目录
1. On the Minimally Almost Periodic Topological Groups 1
*小几乎周期拓扑群
2. 广义函数论 7
On the Theory of Distributions
3. 广义函数的泛函对偶关系 180
Duality Relations in Spaces of Distributions
4. 广义 Mellin 变换 I 188
Mellin Transform in Distributions， I
5. Di.erence Schemes Based on Variational Principle 215
基于变分原理的差分格式
6. 有限元方法 250
The Finite Element Method
7. 组合流形上的椭圆方程与组合弹性结构 299
Elliptic Equations on Composite Manifold and Composite Elastic Structures
8. 论间断有限元的理论 310
On the Theory of Discontinuous Finite Elements
9. 论微分与积分方程以及有限与无限元 319
Di.erential Versus Integral Equations and Finite Versus Infinite Elements
10. 中子迁移方程的守恒差分方法与特征值问题 326
Conservative Di.erence Method for Neutron Transport Equation and Eigenvalue Problem
11. Canonical Boundary Reduction and Finite Element Method 336
正则边界归化与有限元方法
12. Canonical Integral Equations of Elliptic Boundary-Value Problems and Their Numerical Solutions 352
椭圆边值问题的正则积分方程及其数值解法
13. Finite Element Method and Natural Boundary Reduction 379
有限元方法和自然边界归化
14. Asymptotic Radiation Conditions for Reduced Wave Equation 392
约化波动方程的渐近辐射条件
15. 现代数理科学中的一些非线性问题 403
Some Nonlinear Problems in Modern Mathematics and Physics
16. 非协调有限元空间的 Poincar.e， Friedrichs 不等式 406
Poincar.e and Friedrichs Inequalities in Nonconforming Finite Element Spaces
17. 非协调元空间的一个 Sobolev 嵌入定理 425
A Sobolev Imbedding Theorem in Nonconforming Finite Element Spaces
18. 非协调元的分数阶 Sobolev 空间 429
Fractional Order Sobolev Spaces for Nonconforming Finite Elements
19. 关于调和方程自然积分算子的一个定理 437
A Theorem for the Natural Integral Operator of Harmonic Equation
20. 自然边界归化与区域分解 443
Natural Boundary Reduction and Domain Decomposition免费在线读1. On the Minimally Almost Periodic Topological Groups①
*小几乎周期拓扑群
摘要
在拓扑群上如果对任意二不同元素必定有一个几乎周期函数在这二元素上 取不等值， 这个群就叫做**几乎周期群. 如果群上所有的几乎周期函数都是 常数， 它就叫做*小几乎周期群. Freudenthal 及 Weil 解决了**几乎周期群的 问题， 它就是一个封闭群和一个向量加群的直接乘积. 本文系致力于*小几乎周 期群的问题， 阐明一些*小几乎周期性的特征， 知道它们相当于根本上不封闭和 不可换的群. 主要的结果是： 线性 或单连通 连通李群是*小几乎周期群的充 要条件是 一 它与它的换位群相重合， 二 它的**半单李代数不包含相当于 封闭群的直接因子. 由此可见， 对线性李群而言， *小几乎周期性可由局部完全 决定. 此外还列举若干*小几乎周期群的实例， 并应用*小几乎周期性证明一 个关于复数李群的定理.
The theory of almost periodic a.p. functions in arbitrary groups was first established by von Neumann[1]. In the following we shall confine ourselves to the case of topological groups， thus the a.p. functions and the representations are required to be continuous. The a.p. functions are intimately related with the representations by unitary matrices， in fact， a representation is equivalent to a unitary one if and only if all its matrix coe±cients are a.p. functions， and every a.p. function generates a unitary representation[1];[2]. As to the admissibility of the a.p. functions， i.e.， of the unitary representations， we have， after von Neumann， the following two extreme classes of groups： 1. A topological group is called maximally almost periodic if to each pair of distinct elements there is an a.p. function which takes di.erent values at these two elements， or equivalently， to each non-identity element there is a unitary representation which carries it into a matrix di.erent from the unit matrix. 2. A topological group is called minimally almost periodic if every a.p. function is a constant， or equivalently， every unitary representation is trivial. The maximally a.p. case was characterized by Freudenthal and Weil： a connected locally compact group is maximally a.p. if and only if it is a direct product of a compact group and an Euclidean vector group[2];[3]. The present note is devoted to the characterization of the minimally a.p. groups. We obtain conditions， some necessary， some su±cient， for a connected Lie group to be minimally a.p.， and in particular， a necessary and su±cient condition for a connected linear Lie group to be minimally a.p. We see that the minimal case， in contradistinction to the maximal one， corresponds to the \essentially" non-compact and non-abelian groups.
Let G be a topological group， and K be the subset of G which consists of all the elements a such that fa is the unit matrix for every unitary representation f of G. K is called the unitary kernel of G and is a closed normal subgroup of G this was first introduced by Weil[4]， cf. also [5]. With this in view， the maximal and minimal cases correspond to K = e and K = G respectively. The factor group here and henceforth the factor groups are understood in the topologico-group-theoretic sense G=K is obviously maximally a.p.， and every closed normal subgroup H of G such that G=H is maximally a.p. contains K： Thus it follows immediately that a topological group is minimally a.p. if and only if it has no proper closed normal subgroup whose corresponding factor group is maximally a.p. It is also evident that the direct product of a finite number of minimally a.p. topological groups is minimally a.p. and every factor group of a minimally a.p. topological group is minimally a.p.
Lemma 1 Every connected semi-simple Lie group whose Lie algebra contains no sim- ple ideal corresponding to a compact group is minimally a.p. Thus， in particular， all semi- simple complex Lie groups are minimally a.p.
Proof. Let G be a non-compact， non-abelian， simple Lie group. All possible proper closed normal subgroups of G are discrete. Thus all possible non-trivial factor groups are locally isomorphic to G; they are also non-compact， non-abelian， simple Lie groups for which the Freudenthal-Weil decompositions are impossible. Therefore G is minimally a.p. Every connected semi-simple Lie group whose Lie algebra contains no simple ideal corresponding to a compact group is a factor group of a direct product of groups of the above type modulo a discrete normal subgroup. Therefore it is also minimally a.p.
Lemma 2 Let G be a connected Lie group which coincides with its commutator sub- group，and G1 be the maximal semi-simple subgroup of G which corresponds to the maximal semi-simple subalgebra of a Levi decomposition of the Lie algebra of G. Then every closed connected normal subgroup of G containing G1 coincides with G.
Proof. Let A be the Lie algebra of G;A = A1 + A2 be a Levi decomposition of A， where A1 is a semi-simple subalgebra of A and A2 is the maximal solvable ideal of A; and G1 be the subgroup of G which corresponds to the subalgebra A1： Let G0 be a closed connected normal subgroup of G containing G1;A0 be an ideal of A corresponding to G0： Since G0 is itself a connected Lie group， so the factor group G=G0 has a Lie algebra isomorphic with A=A0： Let be the natural homomorphism of G onto G=G0; induces a homomorphism . of A onto A=A0： It is easily seen that the contraction of
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