非紧黎曼曲面上关于方程u+au=0的Runge逼近定理与Mittag-Leffler定理

本文给出了非紧黎曼曲面 R 上关于方程(?)=(?)+(?)=0的 Runge 逼近定理,并证明了消没定理 H~1(R,Ω(?))=0,这里 H~1(R,Ω(?))为开黎曼曲面 R上方程(?)=(?)+au=0的正则解的芽层Ω(?)的一阶上同调群,从而解决于开黎曼曲面上关于方程(?)u=0的 Mittag-Leffler 问题.     

关于双曲空间形式的一个注记

本文主要证明一个具有光滑边界的紧黎曼流形,如果有非平凡解,则就等度量同构与双曲空间形式 会的紧区域,这里Ｄ～２■是■的Ｈｅｓｓｉａｎ与ｇ是Ｍ上的黎曼度量． 还证明关于上述方程的边值问题,只有混合边值问题,而且当ｃ＜－１时有解．     

关于双曲空间形式的一个注记

本文主要证明一个具有光滑边界的紧黎曼流形,如果有非平凡解,则就等度量同构与双曲空间形式Hn(-1)={(x0,x1,.…,xn)∈Rn,1l(x0)2-n∑i=1(xi)2=1,x0＞0}中1≤x0≤(1-c-2)-1/2的紧区域,这里D2ψ是ψ的Hessian与g是M上的黎曼度量.还证明关于上述方程的边值问题,只有混合边值问题,而且当c＜-1时有解.     

非紧黎曼曲面上关于方程u+au=0的Runge逼近定理与Mittag-Leffler定理

本文给出了非紧黎曼曲面 R 上关于方程(?)=(?)+(?)=0的 Runge 逼近定理,并证明了消没定理 H~1(R,Ω(?))=0,这里 H~1(R,Ω(?))为开黎曼曲面 R上方程(?)=(?)+au=0的正则解的芽层Ω(?)的一阶上同调群,从而解决于开黎曼曲面上关于方程(?)u=0的 Mittag-Leffler 问题.     

关于Finsler曲面的几何和拓扑

本文在Ｆｉｎｓｌｅｒ曲面上定义了一个新的不变量Ｈ。该不变量等于零刻画了Ｒｉｅｍａｎｎ流形。文章给出了Ｈ的一个上界并且构造了Ｈ为常值的非Ｒｉｅｍａｎｎ的Ｆｉｎｓｌｅｒ曲面。此外，本文还推广了Ｌａｎｄｓｂｅｒｇ曲面的Ｇａｕｓ－Ｂｏｎｎｅｔ－Ｃｈｅｒｎ定理并分类了非正曲率的Ｆｉｎｓｌｅｒ曲面。     

黎曼曲面上的φ-调和函数和φ-次调和函数

(M,g)是黎曼曲面,该文给出了M上函数的φ-Dirichlet积分的定义,并在此基础上 得到了一个关于具有有限的φ-Dirichlet积分的φ-次调和函数的有界性定理．     

黎曼曲面上的&Phi|-调和函数和&Phi|-次调和函数

(M, g)是黎曼曲面,该文给出了M上函数的Φ- Dirichlet积分的定义,并在此基础上得到了一个关于具有有限的Φ - Dirichlet积分的Φ -次调和函数的有界性定理.     

关于街接定理

<正> 如所周知,衔接定理对研究黎曼曲面的共形映射有重要作用,本文利用边值问题的方法,给出两种型式的衔接定理可以分别化为Riemann-Haseman边值问题,及Carleman边值问题,从而在较弱条件下给出定理的证明。     

带边紧流形上热核的Harnack不等式

１９８６年,Ｐ．Ｌｉ与丘成桐给出了带凸边界的紧黎曼流形上关于热核的一个Ｈａｒｎａｃｋ不等式（可参看［６］）,而该文的目的正是将他们的工作推广到可能带非凸边界的紧黎曼流形上．     

紧黎曼曲面上广义解析函数的若干性质

本文对紧黎曼曲面上的关于算子 ■u=■u+au+b■的 Dolbeault 定理、Serre对偶定理给出了一个清晰的证明.并给出了方程 ■u=0的解的一种表示,利用这种表示得到了方程■u=0的解空间的一系列性质,证明了消没定理.     

Riemann surfaces

Since the classical work of Riemann, Plein, Chobe, and Poincaré, in mathematics the interest in the theory of Riemann surfaces and groups has not abated. The present survey covers papers reviewed in RZhMat during the period 1967–1976 primarily in the sections Algebra. Topology. Geometry. The following topics are considered most completely and thoroughly: the topology of Riemann surfaces and their automorphisms, Fuchsian groups, Teichmüller spaces, and spaces of moduli.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 16, pp. 191–245, 1978,     

Almost geodesic mappings of Riemann spaces onto symmetric Riemann spaces

It is shown that if a Riemann space V_n admits a reduced almost geodesic mapping Π₂ onto a symmetric Riemann space ¯V_n, then ¯V_n has constant curvature, and V_n is itself a symmetric space.     

Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities” argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the “acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.     

The Riemann problem on a finite-sheeted Riemann surface of infinite genus

LetR be the Riemann surface of the functionu(z) specified by the equationu ⁿ=P(z) withn ε ℕ,n ≥ 2, andz ε ℂ, whereP(z) is an entire function with infinitely many simple zeros. OnR, the Riemann boundary-value problem for an arbitrary piecewise smooth contour Γ is considered. Necessary and sufficient conditions for its solvability are obtained, and its explicit solution is constructed. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 25–35, January, 2000.     

黎曼积分的完备化

综述了黎曼可积函数的基本特征,并指出黎曼可积函数列的极限运算在积分意义下是不封闭的.在构造了完备化空间之后,证明了该空间就是勒贝格可积函数空间,从而说明了黎曼积分的完备化形式是勒贝格积分.     

Computing Riemann theta functions

The Riemann theta function is a complex-valued function of complex variables. It appears in the construction of many (quasi-)periodic solutions of various equations of mathematical physics. In this paper, algorithms for its computation are given. First, a formula is derived allowing the pointwise approximation of Riemann theta functions, with arbitrary, user-specified precision. This formula is used to construct a uniform approximation formula, again with arbitrary precision.