矩阵空间C~(m×n)（m≤n）中的域上全纯映照的Schwarz导数及高阶Schwarz导数,证明：如果它们存在的话,那么它们是在R_I（m,n）的紧对偶空间CG（m,n）的全纯自同构群下的相似不变量．并证明：这样得到的Schwarz导数与前几文［1－4］中由Ahlfors的观点得到的Schwarz导数是相一致的．此外,还应用这种观点定义与讨论了C~N中的域上全纯映照的Schwarz导数．     

SPH方法中的Riemann解与人工粘性

本文描述了光滑粒子动力学方法的人工粘性和Riemann解方法,分析了Godunov方法与传统人工粘性方法的耗散项．给定一种人工粘性,总可以找到一种相应的Riemann解法器,使得它们的耗散项在形式上几乎相同．本文利用各种近似Riemann解构造了相应的新的人工粘性．这些新的粘性,无需人工调节粘性系数．本文完成了多个数值试验,比较了使用传统人工粘性方法与Riemann解方法的不同．采用新的人工粘性,辅助热通量粘性,可以获得令人满意的计算结果．     

Riemann—Liouville型分数阶微分方程的微分变换方法

本文在Riemann-Liouville分数阶导数的广义Taylor公式的基础上,建立了求解Riemann-Liouville型分数阶微分方程的微分变换方法.本文所建立的基于Riemann-Liouville分数阶导数微分变换方法给求解Riemann-Liouville分数阶导数的微分方程提供了一种新工具。     

紧对称空间上的最佳逼近和Fejér-Korovkin算子

引进了一秩紧对称空间上的Ｆｅｊéｒ－Ｋｏｒｏｖｋｉｎ 算子,通过建立Ｋ－泛函和光滑模的强渐近等价估计,研究了Ｆｅｊéｒ－Ｋｏｒｏｖｋｉｎ 算子的逼近性质,并由此给出了一秩紧对称空间上最佳逼近的基本定理．     

磁气体动力学欧拉方程组Riemann解磁场消失的极限

证明了理想非等熵磁气体动力学守恒律方程组当磁场作用消失时,其Riemann问题的解收敛于相应的绝热流可压缩欧拉方程组的解,即气体动力学欧拉方程组Riemann解关于磁场强度的稳定性.     

On Conformal Invariants Built from Spinor Fields

By means of a conformal covariant differentiation process we construct generating systems for conformally invariant symmetric (r, s)–spinors in an arbitrary curved space–time. Extending this method to conformally invariant linear differential operators acting on symmetric spinor fields some classes of such operators are derived.     

K(a)hler流形上的不变形式和积分不变量

用现代微分几何理论和高等微积分把Poincare和Cartan-Poincare积分不变量的晕要思想和结果以及E．Cartan在经典力学中首先建立的积分不变量和不变形式的关系推广到Kahler流形上的Hamilton力学中去，得到相应的更广泛的结果．     

Invariant tensors and partial differential equations

We consider tensors with coefficients in a commutative differential algebra A. Using the Lie derivative, we introduce the notion of a tensor invariant under a derivation on an ideal of A. Each system of partial differential equations generates an ideal in some differential algebra. This makes it possible to study invariant tensors on such an ideal. As examples we consider the equations of gas dynamics and magnetohydrodynamics.     

Differential invariants of the transformation group of a homogeneous space

We demonstrate that the invariant operators on a homogeneous space generate differential invariants and invariant differentiation operators. The coordinate-free method of this article makes it possible to simply the computations essentially, namely to reduce them to operations of linear algebra. Some examples are exhibited.     

On the number of invariant lines for polynomial systems

In this paper we will revise the mistakes in a previous paper of Zhang Xikang (Number of integral lines of polynomial systems of degree three and four, J. Nanjing Univ. Math. Biquarterly, Supplement, 1993, pp. 209-212) for the proof of the conjecture on the maximum number of invariant straight lines of cubic and quartic polynomial differential systems; and also prove the conjecture in a previous paper of the second author (Qualitative theory of polynomial differential systems, Shanghai Science-Technical Publishers, Shanghai, 1995, p. 474) for a certain special case of the degree polynomial systems. Furthermore, we will prove that cubic and quartic differential systems have invariant straight lines along at most six and nine different directions, respectively, and also show that the maximum number of the directions can be obtained.

     

Conformally invariant powers of the Laplacian -- A complete nonexistence theorem

We show that on conformal manifolds of even dimension there is no conformally invariant natural differential operator between density bundles with leading part a power of the Laplacian for n/2$">. This shows that a large class of invariant operators on conformally flat manifolds do not generalise to arbitrarily curved manifolds and that the theorem of Graham, Jenne, Mason and Sparling, asserting the existence of curved version of for , is sharp.

     

Coexistence of algebraic and nonalgebraic limit cycles in Kukles systems

A class of Kukles differential systems of degree five having an invariant conic is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, under perturbations of the coefficients of the systems. Financed partially by: USM Grant No. 12.06.28 and 12.06.27.     

Reduction of exterior differential systems with infinite dimensional symmetry groups

A symmetry-based method for constructing solutions to systems of differential equations founded on the reduction of exterior differential systems invariant under the action of an infinite dimensional pseudogroup is proposed. One can associate to any system of differential equations Δ=0 with a symmetry group an exterior differential system invariant under so that solutions of Δ=0 correspond to integral manifolds . The -invariant exterior differential system gives rise to a reduced system specified on a cross section to the pseudogroup orbits, and it is shown that solutions to Δ=0 can be reconstructed from integral manifolds by solving an equation of generalized Lie type for the jets of pseudogroup transformations. In particular, as opposed to the classical method of symmetry reduction, every solution to the system of differential equations can, under some mild regularity assumptions, be constructed by the present algorithm. AMS subject classification (2000)  58A15, 58A20, 58H05, 58J70     

Solution to n-dimensional Sturm-Liouville-like equations using multi-dimensional Schwarzian

In the present work we produce the solution to the n-dimensional Sturm-Liouville-like equations in Rn. To make it, we define the multi-dimensional Schwarzian derivative of a real-valued function of n variables and show that its basic properties related to its invariance under the action of a group of multi-dimensional Möbius transformations defined in Rn correspond to a straightforward generalization of those of the one-dimensional Schwarzian.     

THE POSITIVE INVARIANT SET OF THE DIFFERENTIAL EQUATION ON BANACH SPACE

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