Complements and the Krull-Schmidt Theorem in arbitrary categories

We study direct product decompositions of objects in a finitely complete and cocomplete category with zero object and certain axioms for a coimage factorization of morphisms. Direct productsC=A×B can be characterized by inner properties ofC and its subobjectsA andB. We also show that the Fitting Lemma and the Krull-Schmidt Theorem hold.     

Lie环分解中的Krull—Schmidt定理

该文得到了Lie环分解的Krull-Schmidt定理:若L是在理想上满足极大、极小条件的Lie环,如果L=H1⊕…⊕Hr=K1⊕K2⊕…Ks是L的两个Remak分解,即Hi和Kj是不可分解的,那么r=s,并且存在L的一个中心自同构a,使在适当排列Kj的顺序后,Kai=Ki,进一步地,对任意的k=1,2,…,r,L=K1⊕K2⊕…⊕Kk+1⊕…Hr.如果L=H1⊕…⊕Hr是L的一个Remak分解,那么这个分解是L的唯一Remak分解当且仅当对L的任意正规自同态θ有Hθi≤Hi,i=1,2,…,r.     

The Krull-Schmidt Theorem Holds for Finite Direct Products of Biuniform Groups

We prove that the Krull-Schmidt Theorem holds for finite direct products of biuniform groups, that is, groups G whose lattice of normal subgroups \(\mathcal {N}(G)\) has Goldie dimension and dual Goldie dimension 1. More generally, it holds for the class of completely indecomposable groups.     

On a Generalization of Nikolskij's Extension Theorem in the Case of Two Variables

A modification of the Nikolskij extension theorem for functions from Sobolev spaces H ^k() is presented. This modification requires the boundary to be only Lipschitz continuous for an arbitrary k however, it is restricted to the case of two-dimensional bounded domains.     

Krull-Schmidt theorems in dimension 1

Let be a semiprime, module-finite algebra over a commutative noetherian ring of Krull dimension 1. We find necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated -modules, and necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated torsionfree -modules (called ``-lattices' in integral representation theory, and ``maximal Cohen-Macaulay modules' in the dimension-one situation in commutative algebra).

     

关于替换定理的两点注记

替换定理是线性空间(数域!上的线性空间V!,特别地,!n)理论中涉及线性相关性的最基本的一个事实.为了引导学生更好地理解它,我们在教学实践中形成了关于它的两点注记.     

带二阶导数的插值定理

用B_(δ,p)(1P∞)表示P次可积Fourier变換具有紧支集[-δ,δ]的带有限函数。证明了对于B_(3δ,p)中的函数,可以在L_p(R)尺度下,由序列{f(κπ/δ)},{f'(κπ/δ)}以及{f'(κπ/δ)}的Hermite型插值进行重构.     

A Two Function Metaminimax Theorem

A generalization of Simons's metaminimax theorem to a metaminimax theorem involving two functions is given.     

条件风险下的两基金分离定理

鉴于条件风险价值CVaR具有风险度量的合理性以及两基金分离定理对证券投资的重要意义,以CVaR作为风险度量研究两基金分离定理.在组合收益率服从正态分布的假设下,分别就投资组合含有或没有无风险资产的情形提出并证明了两基金分离定理;放开方差-协方差矩阵为非奇异这一通常假设,证明了CVaR风险度量下的两基金分离定理依然成立.     

两类非线性控制系统的绝对稳定性

本文应用二次型加积分项的 V函数研究了两类非线性控制系统的绝对稳定性 ,给出了这两类系统的绝对稳定性判据 ,推广了有关文献的研究     

Two Extensions to Finsler's Recurring Theorem

Finsler's theorem asserts the equivalence of (i) and (ii) for pairs of real quadratic forms f and g on R ⁿ : (i) f( ξ ) >0 for all ξ≠ 0 with g( ξ ) =0; (ii) f-λ g>0 for some λ∈ R. We prove two extensions: 1. We admit a vector-valued quadratic form g: R ⁿ → R ^k , for which we show that (i) implies that f-λ . . . g>0 on an ( n-k+1 ) -dimensional subspace Y R ⁿ for some λ∈ R ^k . 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R ⁿ by a real vector space X . Accepted 22 February 1998     

二维平移不变子空间中的广义抽样定理

抽样定理在数字信号处理和图像处理中具有重要的作用,古典Shannon抽样定理因其局限性而限制了它的应用.本文研究了二维平移不变子空间中以任意点作为抽样点的规则抽样定理.首先,抽样空间的一些特征被给出;接着,平移不变子空间的决定集的一个刻画被得到.然后,通过平移不变子空间的决定集,函数属于一个抽样空间的充要条件被证明.     

Cayley-Hamilton定理的两种简单证法和两个表现定理

本文对Cayley-Hamilton定理提出两种简单证法并应用该定理和罗必塔法则给出两个有关二阶张量函数的表现定理.     

证券收益率的极大线性无关组及两基金分离定理

在无套利假设下,利用概率论结合线性代数的方法进一步研究了当n种风险资产的协方差矩阵∑是奇异时的证券投资组合问题,在均值-方差模型的框架下得到模型的一些本质特征,并证明了此时的两基金分离定理仍然成立的.     

Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem

Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a commutative ring R we associate the subcategory of the category of Chow motives with coefficients in R, that is, the Tate pseudo-abelian closure of the category of motives of projective homogeneous G-varieties. We show that is a symmetric tensor category, i.e., the motive of the product of two projective homogeneous G-varieties is a direct sum of twisted motives of projective homogeneous G-varieties. We also study the problem of uniqueness of a direct sum decomposition of objects in We prove that the Krull--Schmidt theorem holds in many cases.     

两个函数的拓扑极大极小定理

A topological minimax theorem involving two functions is derived. It generalizes Greco and Horvath＇minimax theorem given by them in 2002.     

On a Special Case of the Herbert Stahl Theorem

The BMV conjecture states that for n ×  n Hermitian matrices A and B the function f_A,B(t) = trace e^tA+B is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions. In the present paper we give a purely “matrix” proof of the BMV conjecture for the special case rank A = 1. This proof is based on the Lie product formula for the exponential of the sum of two matrices and does not require complex analysis.     

Lax等价定理在非线性方面的推广

本文证明了，用差分法求解非线性发展方程的初值问题，当方程适定，在差分格式相容的条件下，稳定性等价于收敛性和逐点Lipschitz条件。从而推广了对线性发展方程成立的Lax等价定理。