模糊集单调型分解定理

模糊集的分解集定理有两种形式，第一种形式与模糊集的λ—截集有关：Ａ＝其中Ａ＿λ是的λ—截集￣［１］．第二种形式与集合套有关：其中Ｈ（λ）为集合套￣［２］或者Ａ＝其中Ｈ（λ）为集合套，Ｑ为（０，１）的可列稠密子集＿［３］，．其实，这两种形式在本文提出的基本模糊集的概念下，可以统一起来成为任何模糊集可以分解为一些（可数或不可数）基本模糊集的和。不仅如此，本文还提出简单模糊集的概念，并证明了任何模糊集都可以表示成为可数个单调上升的简单模糊集的和，或等价地，可以表示成可数个单调上升的简单模糊集的极限，所用的证明方法是构造性的，所以对模糊集的结构也得出了一个清晰的认识．     

A Dynamical Decomposition Theorem

By the decomposition theorem dim X ≦ n if and only if X admits a decomposition into n+1 zero-dimensional subspaces Z _i for i = 0,... n. If f: X → X is a homeomorphism, then under some dimensional restrictions on the set of periodic points, the Z _i can be chosen to be images of Z ₀ under iterates of f.     

点是Gδ^－集的∑^＊－空间的构成定理

在林寿与我最近合作的一篇文章中指出了∑^*-空间的构成定理需重新考虑．本文就是要证明在空间X的每个点是Gδ^-集的条件下该构成定理是成立的，所得的结论是：X是T1且每个点是Gδ^-集的∑^ -空间，如果f：X→Y是闭的满连续映射，则在Y中有-σ-闭离散子空间Z，使得对每个y∈Y＼Z，f^-1(y)是X的ω1^-紧子空间．为得到该主要结果，本文证明了若空间X是每个点是Gδ^-集的次亚紧空间．则X中的每个闭离散子集是X中的Gδ^-集．     

g-上鞅的Riesz分解定理

本文给出了当终端时间趋于无穷时一类有限时间区间上的倒向随机微分方程的解的收敛性,并且证明了这类解平方收敛到特定的无穷时间区间上的倒向随机微分方程的解.本文主要研究了由倒向随机微分方程生成的非线性期望及其鞅的性质,证明了当生成元g是超线性时的g-上鞅Riesz分解定理.并且指出经典鞅论中的Riesz分解定理和下期望(又称最小期望)对应的上鞅Riesz分解定理是g-上鞅Riesz分解定理的两种特殊情况.     

极分解定理的注记

设H和K是Hilbert空间.首先,对给定的算子T∈B（H,K）,刻画了集合U_T={U∈B（H,K）：U是部分等距算子并且T=U（T^*T）^1/2}. 其次,对部分等距算子U∈B（H,K）,还给出集合T_U={T∈B（H,K）：N（T）=N（U）,R（T）=R（U）,T=U（T^*T）^1/2}的刻画. 最后,作为主要结果的应用得到了相关结论.     

A Decomposition Theorem for Group Representations

In this paper we establish a decomposition theorem for an ordinary representation of a finite group G in any category C{\mathcal C} which expresses a suitable irreducible representation of G as the tensor product of two projective ones. The celebrated theorem due to Clifford for a linear representation turns out to be a particular case of it. For that purpose, a definition of projective extension of an ordinary representation of a normal subgroup of G is introduced, as well as a tensor product between two of them.     

On the Parametric Decomposition Theorem in Multiobjective Optimization

An example is given to show the inadequacy of a well-known result, concerning the parametric decomposition theorem for multiobjective optimization problems. We also give an accurate decomposition theorem, which generalizes a corrected reformulation of this result.     

循环置换分解定理的一个证明及其应用

循环置换分解定理:每一个n元置换π都可以写成若干个不相连的循环置换的乘积,是置换群理论最基本的定理之一.在一些教材中该定理的证明用了数学归纳法,本文提供了一个直观的证明方法,并给出了置换的一种表示方法以及一道关于穿珠子的排列组合问题的解法.     

Infinite Dimensional Bicomplex Spectral Decomposition Theorem

This paper presents a bicomplex version of the Spectral Decomposition Theorem on infinite dimensional bicomplex Hilbert spaces. In the process, the ideas of bounded linear operators, orthogonal complements and compact operators on bicomplex Hilbert spaces are introduced and treated in relation with the classical Hilbert space M′ imbedded in any bicomplex Hilbert space M.     

The Lebesgue Decomposition Theorem for Arbitrary Contents

The decomposition theorem named after Lebesgue asserts that certain set functions have canonical representations as sums of particular set functions called the absolutely continuous and the singular ones with respect to some fixed set function. The traditional versions are for the bounded measures with respect to some fixed measure on a σ algebra, in final form due to Hahn 1921, and for the bounded contents with respect to some fixed content on an algebra, due to Bochner-Phillips 1941 and Darst 1962. Then came the version for arbitrary measures, due to R.A.Johnson 1967 and N.Y.Luther 1968. The unpleasant fact with these versions is that each one requires its particular notions of absolutely continuous and singular constituents. It seems mysterious how a common roof for all of them could look, and therefore how a universal version for arbitrary contents could be achieved - and all that while several abstract extensions of particular versions appeared in the subsequent decades, for example due to de Lucia-Morales 2003. After these decades now the present article claims to arrive at the final aim in the original context of arbitrary contents. The article will be based on the author’s new difference formation for arbitrary contents 1999. This difference formation even furnishes simple explicit formulas for the two constituents. Horst Tietz in alter Freundschaft gewidmet     

二次李超三系的分解定理

讨论了二次李超三系的唯一分解,得到了二次李超三系分解为非退化不可约阶化理想的分解定理.     

A Decomposition Theorem for Compact-Valued Henstock Integral

We prove that if X is a separable Banach space, then a measurable multifunction Γ : [0, 1] → ck(X) is Henstock integrable if and only if Γ can be represented as Γ = G + f, where G : [0, 1] → ck(X) is McShane integrable and f is a Henstock integrable selection of Γ.     

关于稳定矩阵分解定理的一个简单证明

一个矩阵称为稳定的,如果这个矩阵的特征值全包含在单位开圆盘内.利用Parker关于复方阵的分解定理给出了稳定矩阵分解定理的一个简单证明,并对奇异值全部严格小于1的矩阵给出了类似的结论.     

广义Kato分解与Weyl型定理的摄动

称T∈B(H)有广义Kato分解,若存在一对T的闭的不变子空间(M,N)使得H=M⊕Ⅳ,其中T|_M为上半Fredholm算子且具有非负的指标,T|_N是幂零的本文利用算子的广义Kato分解性质,研究了Weyl型定理在紧摄动下的稳定性.此外还研究了2×2上三角算子矩阵的Weyl型定理在紧摄动下的稳定性.     

S-harmonic Function and RieszDecomposition Theorem of Superprocesses

RecentlyItheremanyprogressinthetheoryofsuperprocesses(seeLZ'4--gi).Inthispaper,wewillestablishtheRicszdecompositiontheoremofexcessivefunctionsofaclassofsuperprocesses,andgeneralizetheclassicalRicszdecompositiontheoremofHuntprocess.FortheconstructionofsupcrprocesscswecanseeL2,4,sj.ConsiderthefollowingrealfunctionPX,9(x,z)=a(x)z b(x)zZ J,(e":--I "z)n.(du),(1)Pcowheren.(du)isakernelfromR'lto(0,~)and'a(x),b(x)andA(x)~I(uZAu)n.(du)areJ0positiveboundedBorclfunctions.AssumingthatE~{Et,11.}is…     

The De Rham Decomposition Theorem For Metric Spaces

We generalize the classical de Rham decomposition theorem for Riemannian manifolds to the setting of geodesic metric spaces of finite dimension. Received: May 2006 Accepted: December 2006     

A Decomposition Theorem for the Solutions to the Interface Problems of Quasi-Linear Elliptic Equations

Interface problems for second order quasi-linear elliptic partial differential equations in a two-dimensional space are studied. We prove that each weak solution can be decomposed into two parts near singular points, one of which is a finite sum of functions of the form cr^a log^m rφ(θ), where the coefficients c depend on the H^1-norm of the solution, the C^(0,δ) -norm of the solution, and the equation only; and the other one of which is a regular one, the norm of which is also estimated.     

矩阵秩分解定理的一个证明及其应用

矩阵的秩分解定理是矩阵论中的一个基础性定理.本文给出了秩分解定理的一个证明,描述了其中可逆矩阵P,Q的构造,讨论了秩分解定理及其P,Q的构造在解线性方程组中的应用,以及在判别Sylvester不等式等号成立中的应用.     

A Decomposition Theorem for Chordal Graphs and its Applications

We introduce a special decomposition, the so-called split-minors, of the reduced clique graphs of chordal graphs. Using this notion, we characterize asteroidal sets in chordal graphs and clique trees with minimum number of leaves.